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Code No: 43087 Set No. 1

II B.Tech I Semester Supplimentary Examinations, May/Jun 2009PROBABILITY THEORY AND STOCHASTIC PROCESSES

( Common to Electronics & Communication Engineering, Electronics &Telematics and Electronics & Computer Engineering)

Time: 3 hours Max Marks: 80Answer any FIVE Questions

All Questions carry equal marks⋆ ⋆ ⋆ ⋆ ⋆

1. (a) What is Bayes’ theorem? Explain.

(b) Determine probabilities of system error and correct system transmission ofsymbols for an elementary binary communication system shown in figure 1bconsisting of a transmitter that sends one of two possible symbols (a 1 or a 0)over a channel to a receiver. The channel occasionally causes errors to occurso that a ’1’ show up at the receiver as a ’0?, and vice versa. Assume thesymbols ‘1’ and ‘0’ are selected for a transmission as 0.6 and 0.4 respectively.[6+10]

Figure 1b

2. Define and explain the following density functions

(a) Binomial

(b) Exponential. [8+8]

3. (a) In an experiment when two dice are thrown simultaneously, find expectedvalue of the sum of number of points on them.

(b) The exponential density function given by

fx(x) = (1/b)e−(x−a)/b x > a= 0 x < a

Find out variance and coefficient of skewness. [6+10]

4. (a) Define and explain conditional probability mass function. Give its properties.

(b) The joint probability density function of two random variables X and Y is

given byf(x, y) = C(2x + y), 0 ≤ x ≤ 1, 0 ≤ y ≤ 2

= 0, elsewhere

Find:

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i. the value of ‘C’

ii. Marginal distribution functions of X and Y. [8+8]

5. (a) Define cross correlation function of two random processes.

(b) Comment on cross correlation function of two orthogonal random processes.

(c) State and prove the properties of cross correlation function of two randomprocesses. [3+3+10]

6. Given the random process X(t) = A sin (ω0t + Θ) A, ω0 are constants and Θ is arandom variable uniformly distributed in the interval (-π, π). Define a new randomprocess Y(t) = X2(t).

(a) Find the autocorrelation function of Y(t)

(b) Find the cross correlation function of X(t) and Y(t)

(c) Are X(t) and Y(t) wide sense-stationary. [6+6+4]

7. (a) A WSS random process X(t) has RXX (τ) = A0

[

1 −|τ |τ

]

− τ ≤ t ≤ τ

= 0 else whereFind power density spectrum.

(b) RXX (τ) =A2

0

2sin ω0τ. Find Sxx (ω) [8+8]

8. (a) If x(t) is ensemble member of a input random process X(t) and Y(t) is theensemble member of a output random process of an LTI system, obtain therelationships for y(t) and Y(t).

(b) Derive the mean and mean squared value of system response to a input randomprocess. [4+12]

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1. (a) If A is an arbitrary event, then show that P(A) = 1 - P(A)

(b) An experiment consists of rolling a single die, two events are defined as, A =a 6 showa up , B = a 2 or a 5 shows up

i. Find P(A) & P(B)

ii. Define third event C so that P(c) = 1- P(A)-P(B). [6+10]

2. (a) Sketch the probability density function and probability distribution functionsof:

i. Exponential distribution

ii. Rayleigh distribution

iii. Uniform distribution.

(b) List the properties of Gaussian curve.

(c) Mention a few applications of Binomial distribution. [9+4+3]

3. (a) Explain about the moment generating function of a random variable.

(b) Find the moment generating function of the following.

i. Y = ax+b

ii. Y = x+ab

[8+8]

4. (a) Two random variables X and Y have the joint pdf:

fx,y (x, y) = Ae−(2x+y), x, y ≥ 0= 0, elsewhere

i. Evaluate A

ii. Find the marginal pdf’s

iii. Find the marginal pdf’s

iv. Find the joint cdf

v. Find the distribution functions and conditional cdf’s.

(b) Consider a linear amplifier defined by input-output relation Y = aX + b wherea and b are constants. The input X is a Gaussian random variable with meanm and Variance σ2. Determine the probability density function of the randomvariable Y at the amplifier output. [8+8]

5. (a) Write the expression for expected value of a function of random variables andprove that the mean value of a weighted sum of random variables equals theweighted sum of mean values.

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(b) X is a random variable with mean X = 3, variance σ2X = 2.

i. Determine the second moment of X about origin

ii. Determine the mean of random variable y = where y = -6X +22. [8+8]

6. Find the time average mean and time autocorrelation function of the randomprocess for the random process X(t) = A cos (ω0t + Θ), where A, ω0 are constantsand Θ is a uniformly distributed random variable in the interval (0, 2π). [16]


[

1 −|τ |τ

]

− τ ≤ t ≤ τ


(b) RXX (τ) =A2

0


8. (a) What are the precautions to be taken in cascading stages of a network fromthe point of view of noise reduction?

(b) What is the need for band limiting the signal towards the direction of increas-ing SVR? [8+8]

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1. (a) Define and explain random experiment with an example.

(b) Four cards are drawn from a well shuffled pack of playing cards. Find theprobability that

i. All are clubs

ii. Two spades & two hearts

iii. Four cards from a different suit. [4+12]

2. (a) Define Binomial distribution. Write the expressions for distribution functionand density function of Binomial distribution.

(b) Find the probability that in tossing a fair coin five times, there will appear:

i. 3 heads

ii. 3 tails and 2 heads

iii. At least 1 head

iv. Not more than 1 tails. [8+8]

3. (a) For the binomial density, show that:E[X] = Npand σ2

x = Np(1-p)

(b) Explain in detail the non-monotonic transformation of a continuous randomvariable. [8+8]

4. (a) Find the Conditional density functions fX (xIy1), fX (xIy2), fX (yIx1) andfX (yIx2) for the joint function defined by P (x1, y1) = 2/15, P (x2, y1)=3/15,P (x2, y2)=1/15, P (x1, y3)=4/15 and P (x2, y3)=5/15.

(b) Explain the method of finding the distribution and density functions for a sumof statistically independent random variables statistically independent randomvariables. [10+6]





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6. (a) Calculate the PSD of a stationary random process for which the auto corre-lation is RXX (τ) = 6V , e−α|τ | auto correlation function of an aperiodic power

signal is RXX (τ) = exp(

−τ 2/262

)

. Find the PSD and normalized average

power content of the signal.

(b) Which of the following are suitable autocorrelation functions:

i. Acosω0τ ,

ii. Asinω0τ ,

iii. Aπτ/τ0where π(x) is a unit area rectangular function.

iv. A ∧

(

τ/τ0

)

where ∧(x) is the unit triangular function. [8+8]


[

1 −|τ |τ

]

− τ ≤ t ≤ τ


(b) RXX (τ) =A2

0


8. (a) Obtain the expression of auto correlation function of the white noise and writeits significance.

(b) Differentiate between narrow band and broad band noises. [8+8]

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1. (a) Define and explain following with example

i. Random Experiment

ii. Out come

iii. Trial and Event

(b) A die is tossed. Find the probabilities of the event A = odd number showsup, B = number larger than 3 shous up , A ∪ B and A ∩ B. [8+8]

2. (a) Define density function. Derive an equation for binomial distribution function.

(b) What is the distribution function of mixed random variable? Discuss what doyou mean by density function. [8+8]

3. (a) State the Chebyshev’s inequality and prove the same for k > 0.

(b) Prove that if ‘X’ and ‘Y’ are random variables taking real values then[

E (XY )2]≤ E [X2] .E [Y 2]. [8+8]

4. For two random variables X and Y, fX,Y (x, y) = 0.5δ(x + 1)δ(y) + 0.1δ(x)δ(y) +0.1δ(x)δ(y - 2) + 0.4 δ(x - 1)δ(y + 2) + 0.2δ(x - 1)δ(y - 1) + 0.5δ(x - 1)δ(y - 3)Find:

(a) the correlation

(b) the covariance and

(c) the correlation Coefficient of X and Y

(d) are X and Y either uncorrelated or orthogonal. [16]

5. (a) Obtain the relationship between marginal and joint characteristics functions.

(b) The joint characteristic function of two random variables X and Y is given byφXY (ω1, ω2) = exp (−2ω2

1,−8ω22). Show that X and Y are both zero - mean

random variables and they are uncorrelated. [16]

6. (a) Determine whether the random process X(t) = Acos(ω0t+Θ) is wide station-ary or not where A, ω0 are constants and Θ is a uniformly distributed randomvariable on the interval (0,2π)

(b) What is an ergodic random process, present the necessary expressions to sup-port the argument? [10+6]

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[

1 −|τ |τ

]

− τ ≤ t ≤ τ


(b) RXX (τ) =A2

0


8. (a) Write the quadrature representation of narrow band noise and mention fewproperties of narrow band noise.

(b) Write the characteristics of white noise. [10+6]

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Code No: 07A3EC10 Set No. 1

II B.Tech I Semester Regular Examinations, November 2008PROBABILITY THEORY AND STOCHASTIC PROCESSES




1. (a) Discuss Joint and conditional probability.

(b) When are two events said to be mutually exclusive? Explain with an example.

(c) Determine the probability of the card being either red or a king when one cardis drawn from a regular deck of 52 cards. [6+6+4]

2. (a) Define Random variable and give the concept of random variable.

(b) In an experiment of rolling a die and flipping a coin. The random variable(X)is chosen such that:

i. A coin head (H) outcome corresponds to positive values of X that areequal to the numbers that show upon the die and

ii. A coin tail (T) outcome corresponds to negative values of X that are equalin magnitude to twice the number that shows on die. Map the elementsof random variable X into points on the real line and explain.

(c) In experiment where the pointer on a wheel of chance is spun. The possibleoutcomes are the numbers from 0 to 12 marked on the wheel. The samplespace consists of the numbers in the set 0 < S < = 12 and if the randomvariable X is defined as X = X(S) = S2, map the elements of random variableon the real line and explain. [4+6+6]

3. (a) State and prove properties of variance of a random variable.

(b) Let X be a random variable defined by the density functionfX(x) = (π/16)cos(πx/8), −4 ≤ x ≤ 4

= 0, elsewhere

Find E [3X] and E[X2]. [8+8]

4. (a) Find the density function of W = X + Y where the densities of X and Y areassumed to be

fX (x) = 1a[u (x) − u (x − a)]

fY (y) = 1b[u (y) − u (y − b)]

Where 0 < a < b.

(b) Given the functionG (x, y) = u (x) u (y)

[

1 − e−(x+y)]

Show that this function satisfies the first four properties of joint probabilitydistribution function but fails the fifth one. The function is therefore not avalid joint probability distribution function. [8+8]

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6. Discuss in detail about:

(a) First order stationary random process

(b) Second order & Wide - Sense Stationary Random Process. [8+8]


[

1 −|τ |τ

]

− τ ≤ t ≤ τ


(b) RXX (τ) =A2

0




⋆ ⋆ ⋆ ⋆ ⋆

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1. (a) Is probability relative frequency of occurrence of some event? Explain withan example.

(b) One card is drawn from a regular deck of 52 cards. What is the probability ofthe card being either red or a king? [8+8]


(b) In an experiment of rolling a die and flipping a coin. The random variable(X)is chosen such that:

i. A coin head (H) outcome corresponds to positive values of X that areequal to the numbers that show upon the die and

ii. A coin tail (T) outcome corresponds to negative values of X that are equalin magnitude to twice the number that shows on die. Map the elementsof random variable X into points on the real line and explain.

(c) In experiment where the pointer on a wheel of chance is spun. The possibleoutcomes are the numbers from 0 to 12 marked on the wheel. The samplespace consists of the numbers in the set 0 < S < = 12 and if the randomvariable X is defined as X = X(S) = S2, map the elements of random variableon the real line and explain. [4+6+6]


(b) The exponential density function given by

fx(x) = (1/b)e−(x−a)/b x > a= 0 x < a

Find out variance and coefficient of skewness. [6+10]

4. (a) Consider a probability space S = ( Ω, F, P). Let Ω = ξ1....ξ5 = -1, -1/2, 0,1/2, 1 with Pξi = 1/5 i = 1..5. Define two random variables on S as follows:

X(ξ) =ξ and Y(ξ) =ξ2

i. Show that X and Y are dependent random variables

ii. Show that X and Y are uncorrelated.

(b) Let X and Y be independent random variables each N (0, 1). Find the meanand variance of Z =(X2+Y2)1/2. [8+8]

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6. (a) Given X(t) = A cos ω0t + B sin ω0t where A & B are R.V’s & ω0 is a const.S.T X(t) is Wss if A & B are uncorrelated zero mean R.V’s having differentdensity functions but the same variancess σ2.

(b) State the properties of cross correlation. [8+8]

7. (a) For X(t) and Y(t) are random process and prove that SXX (ω) = limt→∞

E[X∗

T(ω)YT (ω)]2T

(b) RXY (τ) = 4u (τ) e−ατ . Find SXY (ω) [12+4]

8. (a) Explain how the available noise power in an electronic circuit can be estimated.

(b) What are the different noise sources that may be present in an electron devices?[8+8]

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1. (a) What is sample space? Explain the Discrete sample space and Continuoussample space with suitable example each.

(b) In a game of dice a “shooter” can win outright if the sum of the two numbersshowing up is either 7 or 11 when two dice are thrown. What is his probabilityof winning outright? [8+8]

2. (a) What are point conditioning and interval conditioning distribution function?Explain.

(b) If P(x) = 0.1x, x = 1,2,3,4= 0, otherwise

Find:

i. PX = 1 or 2

ii. P(1/2) < X (5/2)X > 1. [8+8]

3. (a) If the random variable X has the moment generating function MX (t) = 22−t

,determine the variance of X.

(b) Show that the distribution function for which the characteristic function e−|t|

has the density:fX (x) = 1

π(1+x2),−∞ < x < ∞

(c) Explain the nonmonotonic transformation of a random variable. [6+6+4]



given byf(x, y) = C(2x + y), 0 ≤ x ≤ 1, 0 ≤ y ≤ 2

= 0, elsewhere

Find:


ii. Marginal distribution functions of X and Y. [8+8]




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6. Discuss in detail about:

(a) First order stationary random process

(b) Second order & Wide - Sense Stationary Random Process. [8+8]

7. The auto correlation function of a random process X(t) is RXX (τ) = 3+2 exp (−4τ 2).

(a) Find the power spectrum of X(t).

(b) What is the average power in X(t)

(c) What fractional power lies in the frequency band −1√2≤ ω ≤

1√2. [6+4+6]



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1. (a) Explain the terms Joint probability and Conditional probability.

(b) Show that Conditional probability satisfies the three axioms of probability.

(c) Two cards are drawn from a 52-card deck (the first is not replaced):

i. Given the first card is a queen. What is the probability that the secondis also a queen?

ii. Repeat part (i) for the first card a queen and second card a 7.

iii. What is the probability that both cards will be the queen? [4+6+6]

2. (a) Define cumulative probability distribution function. Discuss distribution func-tion specific properties.

(b) The random variable X has the discrete variable in the set -1,-0.5, 0.7, 1.5, 3the corresponding probabilities are assumed to be 0.1, 0.2, 0.1, 0.4, 0.2. Plotits distribution function and state is it a discrete or continuous distributionfunction. [8+8]

3. (a) A random variable X is uniformly distributed in the interval (-5 , 15). Anotherrandom variable Y = e−x/5 is formed. Find E[Y] and fY (y).

(b) Explain the following terms:

i. Conditional Expected value

ii. Covariance.

(c) Find the Expected value of the number on a die when thrown. [8+6+2]

4. (a) State and prove central limit theorem.

(b) Find the density of W = X + Y, where the densities of X and Y are assumedto be:

fX(x) = [u(x) - u(x - 1)], fY (y) = [u(y) - u(y - 1)] [8+8]





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6. (a) Prove that autocorrelation function of a random process is even function of τ .

(b) Prove that RXX (τ) = RXX (0). [8+8]

7. (a) A WSS noise process N(t) has an autocorrelation function RNN (τ) = Pe−3|τ |

where p is a constant. Find and sketch its power spectrum.

(b) Consider the figure shown in figure 7.

where X(t), Y(t) are random processes & X(t) is WSS. Find the relationbetween SY Y (ω) and SXX(ω). [8+8]

Figure 7

8. (a) A Signal x(t) = u(t) exp (-αt ) is applied to a network having an impulseresponse h(t)= ω u(t) exp (-ω t). Here α & ω are real positive constants.Find the network response?

(b) Two systems have transfer functions H1( ω) & H2( ω). Show the transferfunction H(ω) of the cascade of the two is H(ω) =H1(ω) H2 (ω).

(c) For cascade of N systems with transfer functions Hn(ω) , n=1,2,........N showthat H(ω) = πHn(ω). [6+6+4]

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Code No: NR210402 NR

II B.Tech I Semester Supplimentary Examinations, November 2007PROBABILITY AND RANDOM VARIABLES

( Common to Electronics & Communication Engineering and Electronics &Telematics)


All Questions carry equal marks? ? ? ? ?

1. (a) State and prove Bayes theorem of probability.

(b) In a single throw of two dice, what is the probability of obtaining a sum of atleast 10? [8+8]

2. (a) Define the joint distribution function. Explain how marginal density functionsare Computed given their joint distribution functions.

(b) A product is classified according to the number of defects it contains (X1) andthe factory that produces it (X2). The joint probability distribution is givenby:

X2 → 1 2X1 ↓0 1/8 1/161 1/16 1/162 3/16 1/83 1/8

(a) Find the marginal distribution of X1

(b) Find the conditional distribution of X1 when X2 is equal to 1

(c) Are the variables X1 and X2 independent? [7+9]

3. (a) If X is random variable, show that var(aX+b) = a2 var(X)

(b) A random variable z is uniformly distributed having Probability density func-tionfZ(z) = 1/2, −1 <= Z <= 1

= 0, otherwiseShow that the random variables X=Z and Y=Z2 are un-correlated despite ofthe fact that they are statistically dependant. [6+10]

4. Let the Random process be given as = Z(t) = x(t) cos [$0t + θ] where x(t) instationary Random process with E[x(t)]=0 and E[x2(t)] = σ2

x

(a) If θ = 0 find E[Z(t)] and E[Z2] if Z(t) stationary.

(b) If θ is a random variable independent of x(t) and uniformly distributed over

the interval (−Π, Π) show that E[Z(t)] = 0 and E[Z2(t)] = σ2x

2[8+8]

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5. A random process x(t) is applied to a network with impulse response

h(t) = u(t)texp(−bt)

where b>0 is a constant. The cross-correlation of x(t) with the o/p y(t) is knownto have the same form.

Rxx(τ) = u (τ) τexp(−bτ)

(a) Find the auto correlation of y(t)?

(b) What is the average power in y(t)? [8+8]

6. (a) What are the causes of thermal noise?

(b) What are the causes of shot noise? [8+8]

7. In TV receivers, the antenna is often mounted on a tall mask and a long lossy cableis used to connect the antenna and receiver. To overcome the effect of noisy cable,a preamplifier is mounted on the antenna. The parameters of the different stagesarePreamplifier gain = 20 dBPreamplifier Noise figure = 6 dBLossy cable noisy figure = 3 dBCable Loss = -20 dBReceiver front end gain = 60 dBReceiver Noise figure = 16 dBDetermine the overall noise figure of the system. [16]

8. (a) State and prove Shannon’s theorem with respect to channel capacity.

(b) Find the entropy corresponding to the random variable whose density functionshown below figure8b, [8+8]

Figure 8b

? ? ? ? ?

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II B.Tech I Semester Supplimentary Examinations, May/Jun 2009PROBABILITY AND RANDOM VARIABLES




1. (a) Distinguish between mutually exclusive events and independent events.

(b) A letter is known to have come either from LONDON or CLIFTON. On thepostmark only the two consecutive letters ‘ON’ are legible. What is the Chancethat it came from London? Give step-by-step answer.

(c) Show that the chances of throwing six with 4,3 or 2 dice respectively are as1:6:18. [4+6+6]

2. (a) The number of newspapers that a certain delivery boy is able to sell in a dayis found to be a numerical valued random phenomenon, with a probabilityfunction specified by the p.m.f.p(.) given by:

p(x) = Ax x = 1, 2, 3, ..................., 50

= A(100− x) x = 51, 52, ..............100

= 0 otherwise

i. Find the value of A that makes p (.) a p.m.f. and sketch its graph.

ii. What is the Probability that the number of newspapers sold tomorrow is

A. more than 50

B. less than 50

C. equal to 50

D. between 25 and 75 exclusive

E. an odd number.

iii. Let the events indicated in (ii) be denoted respectively by A, B, C, D andE. Find P(A|B), P(A|C), P(A|D)andP(C|D). Are A and B independent?Are A and D independent? Are C and D independent events? [4+5+7]

3. (a) Find the moment generating function of the random variable having proba-bility density functionfX (x) = x, 0 ≤ x ≤ 1

= -2-x, 1 ≤ x ≤ 2

= 0, else where

(b) Find the moment generating function of the random variable whose momentsare mr= (r + 1)!2r. [8+8]

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4. A class of modulation signal is modulated by

Xc(t) = AX(t)Cos($ct+θ)

Where x(t) is the message signal and A Cos ($ct + θ) is the carrier. The messageSignal x(t) is modeled as a zero mean stationary random process with the autocorre-lation function Rxx(τ) and the PSD Gx(f). The carrier amplitude A and frequency$c are assumed to be constants and the initial carrier phase θ is assumed to be arandom variable uniformly distributed in the interval (−Π, Π). Further more x(t)and θ are assumed to be independent.

(a) Show that Xc(t) is stationary

(b) Find the PSD of Xc(t). [8+8]

5. (a) Find the PSD of a random process z(t) = X(t) + y(t) where x(t) and y(t) arezero mean, individual random process.

(b) A wss random process x(t) is applied to the input of an LTI system whoseimpulse response is 5t.e−2t The mean of x(t) is 3. Find the output of thesystem. [8+8]

6. (a) What are the characteristics of White noise?

(b) Discuss the spectral distribution of thermal noise. [8+8]

7. (a) The noise figure of an amplifier at room temperature (T=29.K) 0.2db. Findthe equivalent temperature.

(b) Explain the concept of effective input noise temperature [8+8]

8. (a) A code is composed of dots and dashes. Assume that a dash is three times aslong as the dot and has one-third the probability of occurrence.Find,

i. The information in a dot and that in a dash, and

ii. The entropy in the dot - dash code.

(b) Suppose 100 voltage levels are employed to transmit 100 equally likely mes-sages. Assume the system to be a Gaussian channel with λ = 3.5 and band-width B = 104 Hz. Find S/N. [8+8]

? ? ? ? ?

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II B.Tech I Semester Supplimentary Examinations, May/Jun 2009PROBABILITY AND RANDOM VARIABLES








p(x) = Ax x = 1, 2, 3, ..................., 50

= A(100− x) x = 51, 52, ..............100

= 0 otherwise



A. more than 50

B. less than 50

C. equal to 50


E. an odd number.



= -2-x, 1 ≤ x ≤ 2

= 0, else where


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Xc(t) = AX(t)Cos($ct+θ)

Where x(t) is the message signal and A Cos ($ct + θ) is the carrier. The messageSignal x(t) is modeled as a zero mean stationary random process with the autocorre-lation function Rxx(τ) and the PSD Gx(f). The carrier amplitude A and frequency$c are assumed to be constants and the initial carrier phase θ is assumed to be arandom variable uniformly distributed in the interval (−Π, Π). Further more x(t)and θ are assumed to be independent.













? ? ? ? ?

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II B.Tech I Semester Supplementary Examinations, May 2005PROBABILITY & RANDOM VARIABLES





(b) In a single throw of two dice, what is the probability of obtaining a sum of atleast 10?

2. (a) Explain the Rayleigh probability density function.

(b) Find the mean value, the mean squared value and the cumulative distributionfunction for the Rayleigh distribution with parameter α ¿0, specified by thepdf f(x) = x

α2 exp

−12

xα

.

3. (a) Prove that |Rxy(τ)| <= sqrt[Rxx(0)Ryy(0)] .

(b) Find the mean and characteristic function of Binomial distribution.

4. (a) State the condition for wide sense stationary Random process.

(b) Find the Auto Correlation function for white noise shown in the figure below.

5. Find the input auto correlation function, output autocorrelation and o/p spectraldensity of RC low pass filter, where the filter is subjected to a white noise of spectraldensity N0/2.


(b) Discuss the spectral distribution of thermal noise.

7. (a) What are the precautions to be taken in cascading stages of a network in thepoint of view of noise reduction?

(b) What is the need for band limiting the signal towards the direction increasingSNR.

8. (a) Compare discrete and continuous channel with respect to information trans-mission.

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(b) Derive an expression for channel capacity of a continuous channel in the pres-ence of White Gaussian noise.

? ? ? ? ?

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II B.Tech I Semester Supplementary Examinations, November 2005PROBABILITY & RANDOM VARIABLES




1. (a) List and explain the properties of probability density function

(b) Three newspapers A, B and C are published in a city and a survey of readersindicates the following:20% read A, 16% read B, 14% read C, 8% read A and B, 5% read A and C,2% read A, B and C.For one adult chosen at random, compute the probability that:

i. he reads none of the papers.

ii. he reads exactly one of the papers

iii. he reads at least A and B if it is known that he reads at least one of thepapers published.

[7+9]

2. (a) Explain the Gaussian distribution with a neat sketches of pdF and cdF.

(b) An analog signal received at the detector (measured in microvolts) may bemodeled as a Gaussian random variable N (200,256) at a fixed point in time.What is the probability that the signal will exceed 240 µ vs. what is theprobability that the signal is larger than 240 µ V, given that it is larger than210 µ Vs?

[8+8]


= -2-x, 1 ≤ x ≤ 2

= 0, else where

(b) Find the moment generating function of the random variable whose momentsare mr= (r + 1)!2r.

[8+8]

4. (a) Which are the following are suitable auto correlation functions?

i. Acos$0τ

ii. AΠ(τ/τ0) where Π(x) is a unit area rectangular function

1 of 2


(b) Suppose we are given a cross power spectrum defined by

Sxx($) = a + ((jb$/w));−W < $ < W

= 0 : ElsewhereWhere W > 0, a and b are real constants. Find the cross correlation function.

[8+8]

5. Find the input auto correlation function, output autocorrelation and o/p spectraldensity of RC low pass filter, where the filter is subjected to a white noise of spectraldensity N0/2. [16]

6. (a) Explain how partition noise is present in electron devices?

(b) Explain the usefulness of knowing the noise power spectral density of a net-work.

[8+8]

7. (a) Derive the equation for narrow band noise and illustrate all its properties

(b) Show their noise figure F of a n/w is given by F= Go(f)K2Gin(f)

where Go(f),

Gin(f), and K are respectively open circuited voltage, spectral density andthe voltage gain of n/w.

[10+6]

8. (a) Obtain the Shannon - Hartley law giving the, relation amongst channel ca-pacity, bandwidth and signal to noise ratio of a continuous system.

(b) Consider a message sequence having alphabets Q1, Q2, Q3 and Q4 with prob-abilities 1/2, 1/4, 1/8 and 1/8 respectively.

i. Calculate the entropy of the message sequence,

ii. Find the information rate of the message rate is 1 message / second,

iii. What is the rate at which binary signals are transmitted if the signal issent after encoding Q1, Q2, Q3 and Q4 is 00, 01, 10 and 11.

[8+2+3+3]

? ? ? ? ?

2 of 2


II B.Tech II Semester Supplimentary Examinations, Apr/May 2009PROBABILTY AND RANDOM VARIABLES

(Bio-Medical Engineering)Time: 3 hours Max Marks: 80

Answer any FIVE QuestionsAll Questions carry equal marks

? ? ? ? ?





(b) Find the mean value, the mean squared value and the cumulative distributionfunction for the Rayleigh distribution with parameter α> 0, specified by thepdf [8+8]

f(x) =x

α2exp

− 1

2

x

α

3. (a) Let x be the random Variable with probability law P (X = r) = qr−1p, r =

1, 2, 3..... Find the moment generating function & hence mean & Variance,Assume p+q=1

(b) The random Variable X has characteristic function is given by′

φ (t) = 1− |t|, |t| <= 1

= 0,|t|>1Find the density function of random variable X [9+7]

4. (a) Explain the concept of Random process.

(b) An ergoic random process is known to have an auto correlation function ofthe form [8+8]

Rxx(τ) = 1− |τ |, |τ | ≤ 1

= 0,|τ | >1Show the spectral density is given by

Sxx (ω) =

[Sinω/2

ω/2

]2

1 of 2


5. A Random process n(t) has a power spectral density g(f) = η/2 for α ≤ f ≤ αRandom process is passed through a low pass filter which has transfer functionH(f)=2 for −fm ≤ f ≤ fm and H(f)= 0 otherwise. Find the PSD of the waveformat the o/p of the filter. [16]

6. (a) What are the characteristics of shot noise?

(b) What are the important requirements of the front-end stage of a communica-tion receiver in the point of view of noise? [8+8]


8. (a) State the axioms of an entropy function. Show that only function whichsatisfies the above axiom is,

H (P1, P2, .........Pn) = λn∑

i=1

pi log2 pi

(b) A man is informed that when a pair of dice was rolled, the result was a seven.How much information is there in this message? [8+8]

? ? ? ? ?

2 of 2


II B.Tech II Semester Supplementary Examinations, April/May 2005PROBABILTY & RANDOM VARIABLES



? ? ? ? ?



2. (a) Explain random variables and probability mass function each with an example.

(b) Define probability density function. List its properties.

3. (a) State and Prove any four properties of characteristic function.

(b) Find the density function of the distribution for which the characteristic func-

tion is φ(t) = e−−t

2σ2

2


(b) An ergoic random process is known to have an auto correlation function of

the formRxx(τ) = 1− |τ | , |τ | 1

= 0, |τ |> 1

Show the spectral density is given by

Sxx (ω) =[

Sinω/2ω/2

]2

5. A Random process n(t) has a power spectral density g(f) =η/2for−α ≤ f ≤ αRandom process is passed through a low pass filter which has transfer functionH(f) = 2for − fm ≤ f ≤ fmandH(f) = 0 otherwise. Find the PSD of the waveformat the o/p of the filter.



7. In TV receivers, the antenna is often mounted on a tall mask and a long lossy cableis used to connect the antenna and receiver. To overcome the effect of noisy cable,a preamplifier is mounted on the antenna. The parameters of the different stagesarePreamplifier gain = 20 dBPreamplifier Noise figure = 6 dBLossy cable noisy figure = 3 dBCable Loss = -20 dBReceiver front end gain = 60 dBReceiver Noise figure = 16 dBDetermine the overall noise figure of the system.

1 of 2


8. (a) A code is composed of dots and dashes. Assume that a dash is three times aslong as the dot and has one-third the probability of occurrence.

(b) Suppose 100 voltage levels are employed to transmit 100 equally likely mes-sages. Assume the system to be a Gaussian channel with λ= 3.5 and band-width B = 104Hz Find S/N.

? ? ? ? ?

2 of 2


II B.Tech II Semester Supplementary Examinations,November/December 2005

PROBABILTY & RANDOM VARIABLES(Bio-Medical Engineering)





(c) Show that the chances of throwing six with 4,3 or 2 dice respectively are as1:6:18.

[4+6+6]

2. (a) If A and B are independent events, prove that the events−

A and B,−

A and B;and A and B are also independent. [6]

(b) A1, A2 and A3 are three mutually exclusive and exhaustive sets of eventsassociated with a random experiment E1. Events B1,B2 and B3 are mutuallyexclusive and exhaustive sets of events associated with a random experimentE2. The joint Probabilities of occurrence of these events and some marginalprobabilities are listed in the table given below:

B1 B2 B3A1 3/36 * 5/36A2 5/36 4/36 5/36A3 * 6/36 *

P(Bj) 12/36 14/36 *

i. Find the missing probabilities (*) in the table.

ii. Find P(B3|A1) and P(A1|B3)

iii. Are events A1 and B1 statistically independent?

[4+4+2]

3. (a) For a function Y=(X − mx)/σx, prove that mean is zero & variance is 1

(b) For the joint distribution of (X,Y) given by

fxy(x, y) = 14a2 [(1 + xy) (x2 − y2] , |x| <= a, |y| <= a, a > 0

= 0, otherwiseShow that the Characteristic function of X+Y is equal to the product of thecharacteristic function of X & Y.

[8+8]

1 of 2


4. Consider a Random binary waveform that consists of a sequence of pulses with thefollowing properties

(a) Each pulse is of duration T0

(b) Pulses are Equally likely to be ±1

(c) All pulses are statistically independent

(d) The pulses are not synchronized, that is, the starting time T of the first pulseis Equally likely to be anywhere between 0 and Tb

Find the Auto correlation and power spectral density function of x(t). [8+8]


6. (a) What are the different man made noises introduced in the communicationsystem?

(b) Write a short note on noises introduced by extra terrestrial objects.

[8+8]

7. (a) An amplifier has input and output impedances of 75 ohm, 60dB power gain,and a noise equivalent bandwidth of 15KHz. When a 75Ω resistor at 290K isconnected to the input, the output rms noise voltage is 75microvolt. Determinethe effective noise temperature of the amplifier assuming that the meter isimpedance matched to the amplifier.

(b) List the devices in which narrowband noise can be present.

[8+8]


H (P1, P2, .........Pn) = λn∑

i=1

pi log2 pi

(b) A man is informed that when a pair of dice was rolled, the result was a seven.How much information is there in this message?

[8+8]

? ? ? ? ?

2 of 2

Code No: OR220454 OR

II B.Tech II Semester Supplementary Examinations, April/May 2005PROBABILITY & RANDOM VARIABLES






2. (a) Explain random variables and probability mass function each with an example.

(b) Define probability density function. List its properties.


(b) Find the density function of the distribution for which the characteristic func-

tion is f (t) = e−−t

2σ2

2


(b) An ergoic random process is known to have an auto correlation function of

the formRxx(τ) = 1− |τ | , |τ | ≤ 1= 0, |τ |> 1

Show the spectral density is given by

Sxx (ω) =[

Sinω/2ω/2

]2

5. A Random process n(t) has a power spectral density g(f) = α ≤ f ≤ α for -Random process is passed through a low pass filter which has transfer functionH(f) = 2for − fm ≤ f ≤ fmandH(f) = 0 otherwise. Find the PSD of the waveformat the o/p of the filter.



7. In TV receivers, the antenna is often mounted on a tall mask and a long lossy cableis used to connect the antenna and receiver. To overcome the effect of noisy cable,a preamplifier is mounted on the antenna. The parameters of the different stagesarePreamplifier gain = 20 dBPreamplifier Noise figure = 6 dBLossy cable noisy figure = 3 dBCable Loss = -20 dBReceiver front end gain = 60 dB

1 of 2


Receiver Noise figure = 16 dBDetermine the overall noise figure of the system.

8. (a) A code is composed of dots and dashes. Assume that a dash is three times aslong as the dot and has one-third the probability of occurrence. Find,


ii. The entropy in the dot dash code.

(b) Suppose 100 voltage levels are employed to transmit 100 equally likely mes-sages. Assume the system to be a Gaussian channel with λ= 3.5and bandwidthB = 104Hz Find S/N.

? ? ? ? ?

2 of 2


III B.Tech I Semester Supplementary Examinations, April/May 2005PROBABILITY & RANDOM VARIABLES



? ? ? ? ?

1. (a) Give the classical and axiomatic definitions of Probability.

(b) If a three digit decimal number is chosen at random, find the probability thatexactly K digits are greater than equal to 5, for 0 ≤ K ≤ 3 .

(c) Three boxes of identical appearance contain two coins each. In one box bothare gold; in the second both are silver and in the third box one is silver and theother is the gold coin. Suppose that a box is selected at random and furtherthat a coin in that box is selected at random. If this coin proves to be gold,what is the probability that the other coin is also gold?


(b) An analog signal received at the detector (measured in microvolts) may bemodeled as a Gaussian random variable N (200,256) at a fixed point in time.What is the probability that the signal will exceed 240 µvs. what is theprobability that the signal is larger than 240 µV, given that it is larger than210 µVs?

3. (a) State and prove any four properties of mean of a random variable X.

(b) Prove that the density function of sum of two statistically independent randomvariables is the convolution of their individual density functions.

4. Find the Auto correlation function and power spectral density of the Randomprocess.x(t) = K Cos (ωot + θ) Where θ is a Random variable over the ensemble and isuniformly distributed over the Range ( 0, 2π)


6. Give reasons for the following:

(a) In any communication system the first stage must have low noise operation.

(b) Describe how FET gives low noise performance compared to BJT.

7. (a) The noise figure of an amplifier at room temperature (T=290oK) is 0.2db.Find the equivalent temperature.

(b) Explain the concept of effective input noise temperature.

1 of 2


8. Explain the following:

(a) Code efficiency

(b) Noiseless-coding theorem

(c) Ideal channel

(d) Hamming codes.

? ? ? ? ?

2 of 2


II B.Tech I Semester Supplimentary Examinations, May/Jun 2009PROBABILITY THEORY AND STOCHASTIC PROCESS




1. (a) What is the meaning of combined sample space? Explain with an example?

(b) In a combined experiment of flipping a coin and rolling a single dice, determinethe combined sample space.

(c) In a combined experiment of flipping a coin twice, what is the sample spaceand what is meant by ordered pairs of objects. [8+4+4]

2. (a) What is poisson random variable? Explain in brief.

(b) What is binomial density and distrbution function?

(c) Assume automobile arrives at a gasoline station are poisson and occur at anaverage rate of 50/hr. The station has only one gasoline pump. If all cars areassumed to require one minute to obtain fuel. What is the probability that awaiting line will occur at the pump? [5+5+6]


(b) The exponential density function given by .fX(x) =

1be−(x−a)/b x > a

0 x < aFind out variance and coefficient of skewness. [6+10]



given by f(x, y) =

C(2x + y) 0 ≤ x ≤ 1 , 0 ≤ y ≤ 20 elsewhere

. Find


ii. marginal distribution functions of X and Y. [8+8]

5. (a) let Y = X1 + X2 + ............+XN be the sum of N statistically independentrandom variables Xi, i=1,2.............. N. If Xi are identically distributed thenfind density of Y, fy(y).

(b) Consider random variables Y1 and Y2 related to arbitrary random variables Xand Y by the coordinate rotation. Y1=X Cos θ + Y Sin θ Y2 = -X Sin θ + YCos θ

i. Find the covariance of Y1 and Y2, CY1Y2

ii. For what value of θ, the random variables Y1 and Y2 uncorrelated. [8+8]

1 of 2


6. A random process Y(t) = X(t)- X(t +τ) is defined in terms of a process X(t) thatis at least wide sense stationary.

(a) Show that mean value of Y(t) is 0 even if X(t) has a non Zero mean value.

(b) Show that σY2= 2[RXX(0) − RXX(τ)]

(c) If Y(t) = X(t) +X(t + τ) find E[Y(t)] and σY 2. [5+5+6]

7. (a) Prove the relation between continuous and discrete power spectral densities.

(b) Draw the ACF and PSD of white noise. and hence derive the expression forACF and PSD of band limited white noise and plot them. [8+8]

8. (a) A Signal x(t) = u(t) exp (-αt ) is applied to a network having an impulseresponse h(t)= ω u(t) exp (-ω t). Here α & ω are real positive constants.Find the network response? (6M)

(b) Two systems have transfer functions H1( ω) & H2( ω). Show the transferfunction H(ω) of the cascade of the two is H(ω ) =H1( ω) H2 (ω ).

(c) For cascade of N systems with transfer functions Hn(ω) , n=1,2,.... .N showthat H( ω) = πHn(ω). [6+6+4]

⋆ ⋆ ⋆ ⋆ ⋆

2 of 2






1. (a) Define probability based on set theory and fundamental axioms.

(b) When two dice are thrown, find the probability of getting the sums of 10 or11. [8+8]


(b) In an experiment of rolling a die and flipping a coin. The random variable(X)is choosen such that

i. a coin head (H) outcome corresonds to positive values of X that are equalto the numbers that show upon the die and

ii. a coin tail (T) outcome corresponds to negative values of X that are equalin magnitude to twice the number that shows on die. Map the elementsof random variable X into points on the real line and explain.

(c) In experiment where the pointer on a wheel of chance is spun. The possibleoutcomes are the numbers from 0 to 12 marked on the wheel. The samplespace consists of the numbers in the set 0 < S <= 12 and if the randomvariable X is defined as X = X(S) = S2, map the elements of random variableon the real line and explain. [4+6+6]

3. (a) Explain the concept of a transformation of a random variable X

(b) A Gaussian random variable X having a mean value of zero and variance one istransformed to an another random variable Y by a square law transformation.Find the density function of Y. [8+8]

4. (a) Define and explain joint distribution function and joint density function oftwo random variables X and Y.

(b) If the function f(x, y) =

be−2x cos(y/2) 0 ≤ x ≤ 1 , 0 ≤ y ≤ π0 elsewhere

, where

‘b’ is a positive constant, is valid joint probability density function, find ‘b’[8+8]

5. Random variables X and Y have the Joint density functionfX,Y(x, y) = (x + y)2/40 −1 < x < 1 and − 3 < y < 3. [6+5+5]

(a) find all the second order moments of X and Y

(b) what are the variances of X and Y.

(c) what is the correlation coefficient.

1 of 3


6. A random process X(t) has periodic sample functions as shown in figure6, whereB,T and 4to ≤ T are constants but ε is a random variable uniformly distributedon the interval (0,T)

(a) find the first order distribution function of X(t)

(b) find the first order density function

(c) find E[X(t)], E[X2(t)] and σX2. [4+4+8]

Figure 6

7. (a) Determine which of the following functions are valid PSDS.

i. ω2

ω63ω2+3

ii. exp[−(ω − 1)2]

iii. ω2

ω4+1− δ(ω)

iv. ω4

1+ω2+jω6 . [4 ×3 = 12]

(b) Define RMS bandwidth of PSD and explain. [4]

8. (a) For the network shown in figure 8 find the transfer function of the system.[8]

(b) Define the following systems

i. LTI system

ii. Causal system

iii. stable system

iv. Noise Bandwidth. [4×2=8]

2 of 3


Figure 8

⋆ ⋆ ⋆ ⋆ ⋆

3 of 3






1. (a) Discuss in brief permutations and combinations.

(b) How many permutations are there for four cards taken from a 52-card deck?

(c) A coach has five athletes from whom a 3-persons team is to be selected for acompetition. How many such teams could be select? [8+4+4]

2. (a) What are the conditions for the function to be a random variable? Discuss

(b) What do you mean by continuous and discrete random variable?

(c) Demonstrate with an examle that the probability of occurrence of any discretevalue of a continuous random variable is zero. [4+6+6]


(b) The exponential density function given by .fX(x) =

1be−(x−a)/b x > a

0 x < aFind out variance and coefficient of skewness. [6+10]

4. The joint space for two random variables X and Y and corresponding probabilitiesare shown in table

Find and Plot

(a) FXY (x, y)

(b) marginal distribution functions of X and Y.

(c) Find P(0.5 < X < 1.5),

(d) Find P(X ≤ 1, Y ≤ 2) and

(e) Find P(1 < X ≤ 2, Y ≤ 3).

X, Y 1,1 2,2 3,3 4,4P 0.05 0.35 0.45 0.15

[3+4+3+3+3]



1 of 2




6. Statistically independent zero mean random processes X(t) and Y(t) have autocorrelations functionsRXY (τ) = e - |τ | andRYY(τ) = cos (2Πτ) respectively.

(a) find the auto correlation function of the sum W1(t) = X(t) + Y(t)

(b) find the auto correlation function of difference W2(t) = X(t) - Y(t)

(c) Find the cross correlation function of W1(t) and W2(t). [5+5+6]


i. ω2

ω63ω2+3

ii. exp[−(ω − 1)2]

iii. ω2

ω4+1− δ(ω)

iv. ω4

1+ω2+jω6 . [4 ×3 = 12]




i. LTI system

ii. Causal system

iii. stable system


Figure 8

⋆ ⋆ ⋆ ⋆ ⋆

2 of 2






1. (a) What is Bayes’ theorem? Explain.

(b) Determine probabilities of system error and correct system transmission ofsymbols for an elementary binary communication system shown in figure 1consisting of a transmitter that sends one of two possible symbols (a 1 or a 0)over a channel to a receiver. The channel occasionally causes errors to occurso that a ‘1’ show up at the receiver as a ‘0’, and vice versa. Assume thesymbols ‘1’ and ‘0’ are selected for a transmission as 0.6 and 0.4 respectively.

[6+10]

Figure 1

2. (a) In an experiment of fair wheel of chance is spun with the numbers 0 to 12 onthe wheel. What is the distribution and density function explain with plot.


3. (a) Find the nth moment of uniform random variable and hence its mean.

(b) Find the density function a random variable X whose characteristic functionis ΦX(ω) = 1

2e−|ω| −∞ ≤ ω ≤ ∞ . [8+8]

4. (a) Joint probabilities of two random variables X and Y are given in table 4a

1 of 3


Table 4aIf the marginal probability of X is P(Xi) = [8/18 6/18 4/18]Find

i. missing probabilities (∗) in the table

ii. marginal probability of Y and

iii. P(Yi /X=2).

(b) The probability density functions of two statistically independent random vari-ables X and Y are given byfX(x) = xe−x x > 0 fY (y) = ye−y y > 0Find the probability distribution and density functions of W = X+Y. [8+8]

5. (a) Define random variables V and W by V=X+aY & W = X-aY where a is a realnumber and X and Y are random variables. Determine a in terms of momentsof X and Y such that V and W are orthogonal.

(b) Define random variables V and W by V = X+aY & W = X-aY where a is a realnumber and X and Y are gaussian random variables, show that W and V arestatistically independent if a2=σX2/σY2 where σX2andσY2 are the variancesof X and Y respectively.

(c) Let Y = aX+b. If ψx (ω) is the characteristic function of X then find thecharacteristic function of Y. [6+6+4]

6. A random process is defined by

(a) Y(t) = X(t)cos(ωot+θ) where X(t) is wide sense stationary random processthat amplitude modulates a carrier of constant angular frequency ωo with arandom phase θ independent of X(t) and uniformly distributed on (−Π,Π)

i. Find E[Y(t)]

ii. Find the auto correlation function of Y(t)

iii. Is Y(t) wide sense stationary.

(b) Given the random process X(t) = Acosω0t + Bsinω0twhereω0 is a constant,and A and B are uncorrelated zero mean random variables having differentdensity functions but the same variances σ2, show that X(t) is wide sensestationary process. [8+8]

2 of 3


7. (a) The PSD of random process is given by

δXX (ω) =

π, |ω| <|0, elsewhere

Find its Auto correlation function.

(b) State and Prove any four properties of PSD. [8+8]

8. (a) Define the following random processes

i. Band Pass

ii. Band limited

iii. Narrow band. [3×2 = 6]

(b) A Random process X(t) is applied to a network with impulse response h(t) =u(t) exp (-bt)

where b > 0 is ω constant. The Cross correlation of X(t) with the output Y(t) is known to have the same form:RXY (τ) = u(τ)τ exp (-bτ)

i. Find the Auto correlation of Y(t)

ii. What is the average power in Y(t). [6+4]

⋆ ⋆ ⋆ ⋆ ⋆

3 of 3

Code No: R05210401 Set No. 1

II B.Tech I Semester Supplimentary Examinations, November 2008PROBABILITY THEORY AND STOCHASTIC PROCESS




1. (a) What is an event and explain discrete and continuous events with an example.

(b) Discuss joint and conational probability.

(c) Determine the probability of a card being either red or a queen. [6+6+4]

2. (a) Define cumulative probability distribution function. And discuss distributionfunctions specific properties.

(b) What are the conditions for the function to be a random variable? Discuss.What do you mean by continuous and discrete random variable? [8+8]

3. (a) A discrete random variable X is an outcome of throwing a fair die. Find meanand variance of X.

(b) Find the skew and coefficient of skewness for a Rayleigh random variable X

fX(x) =

(x−2)5

e−(x−2)/10 x > 20 x < 2

[6+10]

4. Discrete random variables X and Y have a joint distribution functionFXY (x, y) = 0.1u(x + 4)u(y − 1) + 0.15u(x + 3)u(y + 5) + 0.17u(x + 1)u(y − 3)+0.05u(x)u(y − 1) + 0.18u(x − 2)u(y + 2) + 0.23u(x − 3)u(y − 4)+0.12u(x − 4)u(y + 3)

Find

(a) Sketch FXY (x, y)


(c) P(−1 < X ≤ 4,−3 < Y ≤ 3) and

(d) Find P(X ≤ 1, Y ≤ 2). [4+6+3+3]

5. (a) Random variables X and Y have the joint density f XY (x,y)= 1/240 < x < b and 0 < y < 4 what is the expected value of the function g(x,y) =(xy)2

(b) Show that the correlation coefficient satisfies the expression |ρ| = |µ11| /√

(µ11µ20) ≤

1. [8+8]

6. Let X(t) be a stationary continuous random process that is differentiable. Denoteits time derivative by X(t).

(a) Show that E[ •× (t)

]

= 0.

1 of 2


(b) Find R×× (τ) in terms of R×× (τ)sss [8+8]


i. ω2

ω63ω2+3

ii. exp[−(ω − 1)2]

iii. ω2

ω4+1− δ(ω)

iv. ω4

1+ω2+jω6 . [4 ×3 = 12]




i. LTI system

ii. Causal system

iii. stable system


Figure 8

⋆ ⋆ ⋆ ⋆ ⋆

2 of 2






1. (a) Is probability relative frequency of occurrence of some event? Explain withan example.

(b) Define an event and explain discrete and continuous event with an example.

(c) One card is drawn from a regular deck of 52 cards. What is the probability ofthe card being either red or a king? [6+6+4]

2. (a) Define cumulative probability distribution function. And discuss distributionfunctions specific properties.

(b) What are the conditions for the function to be a random variable? Discuss.What do you mean by continuous and discrete random variable? [8+8]

3. (a) The density function of a random variable X is gX(x) =

5e−5x 0 ≤ x ≤ ∞

0 elsewhereFind

i. E[X],

ii. E[(X − 1)2]

iii. E[3X-1]

(b) If the mean and variance of the binomial distribution are 6 and 1.5 respectively.Find E[X − P(X ≥ 3)]. [8+8]

4. If the joint PDF of (X,Y) is given by fXY (x, y) =

Cxye−(x2+y2) x ≥ 0, y ≥ 00 otherwise

Find

(a) constant ‘C’ so that this is a valid joint density function.


(c) show that X and Y are independent random variables.

(d) Find P(X ≤ 1, Y ≤ 1). [4+6+3+3]

5. (a) Show that the variance of a weighted sum of uncorrected random variablesequals the weighted sum of the variances of the random variables.

(b) Two random variables X and Y have joint characteristic functionφX, Y(ω1,ω2) = exp(−2ω2

1−8ω22).

i. Show that X and Y are zero mean random variables.

1 of 2


ii. are X and Y are correlated. [8+8]


6. (a) Define cross correlation function of two random processes X(t) and Y(t) andstate the properties of cross correlation function.

(b) let two random processes X(t) and Y(t) be defined byX(t) = A cos ω0t + B sin ω0tY(t) = B cos ω0t - A sin ω0tWhere A and B are random variables and ω0 is a constant. Assume A andB are uncorrelated, zero mean random variables with same variance. Findthe cross correlation function RXY (t,t+τ) and show that X(t) and Y(t) arejointly wide sense stationary. [6+10]

7. (a) If the PSD of X(t) is Sxx(ω ). Find the PSD of dx(t)dt

(b) Prove that Sxx (ω ) = Sxx (-ω )

(c) If R (τ) = ae|by|. Find the spectral density function, where a and b are con-stants. [5+5+6]

8. (a) A Stationary random process X(t) having an Auto Correlation functionRXX τ = 2e−4|τ | is applied to the network shown in figure 8a find

i. SXX (ω )

ii. IH(ω )I2

iii. SY Y (ω ). [4+4+2]

Figure 8a

(b) Write short notes on different types of noises. [6]

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II B.Tech I Semester Regular Examinations, November 2007PROBABILITY THEORY AND STOCHASTIC PROCESS




1. (a) Define and explain the following with an example:

i. Equally likely events

ii. Exhaustive events

iii. Mutually exclusive events

(b) Give the classical definition of probability.

(c) Find the probability of three half-rupee coins falling all heads up when tossedsimultaneously. [6+4+6]

2. (a) What is poisson random variable? Explain in brief.

(b) What is binomial density and distrbution function?

(c) Assume automobile arrives at a gasoline station are poisson and occur at anaverage rate of 50/hr. The station has only one gasoline pump. If all cars areassumed to require one minute to obtain fuel. What is the probability that awaiting line will occur at the pump? [5+5+6]

3. (a) Define moment generating function.

(b) State properties of moment generating function.

(c) Find the moment generating function about origin of the Poisson distribution.[3+4+9]

4. Given the function f(x, y) =

(x2 + y2)/8π x2 + y2 < b0 elsewhere

(a) Find the constant ‘b’ so that this is a valid joint density function.

(b) Find P(0.5b < X2 + Y2 < 0.8b). [7+9]

5. Three statistically independent random variables X1,X2 and X3 have mean valuesX1= 3, X2= 6 and X3= −2. Find the mean values of the following functions.

(a) g(X1,X2,X3) = X1 + 3X2 + 4X3

(b) g(X1,X2,X3) = X1 X2 X3

(c) g(X1,X2,X3) = −2X1,X2 −3X1 X3 + 4X2 X3

(d) g (X1,X2,X3) = X1+X2+X3. [16]

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6. Statistically independent zero mean random processes X(t) and Y(t) have autocorrelations functionsRXY (τ) = e - |τ | andRYY(τ) = cos (2Πτ) respectively.

(a) find the auto correlation function of the sum W1(t) = X(t) + Y(t)

(b) find the auto correlation function of difference W2(t) = X(t) - Y(t)

(c) Find the cross correlation function of W1(t) and W2(t). [5+5+6]

7. (a) Find the ACF of the following PSD’s

i. Sχχ(ω) = 157+12ω2

(16+ω2)(9+ω2)

ii. Sχχ(ω) = 8(9+ω2)2

(b) State and Prove wiener-Khinchin relations. [8+8]

8. A random noise X(t) having power spectrum SXX(ω) = 349+ω2 is applied to a to a

network for which h(t) = u(t)t2 exp(−7t). The network response is denoted by Y(t)

(a) What is the average power is X(t)

(b) Find the power spectrum of Y(t)

(c) Find average power of Y(t). [5+6+5]

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II B.Tech I Semester Supplimentary Examinations, February 2008PROBABILITY THEORY AND STOCHASTIC PROCESS




1. (a) Define probability with an Axiomatic Approach.

(b) In a box there are 100 resistos having resistance and tolerance as shown intable 1. If a resistor is chosen with same likelihood of being chosen for thethree events, A as “draw a 470 Ω resistor”, B as “draw a 100 Ω resistor”,determine joint probabilities and conditional probabilities.Table 1

Number of resistors in a box having given resistance and tolerance. [6+10]

Resistance (Ω) Tolerance5% 10% Total

22 10 14 2447 28 16 44100 24 8 32

Total 62 38 100

2. (a) Define and explain the following density functions

i. Binomial

ii. Exponential

iii. Uniform

iv. Rayleigh.

(b) What is density function of a random variable x, if x is guassian. [12+4]

3. (a) A random variable X has a characteristic function given by

Φx (ω) =

1 − |ω| |ω| ≤ 10 |ω| > 1

. Find density function

(b) A random variable X has the density function fX(x) = 1ae−b|x| −∞ ≤ x ≤ ∞ .

Find E[X], E[X2] and variance. [8+8]

4. The joint space for two random variables X and Y and corresponding probabilitiesare shown in tableX,Y 1,1 2,2 3,3 4,4P 0.2 0.3 0.35 0.15

Find and Plot

1 of 2


(a) FXY (x, y) ,

(b) marginal distribution functions of X and Y,

(c) Find P(X ≤ 2, Y ≤ 2) and

(d) Find P(1 < X ≤ 3, Y ≥ 3). [5+5+3+3]





6. (a) Consider random processesX(t) = A cos (ω0t +θ)Y(t) = B cos (ω1t +φ)Where A,B, ω1and ω0 are constants, while θ and φ are statistically independentrandom variables each uniform on (0,2Π)

i. Show that X(t) and Y(t) are jointly wide sense stationary

ii. If θ = φ show that X(t) and Y(t) are not jointly wide sense stationaryunless ω1 = ω0. [8+8]

(b) Let X(t) be the sum of a deterministic signal S(t) and a wide sense stationarynoise process N(t). Find the mean value, auto correlation and auto covariancefunctions of X(t). Discuss the stationary of X(t).


δXX (ω) =

π, |ω| <|

0, elsewhereFind its Auto correlation function.







⋆ ⋆ ⋆ ⋆ ⋆

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1. (a) Define and Explain the following with example:

i. Sample Space

ii. Discrete Sample Space

iii. Continuous Sample Space

(b) A pack contains 4 white and 2 green pencils, another contains 3 white and 5green pencils. If one pencil is drawn from each pack, find the probability that

i. both are white and

ii. one is white and another is green. [12+4]

2. (a) What are the conditions for the function to be a random variable? Discuss

(b) What do you mean by continuous and discrete random variable?

(c) Demonstrate with an examle that the probability of occurrence of any discretevalue of a continuous random variable is zero. [4+6+6]

3. (a) Define moment generating function.

(b) State properties of moment generating function.

(c) Find the moment generating function about origin of the Poisson distribution.[3+4+9]

4. (a) State and prove central limit theorem.

(b) Find the density of W = X + Y where the densities of X and Y are assumedto befX(x) == [u(x) − u(x − 1)], fY (y) = [u(y) − u(y − 1)], [8+8]

5. (a) let Xi, i = 1,2,3,4 be four zero mean Gaussian random variables. Use the jointcharacteristic function to show that E X1 X2 X3 X4 = E[X1 X2] E[X3 X4]+ E[X1X3]E[X2X4] + E[X2X3] E[X1X4]

(b) Show that two random variables X1 and X2 with joint pdf.fX1X2(X1, X2) = 1/16 |X1|< 4, 2 < X2< 4 are independent and orthogonal.[8+8]

6. (a) State the properties of cross correlation function

(b) Let X(t) be a wide sense stationary random process with auto correlation

function. RXX(τ) = e−a|τ | where a > 0 is a constant. Assume X(t) “amplitudemodulates” a “carrier” Cos (ωot + θ) as shown in figure 6, where ω0 is a

1 of 2


constant and θ is a random variable uniform on (-Π, Π) that is statisticallyindependent of X(t). Determine the auto correlation function of Y(t). [6+10]

Figure 6


δXX (ω) =

π, |ω| <|

0, elsewhereFind its Auto correlation function.







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1. (a) Define an event and when do we say the events are mutually exclusive. Explainwith an example.

(b) When two dice are thrown determine the probabilities from axiom 3 for thefollowing three events.

i. A = sum = 7

ii. B = 8 ≺ sum ≤ 11

iii. C = 10 ≺ sum and determine

iv. P(B ∩ C)

v. P(B ∩ C) [8+8]



3. (a) Find the nth moment of uniform random variable and hence its mean.

(b) Find the density function a random variable X whose characteristic functionis ΦX(ω) = 1

2e−|ω| −∞ ≤ ω ≤ ∞ . [8+8]


Find and Plot

(a) FXY (x, y)


(c) Find P(0.5 < X < 1.5),

(d) Find P(X ≤ 1, Y ≤ 2) and

(e) Find P(1 < X ≤ 2, Y ≤ 3).

X, Y 1,1 2,2 3,3 4,4P 0.05 0.35 0.45 0.15

[3+4+3+3+3]

5. (a) For two zero mean Gaussian random variables X and Y show that their jointcharacteristic function isφXY(ω1,ω2) = exp−1/2[σX2ω2

1+2ρσXσYω1ω2+σY2ω22].

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(b) Statistically independent random variables X and Y have moments m10 = 2,m20 = 14, m02 = 12 and m11 = -6 find the moment µ 22.

(c) Two Gaussian random variables X and Y have variances σX2= 9 and σY2= 4,respectively and correlation coefficient ρ. It is known that a coordinate ro-tation by an angle Π/8 results in new random variables Y1 and Y2 that areuncorrelated. what is ρ. [8+4+4]

6. A Random process is defined by X(t)= X0+Vt where X0 and V are statisticallyindependent random variables uniformly distributed on intervals . [X01,X02] and[V1,V2]. Respectively. Find

(a) mean

(b) the auto correlation

(c) the auto covariance functions of X(t).

(d) is X(t) stationary in any sense if so state the type. [16]

7. (a) If the PSD of X(t) is Sxx(ω). Find the PSD of dx(t)dt

(b) Prove that Sxx (ω) = Sxx (-ω )

(c) If R (τ) = ae−b|τ |. Find the spectral density function, where a and b areconstants. [5+5+6]



i. LTI system

ii. Causal system

iii. stable system


Figure 8

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1. (a) show that two elements cannot be both mutually exclusive and statisticallyindependent. What is the condition for two events to be independent?

(b) One card is selected from an ordinary 52-card deck with an event A as“selecta king”, B as “select a jack or queen” and c as “select a heart”. Determinewhether A,B and C are independent by pairs.

(c) What is the probability of drawing 3 white and 4 green balls from a bag thatcontains 5 white and 6 green balls, if 7 balls are drawn simultaneously atrandom? [6+6+4]



3. (a) State and prove properties of moment generating function of a random variableX

(b) The characteristic function for a random variable X is given by ΦX(ω) =1

(1−j2ω)N/2. Find mean and second moment of X. [8+8]


Find and Plot

(a) FXY (x, y)


(c) Find P(0.5 < X < 1.5),

(d) Find P(X ≤ 1, Y ≤ 2) and

(e) Find P(1 < X ≤ 2, Y ≤ 3).

X, Y 1,1 2,2 3,3 4,4P 0.05 0.35 0.45 0.15

[3+4+3+3+3]

5. Two random variables X and Y have means X = 1andY = 2, variances σx2= 4 and σY2= 1and a correlation coefficient ρXY = 0.4 New random variables W and V are definedby V=-X+2Y & W = X+3Y. Find

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(a) The means

(b) The variances

(c) The correlations and

(d) The correlation coefficient ρvw of V and W. [16]

6. Let X(t) be a stationary continuous random process that is differentiable. Denoteits time derivative by X(t).

(a) Show that E[ •× (t)

]

= 0.

(b) Find R×× (τ) in terms of R×× (τ)sss [8+8]

7. (a) If the PSD of X(t) is Sxx(ω). Find the PSD of dx(t)dt

(b) Prove that Sxx (ω) = Sxx (-ω )

(c) If R (τ) = ae−b|τ |. Find the spectral density function, where a and b areconstants. [5+5+6]






⋆ ⋆ ⋆ ⋆ ⋆

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Code No: RR210403 Set No. 1









X2 → 1 2X1 ↓

0 1/8 1/161 1/16 1/162 3/16 1/83 1/8






= 0, otherwiseShow that the random variables X=Z and Y=Z2 are un-correlated despite ofthe fact that they are statistically dependant. [6+10]

4. (a) State and prove properties of cross correlation function.

(b) Consider the Random process x(t)= A cos(0t + θ) where A and 0 are realconstants and θ is a random variable uniformly distributed on the interval(0, π/2) find the average power Pxx in x(t). [8+8]

1 of 2




6. Write short notes on

(a) Flicker noise

(b) Partition noise

(c) Johnson’s noise [5+5+6]


8. (a) Define the terms: Information, Self Information and Information rate withrespect to a source.

(b) A source generates eight symbols: A, B, C, D, E, F, G and H with probabilities0.5, 0.2, 0.1, 0.05, 0.05, 0.05, 0.03 and 0.02. Find the entropy and informationrate if the symbols are generated at a rate of 1000 / Sec. [6+10]

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(b) If a three digit decimal number is chosen at random, find the probability thatexactly K digits are greater than equal to 5, for 0 ≤ K ≤ 3.

(c) Three boxes of identical appearance contain two coins each. In one box bothare gold; in the second both are silver and in the third box one is silver and theother is the gold coin. Suppose that a box is selected at random and furtherthat a coin in that box is selected at random. If this coin proves to be gold,what is the probability that the other coin is also gold? [4+6+6]


A and B,−



B1 B2 B3A1 3/36 * 5/36A2 5/36 4/36 5/36A3 * 6/36 *

P(Bj) 12/36 14/36 *



iii. Are events A1 and B1 statistically independent? [4+4+2]

3. (a) Prove that the Characteristic function of a Gaussian random variable havingzero mean value is given by exp(−σ2w2).

(b) Let X be a uniformly distributed random variable in the interval (−π, π). Thisundergoes the transformation Y= cos X, find fy (y) and E[Y]. [8+8]


Xc(t) = AX(t)Cos(ct+θ)

1 of 2


Where x(t) is the message signal and A Cos (ct + θ) is the carrier. The messageSignal x(t) is modeled as a zero mean stationary random process with the autocorre-lation function Rxx(τ) and the PSD Gx(f). The carrier amplitude A and frequencyc are assumed to be constants and the initial carrier phase θ is assumed to be arandom variable uniformly distributed in the interval (−Π, Π). Further more x(t)and θ are assumed to be independent.



5. (a) Derive the relation between PSDs of input and output random process of anLTI system.

(b) X(t) is a stationary random process with zero mean and auto correlationRXX (τ) e−2|τ |is applied to a system of function H (w) = 1

jw+2Find mean

and PSD of its output. [8+8]


(b) What are the different noise sources that may be present in an electron devices?[8+8]

7. (a) Derive the mathematical description of noise figure.

(b) An amplifier with ga = 40dB and BN = 20KHz is found to have Nao = 10K T0

when Ti = T0 . Find Te and noise figure. [10+6]





⋆ ⋆ ⋆ ⋆ ⋆

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2. (a) Let X and Y be jointly continuous random variable with joint p.d.f.

fxy(x, y) = x2 +xy

30 ≤ x ≤ 1, 0 ≤ y ≤ 2

= 0 elsewhere

find

i. fx(x)

ii. fy(y)

iii. are X and Y independent?

iv. fx (x|y)

v. fy (y|x)

(b) Two independent random variables X and Y have the probability densityfunctions respectively)

fx(x) = xe−x, x > 0; fY(y) = 1; 0 ≤ y ≤ 1

= 0 otherwise

Calculate the probability distribution and density functions of the randomvariable Z = XY [10+6]

3. (a) State and prove any four properties of mean of a random variable X

(b) Prove that the density function of sum of two statistically independent randomvariables is the convolution of their individual density functions. [10+6]




1 of 2





6. (a) What do you understand by noise power spectral density?

(b) How the autocorrelation function of the White noise represented? What is itssignificance? [8+8]


(b) What is the need for band limiting the signal towards the direction increasingSNR. [8+8]

8. (a) Write a short notes on the role and use of Information theory to a communi-cation engineer.

(b) A card is drawn at random from ordinary deck of 52 playing cards. Find theinformation in bits that you receive when you are told that the card is a heart,a face card, and a heart face card. [8+8]

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2. (a) Two discrete random variables X and Y have joint p.m.f. given by the followingtableX ↓ 1 2 3 Y

←

1 1/12 1/6 1/122 1/6 1/4 1/123 1/12 1/12 0

(b) Compute the probability of each of the following events :

i. X ≤ 11/2

ii. XY is even

iii. Y is even given that X is even. [5+5+6]

3. (a) Let x be the random Variable with probability law P (X = r) = qr−1p, r =1, 2, 3..... Find the moment generating function & hence mean & Variance,Assume p+q=1


φ (t) = 1− |t|, |t| <= 1






1 of 2





(b) What are the important requirements of the front-end stage of a communica-tion receiver in the point of view of noise? [8+8]


8. (a) A source emits seven messages with probabilities 1/2, 1/4, 1/8, 1/16, 1/32,1/64 and 1/64 respectively. Find the entropy of the source. Obtain the com-pact binary code and find the average length of the code word. Determine theefficiency and redundancy of the code.

(b) Consider AWGN channel with B = 3 KHz, find the minimum value of S/Nin dB for reliable information transmission at R = 2400, 4800 and 9600 bit /sec. [10+6]

⋆ ⋆ ⋆ ⋆ ⋆

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1. A binary communication channel carries data as one of the two types of signalsdenoted by 0 and 1. Owing to noise a transmitted 0 is sometimes received as 1and a transmitted 1 is sometimes received as a 0. For a given channel, assume aprobability of 0.94 that a transmitted 0 is correctly received as a 0 and a probabilityof 0.91 that a transmitted 1 is received as a 1.Further assume a probability of 0.45of transmitting a 0. If a signal is sent, Determine

(a) probability that a 1 is received.

(b) Probability that a 0 was received

(c) Probability that a 1 was transmitted, given that a 1 was received

(d) Probability that a 0 was transmitted, given that a 0 was received

(e) Probability of as error [3+3+4+4+2]


←

1 1/12 1/6 1/122 1/6 1/4 1/123 1/12 1/12 0


i. X ≤ 11/2

ii. XY is even


3. (a) Prove that Rxy(τ)| <= sqrt[Rxx(0)Ryy(0)]

(b) Find the mean and characteristic function of Binomial distribution [8+8]


(b) Consider the Random process x(t)= A cos(0t + θ) where A and 0 are realconstants and θ is a random variable uniformly distributed on the interval(0, π/2) find the average power Pxx in x(t). [8+8]


1 of 2



(a) Flicker noise

(b) Partition noise




8. (a) The voice frequency-modulating signal of a PCM system is quantized in 16levels with the following probabilities,

P1 = P2 = P 3 = P 4 = 0.1P 5 = P 6 = P 7 = P8 = 0.05P 9 = P10 = P 11 = P 12 = 0.075P 13 = P14 = P15 = P 16 = 0.025Find the information rate taking the band limiting frequency of the modulatingsignal at 3 KHz.

(b) For the communication system where channel characteristic is given belowfigure8b, [6+10]

Figure 8b

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(c) Three boxes of identical appearance contain two coins each. In one box bothare gold; in the second both are silver and in the third box one is silver and theother is the gold coin. Suppose that a box is selected at random and furtherthat a coin in that box is selected at random. If this coin proves to be gold,what is the probability that the other coin is also gold? [4+6+6]


A and B,−



B1 B2 B3A1 3/36 * 5/36A2 5/36 4/36 5/36A3 * 6/36 *

P(Bj) 12/36 14/36 *





= -2-x, 1 ≤ x ≤ 2

= 0, else where



1 of 2


(b) Find the Auto Correlation function for white noise shown in the figure 4bbelow. [8+8]

Figure 4b







6. (a) How bandwidth of operation of an electronic system controls the noise contentin it?

(b) How is Plank?s spectrum theory useful in predicting the noise process in com-munication systems? [8+8]

7. (a) Bring out the difference between narrowband and broadband noises

(b) Describe the quadrature representation of narrowband noise. [8+8]

8. A Discrete Message Source (DMS) has four symbols x1, x2, x3 and x4 with proba-bilities p(x1) = 0.4, p(x2) = 0.3, p(x3) = 0.2, and p(x4) = 0.1,

(a) Calculate H (x).

(b) Find the amount of information contained in the messages x1 x2 x3 x4 and x4 x3 x2 x1,and compare with H (x) obtained in part (a). [8+8]

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1. (a) If A and B are any events, not necessarily mutually exclusive events, derivean expression for probability of A Union B. When A and B are mutuallyexclusive, what happens to the above expression derived?

(b) Define the term Independent events. State the conditions for independence of

i. any two events A and B.

ii. any three events A, B and C.

(c) A coin is tossed. If it turns up heads, two balls will be drawn from box A,otherwise, two balls will be drawn from box B. Box A contains three blackand five white balls. Box B contains seven black and one white balls. In bothcases, selections are to be made with replacement. What is the probabilitythat Box A is used, given that both balls drawn are black? [5+6+5]



X2 → 1 2X1 ↓

0 1/8 1/161 1/16 1/162 3/16 1/83 1/8




3. (a) Prove that mean is ‘m’ and variance is σ2 for Gaussian distribution function.

(b) Find the moment generating and Characteristic function of the random vari-able X which has uniform distribution. [8+8]

1 of 2








6. (a) Explain the external noise sources of random noise.

(b) Calculate the rms noise voltage generated in a bandwidth if 15 kHz by aresistor of 2kΩ operating at 200c. Find the noise power over this bandwidth.Find the noise PSD. [8+8]


8. (a) Discuss the necessity for “Source coding”.

(b) A source has an alphabet a1, a2, a3, a4, a5, and a6 with correspondingprobabilities 0.1, 0.2, 0.3, 0.05, 0.15, and 0.2. Find the entropy of its source.Compare the entropy with the entropy of a uniformly distributed source withsame alphabet. [8+8]

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1. (a) The number of times that an electric switch operate before having to be dis-carded is found to be a random variable with probability mass function (pmf):

p(x) = A(1/3)x for x = 0, 1, 2,......= 0 otherwise.

i. Find the value of A that makes p(.) a p.m.f.

ii. Sketch the p.m.f.

iii. What is the probability that the number of times the switch will operatebefore having to be discarded is greater than 5, an even number (regard0 as even.), and an odd number.

(b) A continuous random variable has the p.d.f. given byfx(x) = 5 − Cx; 0 ≤ x ≤ 5.If Y = ax2+b, find the p.d.f.of Y. [8 + 8]

2. The Rayleigh density function is given by

f(x) = x e−x2

/2 x ≥ 0

= 0 x < 0

(a) Prove that f (x) satisfies the properties of the p.d.f.

i. f(x)≥ 0 for all x and

ii.∫ ∞

∞f(x) dx = 1

(b) Find the distribution function F (x)

(c) Find P(0.5 < x ≤ 2)

(d) Find P(0.5 ≤ x < 2). [2+2+4+4+4]

3. (a) For the random variable X whose density function is

f(x) =1

b − a, a ≤ x ≤ b

0, otherwise

Determine

i. Moment generating function

ii. Mean and Variance

1 of 2


(b) Prove that E(X) = E(X/Y), where X and Y are two random variables [8+8]

4. Let the Random process be given as = Z(t) = x(t) cos [0t + θ] where x(t) instationary Random process with E[x(t)]=0 and E[x2(t)] = σ2

x



the interval (−Π, Π) show that E[Z(t)] = 0 and E[Z2(t)] = σ2x

2[8+8]


6. (a) What are the different man made noises introduced in the communicationsystem?

(b) Write a short note on noises introduced by extra terrestrial objects. [8+8]



8. (a) Messages Q1, Q2, Q3 have probabilities P1, P2, P3 such that P1 + P2 + P3 = 1.Find Hmax by first eliminating P3.

(b) Consider five messages have probabilities 1/2, 1/4, 1/8, 1/16, 1/16. UsingShannon - Fano algorithm, develop efficient code. For that code find theaverage number of bits message and compare with H. [8+8]

⋆ ⋆ ⋆ ⋆ ⋆

2 of 2










X2 → 1 2X1 ↓

0 1/8 1/161 1/16 1/162 3/16 1/83 1/8




3. (a) Show that mean of the binomial distribution is the product of the parameterp & the number of times n.

(b) Sketch the probability density function & Probability distribution function of

i. Exponential distribution

ii. Uniform distribution

iii. Gaussian distribution [6+10]



1 of 2


Figure 4b



jw+2Find mean




7. (a) What are the important parameters that determine the overall noise figure ofa multistage filtering?

(b) Bring out the importance of Frii’s formula. [10+6]





⋆ ⋆ ⋆ ⋆ ⋆

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(c) A coin is tossed. If it turns up heads, two balls will be drawn from box A,otherwise, two balls will be drawn from box B. Box A contains three blackand five white balls. Box B contains seven black and one white balls. In bothcases, selections are to be made with replacement. What is the probabilitythat Box A is used, given that both balls drawn are black? [5+6+5]


←

1 1/12 1/6 1/122 1/6 1/4 1/123 1/12 1/12 0


i. X ≤ 11/2ii. XY is even







1 of 2






jw+2Find mean









⋆ ⋆ ⋆ ⋆ ⋆

2 of 2






1. (a) Define Probability density function and obtain the relationship between prob-ability and probability density.

(b) Consider the probability density f (x) = ae−b|x| where x is a random variableWhose allowable values range from x = −∞to∞. Find

i. the CDF F (x)

ii. the relationship between a and b. and

iii. the probability that the out come x lies between 1 and 2. [7+9]

2. (a) Derive an expression for the average value and variance associated with theGaussian probability density function.

(b) The average life of a certain type of electric club Rs.1200 hours What percent-age of this type of bulbs is expected o fail in the first 800 hours of working?What percentage is expected to fail between 800 is 1000 hours? Assume anormal distribution with σ=200 hours. [8+8]

3. (a) State and Prove any four properties of characteristic function. [10+6]

(b) Find the density function of the distribution for which the characteristic func-tion is

∅(t) =e−−t

2σ2

2

4. X(t) is a zero mean stationary Random process with an Auto correction functionRxx(τ). By integration X(t) form a Random variable Y as [8+8]

Y =1

2T

T∫

−T

x(t)dt show thatE(Y ) = µx = 0 and σ2

y

1

T

2T∫

0

(1 −τ

2τ) Rxx(τ)dτ



jw+2Find mean



1 of 2


(a) Flicker noise

(b) Partition noise




8. (a) Obtain the Shannon - Hartley law giving the, relation amongst channel ca-pacity, bandwidth and signal to noise ratio of a continuous system.

(b) Consider a message sequence having alphabets Q1, Q2, Q3 and Q4 with prob-abilities 1/2, 1/4, 1/8 and 1/8 respectively.

i. Calculate the entropy of the message sequence,

ii. Find the information rate of the message rate is 1 message / second,

iii. What is the rate at which binary signals are transmitted if the signal issent after encoding Q1, Q2, Q3 and Q4 is 00, 01, 10 and 11. [8+2+3+3]

⋆ ⋆ ⋆ ⋆ ⋆

2 of 2













4. (a) If the auto correlation function of a wss process is R(τ) = k .e−k(τ), show thatits spectral density is given by S(ω) = 2

1+(ω

k)2

(b) Find the PSD of a random process x(t) if E[x(t)] = 1 and Rxx(τ) = 1 + e−α|τ |

[8+8]


6. (a) Calculate the equivalent noise bandwidth of an RC Low pass filter. How it isrelated to its 3 db bandwidth.

(b) Find the PSD of the noise voltage across the terminals 1 & 2 for the followingckt figure6b. [8+8]

1 of 2


Figure 6b

7. (a) Evaluate the equivalent noise temperature of a two porr device with a matchedsource and a matched load.

(b) Bring out the significance of noise figure in determining the performance of acommunication system. [8+8]

8. (a) Describe the channel capacity of a discrete channel.

(b) Explain Shannon - Fano algorithm to develop a code to increase average in-formation per bit. [8+8]

⋆ ⋆ ⋆ ⋆ ⋆

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←

1 1/12 1/6 1/122 1/6 1/4 1/123 1/12 1/12 0


i. X ≤ 11/2

ii. XY is even


3. (a) Prove that moment generating function of the sum of two independent vari-ables is the product of their moment generating function.

(b) Let X and Y be independent random Variables, prove thatVar(XY) = Var(X)Var(Y) if E[X] = E[Y] = 0 [8+8]

4. (a) Explain the classification of random processes with neat sketches.

(b) The power spectral density of a stationary random process is given bySxx()= A −k < < k

= 0 otherwise.Find the auto correlation function. [8+8]

1 of 2








6. (a) Explain available power of a noise source.

(b) Explain the components of noise power spectral density. [8+8]

7. Which of the following noise parameters is the true representation of noise in elec-trical circuits?

(a) Noise figure

(b) Noise temperature

(c) Noise bandwidth

Support your answer with the help of suitable examples. [5+5+6]



⋆ ⋆ ⋆ ⋆ ⋆

2 of 2














X2 → 1 2X1 ↓

0 1/8 1/161 1/16 1/162 3/16 1/83 1/8




3. (a) State and Prove any four properties of characteristic function. [10+6]


∅(t) =e−−t

2σ2

2

1 of 2










(b) How the autocorrelation function of the White noise represented? What is itssignificance? [8+8]


(b) List the devices in which narrowband noise can be present. [8+8]


(b) A source generates eight symbols: A, B, C, D, E, F, G and H with probabilities0.5, 0.2, 0.1, 0.05, 0.05, 0.05, 0.03 and 0.02. Find the entropy and informationrate if the symbols are generated at a rate of 1000 / Sec. [6+10]

⋆ ⋆ ⋆ ⋆ ⋆

2 of 2











3. (a) Find the rth moment of the random variable X in terms of its characteristicfunction φx().

(b) Prove that σ2x+y = σ2

x + σ2y + 2[E[XY]]-E[X]E[Y]] where X and Y are two

random variables. [8+8]







1 of 2


6. (a) What is thermal noise? How is it quantified?

(b) Describe the behavior of Zero mean stationary Gaussian bandlimited Whitenoise [8+8]






Figure 8b

⋆ ⋆ ⋆ ⋆ ⋆

2 of 2






1. (a) A jar contains two white and three black balls. A sample of size 4 is made.What is the probability that the sample is in the order (white, black, white,black)?

(b) A box contains 4 bad and 6 good tubes. The tubes are checked by drawinga tube at random, testing and repeating the process until all 4 bad tubes arelocated. What is the probability that the fourth bad tube will be located

i. on the fifth test

ii. on the tenth test?

(c) Consider the experiment of tossing four fair coins. The random variable Xis associated with the number of tails showing. Compute and sketch theCumulative distribution function of X [4+6+6]

2. (a) Derive an expression for, the error function of the standard normal Randomvariable

(b) Lifetime of IC chips manufactured by a semiconductor manufacturer is ap-proximately normally distributed with mean = 5x 106 hours and standarddeviation of 5x 105 hours. A mainframe manufacturer requires that at least95% of a batch should have a lifetime greater than 4x106 hours. Will the dealbe made? [8+8]

3. (a) Prove that Rxy(τ)| <= sqrt[Rxx(0)Ryy(0)]

(b) Find the mean and characteristic function of Binomial distribution [8+8]






1 of 2



6. (a) How bandwidth of operation of an electronic system controls the noise contentin it?

(b) How is Plank?s spectrum theory useful in predicting the noise process in com-munication systems? [8+8]

7. (a) Discuss the significance of noise equivalent temperature of an electronic sys-tem.

(b) Evaluate the equivalent noise temperature of a two port device with a matchedsource and a matched load. [8+8]





⋆ ⋆ ⋆ ⋆ ⋆

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Code No: RR210403 Set No.1

II B.Tech. I Semester Supplementary Examinations, May -2005PROBABILITY THEORY & STOCHASTIC PROCESS






(c) Show that the chances of throwing six with 4,3or 2 dice respectively are as1:6:18.


(b) An analog signal received at the detector (measured in microvolts) may bemodeled as a Gaussian random variable N (200,256) at a fixed point in time.What is the probability that the signal will exceed 240 µvs. what is theprobability that the signal is larger than 240 µV, given that it is larger than210 µVs?

3. (a) If the Random variable X has uniform distribution, find its variance.

(b) Let X is a Gaussian random Variable with zero mean and variance σ2. LetY=X2.Find mean of Random Variable Y.

4. Find the Auto correlation function and power spectral density of the Randomprocess.x(t) = K Cos (ωot + θ) Where θ is a Random variable over the ensemble and isuniformly distributed over the Range ( 0, 2π).

5. The input voltage to an RLC series circuit is a stationary Random process X(t) withE[x(t)]=2 and Rxx(τ) = 4 + exp(−2 |τ | ) . Let Y(t) be the voltage across capacitor.Find E[Y(t)] and Gy(f).


(b) How is the autocorrelation function of the White noise represented? What isits significance?

7. (a) Bring out the difference between narrowband and broadband noises.

(b) Describe the quadrature representation of narrowband noise.

8. (a) Explain what do you understand by “Entropy and Channel capacity”.

(b) With respect to information transmission, explain the trade off between “S/Nratio and transmission Band width”. Justify this from Shannon Hartley law.

1 of 2









(b) Find the mean value, the mean squared value and the cumulative distributionfunction for the Rayleigh distribution with parameter α ¿0, specified by thepdf f(x) = x

α2 exp

−12

xα

.

3. (a) Prove that |Rxy(τ)| <= sqrt[Rxx(0)Ryy(0)] .

(b) Find the mean and characteristic function of Binomial distribution.


(b) Find the Auto Correlation function for white noise shown in the figure below.



(b) Discuss the spectral distribution of thermal noise.




1 of 2



? ? ? ? ?

2 of 2






1. (a) Explain the concept of random variable.

(b) What is the probability of picking an ace and a king from a deck of 52 cards?

(c) A box contains 4-point contact diodes and 6 alloy junction diodes. What is theProbability that 3 diodes picked at random contain at least two point contactDiodes?


(b) Lifetime of IC chips manufactured by a semiconductor manufacturer is ap-proximately normally distributed with mean = 5x 106 hours and standarddeviation of 5x 105 hours. A mainframe manufacturer requires that at least95% of a batch should have a lifetime greater than 4x106 hours. Will the dealbe made?


(b) Let X and Y be independent random Variables, prove thatVar (XY) = Var (X) Var (Y) if E[X] = E[Y] = 0.

4. Consider a Random binary waveform that consists of a sequence of pulses with thefollowing properties.


(b) Pulses are Equally likely to be ± 1


(d) The pulses are not synchronized, that is, the starting time T of the first pulseis Equally likely to be anywhere between 0 and Tb.

Find the Auto correlation and power spectral density function of x(t).

5. A Random process n(t) has a power spectral density g(f) =η/2 for -α ≤ f ≤ αRandom process is passed through a low pass filter which has transfer functionH(f)=2 for−fm ≤ f ≤ fm and H(f)= 0 otherwise. Find the PSD of the waveform atthe o/p of the filter.

6. (a) What is shot noise? How is it qualified?

1 of 2


(b) How the spectral density of White noise is denoted.

7. Which of the following noise parameters is the true representation of noise in elec-trical circuits?

(a) Noise figure

(b) Noise temperature

(c) Noise bandwidth

Support your answer with the help of suitable examples.

8. (a) Describe the channel capacity of a discrete channel.

(b) Explain Shannon Fano algorithm to develop a code to increase average infor-mation per bit.

? ? ? ? ?

2 of 2







(b) Consider the probability density f (x) = a e −b|x| where x is a random variablewhose allowable values range from x = - ∞ to ∞. Find:

i. the CDF F (x)


iii. the probability that the outtcome x lies between 1 and 2.

2. (a) Derive an expression for the average value and variance associated with theGaussian probability density function

(b) The average life of a certain type of electric bulb is 1200 hours. What percent-age of this type of bulbs is expected to fail in the first 800 hours of working?What percentage is expected to fail between 800 and 1000 hours? Assume anormal distribution with σ = 200 hours.

3. (a) Find the density function whose characteristic function is exp (-|t|).

(b) Let X be a continuous random variable with pdf fX (x)=8/ x3, x> 2. FindE[W] where W = X/3.

4. (a) Explain Ergodic random process.

(b) State and prove properties of Auto correlation function.


(b) X(t) is a stationary random process with zero mean and auto correlationRXX (τ) e−2|τ | is applied to a system of function H (w) = 1

jw+2. Find mean

and PSD of its output.


(b) Describe the behavior of Zero mean stationary Gaussian bandlimited Whitenoise.

7. (a) The noise figure of an amplifier at room temperature (T=290o K) is 0.2db.Find the equivalent temperature.

(b) Explain the concept of effective input noise temperature.

1 of 2



(b) A card is drawn at random from ordinary deck of 52 playing cards. Find theinformation in bits that you receive when you are told that the card is a heart,a face card, and a heart face card.

? ? ? ? ?

2 of 2


II B.Tech. I Semester Regular Examinations, November -2005PROBABILITY THEORY & STOCHASTIC PROCESS




1. (a) List and explain the properties of probability density function

(b) Three newspapers A, B and C are published in a city and a survey of readersindicates the following:20% read A, 16% read B, 14% read C, 8% read A and B, 5% read A and C,2% read A, B and C.For one adult chosen at random, compute the probability that:

i. he reads none of the papers.

ii. he reads exactly one of the papers

iii. he reads at least A and B if it is known that he reads at least one of thepapers published.

[7+9]

2. (a) A fair coin is tossed three times and the faces showing up are observed

i. Write the Sample description space

ii. If X is the number of heads in each of the outcomes of this experiment,find the probability function.

iii. Sketch the CDF and pdf.

(b) The continuous random variable X has a p.d.f. f(x) = x/2, 0 ≤ x ≤ 2.Two independent determinations of X are made. What is the probabilitythat both these determinations will be greater than one. If three independentdeterminations are made, what is the probability that exactly two of these arelarger than one?

[10+6]

3. (a) State and prove central limit theorem

(b) If fX,Y(X, Y) = 0.5exp(−|X| − |Y|), where X and y are two random vari-able, if Z=X+Y , find fZ(Z)

[8+8]




1 of 2








[8+8]


(b) Evaluate the equivalent noise temperature of a two port device with a matchedsource and a matched load.

[8+8]

8. (a) A source emits seven messages with probabilities 1/2, 1/4, 1/8, 1/16, 1/32,1/64 and 1/64 respectively. Find the entropy of the source. Obtain the com-pact binary code and find the average length of the code word. Determine theefficiency and redundancy of the code.

(b) Consider AWGN channel with B = 3 KHz, find the minimum value of S/N indB for reliable information transmission at R = 2400, 4800 and 9600 bit / sec.

[10+6]

? ? ? ? ?

2 of 2








[8+8]

2. (a) A fair coin is tossed three times and the faces showing up are observed

i. Write the Sample description space

ii. If X is the number of heads in each of the outcomes of this experiment,find the probability function.

iii. Sketch the CDF and pdf.

(b) The continuous random variable X has a p.d.f. f(x) = x/2, 0 ≤ x ≤ 2.Two independent determinations of X are made. What is the probabilitythat both these determinations will be greater than one. If three independentdeterminations are made, what is the probability that exactly two of these arelarger than one?

[10+6]

3. (a) A fair Coin is tossed three times. Let X be the number of tails appearing.Find the probability distribution of X calculate the expected value of X.

(b) Show that E[X −m]3 = E(X)3 − 3mxσ2x −m3

x where mx and σ2x are the mean

and variance of X respectively.

[8+8]







1 of 2



(b) A wss random process x(t) is applied to the input of an LTI system whoseimpulse response is 5t.e−2t The mean of x(t) is 3. Find the output of thesystem.

[8+8]


(b) Explain the components of noise power spectral density.

[8+8]



when Ti = T0 . Find Te and noise figure.

[10+6]


(b) A card is drawn at random from ordinary deck of 52 playing cards. Find theinformation in bits that you receive when you are told that the card is a heart,a face card, and a heart face card.

[8+8]

? ? ? ? ?

2 of 2









[4+6+6]

2. (a) Define Conditional Probability mass function.

(b) If two random variables have the joint probability density

f(x1, x2) =2

3(x1 + 2x2) for 0 < x1 < 1, 0 < x2 < 1

= 0 else whereFind

i. the marginal density of x2.

ii. Conditional density of the first given that the second takes on the valuex2.

(c) A pair of dice is tossed. Define a random variable X to be the difference ofthe face values turned up. Determine the probability mass function of X

[4+6+6]

3. (a) For a function Y=(X − mx)/σx, prove that mean is zero & variance is 1




[8+8]


1+(ω

k)2

1 of 2



[8+8]

5. White noise n(t) with G(f) =η/2 is passed through a low pass RC network with a3dB frequency fc.

(a) Find the autocorrelation R(τ) of the out put noise of the network.

(b) Sketch P(τ) =R(τ)/R(0)

(c) Find $c(τ)such that P (τ) ≤0.1.

[8+4+4]




[8+8]

7. (a) Define noise temperature and noise figure.

(b) Show that the noise figure of a well designed receiver is usually near about thenoise figure of the low noise RF stage amplifier at the antenna input.

[8+8]


(b) A source generates eight symbols: A, B, C, D, E, F, G and H with probabilities0.5, 0.2, 0.1, 0.05, 0.05, 0.05, 0.03 and 0.02. Find the entropy and informationrate if the symbols are generated at a rate of 1000 / Sec.

[6+10]

? ? ? ? ?

2 of 2






1. (a) State and derive the theorem on Total probability.

(b) A shipment of components consists of three identical boxes. One box contains2000 components of which 25% are defective, the second box has 5000 compo-nents of which 20% are defective and the third box contains 2000 componentsof which 600 are defective. A box is selected at random and a component isremoved at random from the box. What is the probability that this compo-

nent is defective? What is the probability that it came from the second

box?

[8+8]

2. (a) Define joint distribution and Joint probability density function for the tworandom variables X and Y.

(b) Let X and Y be jointly continuous random variables with joint density function

f(x, y) = xy exp

h−

12(x2+y2)

i; x > 0, y > 0

= 0 otherwiseCheck whether X and Y are independent. Find

i. P(X ≤ 1, Y ≤ 1) and

ii. P(X + Y ≤ 1)

[4+12]



= 0, otherwiseShow that the random variables X=Z and Y=Z2 are un-correlated despite ofthe fact that they are statistically dependant.

[6+10]



1 of 2








jw+2Find mean


[8+8]


(b) What are the important requirements of the front-end stage of a communica-tion receiver in the point of view of noise?

[8+8]



[8+8]

8. (a) Estimate the communication rate of a binary symmetric system in which thetwo symbols A and B were transmitted at a rate of 1000 symbols / sec, if thesystem noise cause 1 % of the symbols to be received incorrectly.

(b) Determine the maximum information rate in case of a Teletype writer systemcapable of,

i. A rate of 100 words per minute,

ii. A radio speech communication system with a bandwidth of 5000 Hz anda signal to noise power ratio of 30 dB.

[8+8]

? ? ? ? ?

2 of 2


II B.Tech II Semester Supplimentary Examinations, Aug/Sep 2007PROBABILITY THEORY AND STOCHASTIC PROCESS



⋆ ⋆ ⋆ ⋆ ⋆



[8+8]



X2 → 1 2X1 ↓

0 1/8 1/161 1/16 1/162 3/16 1/83 1/8



(c) Are the variables X1 and X2 independent?

[7+9]

3. (a) For the random variable X whose density function is

f(x) =1

b − a a ≤ x ≤ b

=0, otherwise

Determine

i. Moment generating function

ii. Mean and Variance

(b) Prove that E(X) = E(X/Y), where X and Y are two random variables

[8+8]

1 of 3


4. X(t) is a zero mean stationary Random process with an Auto correction functionRxx(τ). By integration X(t) form a Random variable Y as

Y =1

2T

T∫

−T

x(t)dt show thatE(Y ) = µx = 0 and σ2

y

1

T

2T∫

0

(1 −τ

2τ) Rxx(τ)dτ

[8+8]


6. (a) Calculate the equivalent noise bandwidth of an RC Low pass filter. How it isrelated to its 3 db bandwidth.

(b) Find the PSD of the noise voltage across the terminals 1 & 2 for the followingckt figure6b.

Figure 6b

[8+8]


(b) Explain the concept of effective input noise temperature

[8+8]



2 of 3


(b) For the communication system where channel characteristic is given belowfigure8b,

Figure 8b

[6+10]

⋆ ⋆ ⋆ ⋆ ⋆

3 of 3





⋆ ⋆ ⋆ ⋆ ⋆




[4+6+6]



X2 → 1 2X1 ↓

0 1/8 1/161 1/16 1/162 3/16 1/83 1/8

(i) Find the marginal distribution of X1

(ii) Find the conditional distribution of X1 when X2 is equal to 1

(iii) Are the variables X1 and X2 independent?

[7+9]


= -2-x, 1 ≤ x ≤ 2

= 0, else where

1 of 2


(b) Find the moment generating function of the random variable whose momentsare mr= (r + 1)!2r.

[8+8]

4. (a) Distinguish between stationary and non stationary Random process.

(b) Consider a train of Rectangular pulses having as amplitude of 2 volts andwidths which are either 1 µs or 2µs with equal probability. The meantimebetween pulses is 5 µs. Find the PSD Gn(f) of the pulse train.

[8+8]


6. (a) Write short notes on

i. Noise power spectral density.

ii. Noise suppression in semiconductor devices.

(b) Discuss the role of temperature variation in the operation of electronic devicesin the point of view of noise generation.

[5+5+6]



[8+8]

8. A Discrete Message Source (DMS) has four symbols x1, x2, x3 and x4 with proba-bilities p(x1) = 0.4, p(x2) = 0.3, p(x3) = 0.2, and p(x4) = 0.1,

(a) Calculate H (x).

(b) Find the amount of information contained in the messages x1 x2 x3 x4 and x4 x3 x2 x1,and compare with H (x) obtained in part (a).

[8+8]

⋆ ⋆ ⋆ ⋆ ⋆

2 of 2





⋆ ⋆ ⋆ ⋆ ⋆





(c) Consider the experiment of tossing four fair coins. The random variable Xis associated with the number of tails showing. Compute and sketch theCumulative distribution function of X

[4+6+6]


←

1 1/12 1/6 1/122 1/6 1/4 1/123 1/12 1/12 0


i. X ≤ 11/2

ii. XY is even




φ (t) = 1− |t|, |t| <= 1

= 0,|t|>1Find the density function of random variable X

[9+7]

1 of 2






(b) Find the PSD of Xc(t).

[8+8]



(a) Flicker noise

(b) Partition noise

(c) Johnson’s noise

[5+5+6]


8. Consider a source with M = 3 and symbol probabilities 0.5, 0.4 and 0.1,

(a) Obtain the Shannon - Fano code and calculate its efficiency.

(b) Repeat part (a) for the second extension code, grouping the symbols in blocksof two.

[8+8]

⋆ ⋆ ⋆ ⋆ ⋆

2 of 2





⋆ ⋆ ⋆ ⋆ ⋆



(c) Show that the chances of throwing six with 4,3 or 2 dice respectively are as1:6:18.

[4+6+6]


f(x) = x e−x2

/2 x ≥ 0

= 0 x < 0



ii.∫ ∞

∞f(x) dx = 1


(c) Find P(0.5 < x ≤ 2)

(d) Find P(0.5 ≤ x < 2).

[2+2+4+4+4]



φ (t) = 1 − |t|, |t| <= 1

= 0,|t|>1Find the density function of random variable X

[9+7]



1 of 2




(b) Find the PSD of Xc(t).

[8+8]



[8+8]


(b) How the autocorrelation function of the White noise represented? What is itssignificance?

[8+8]



[8+8]



[8+8]

⋆ ⋆ ⋆ ⋆ ⋆

2 of 2





⋆ ⋆ ⋆ ⋆ ⋆



i. the CDF F (x)


iii. the probability that the out come x lies between 1 and 2. [7+9]


(b) Lifetime of IC chips manufactured by a semiconductor manufacturer is ap-proximately normally distributed with mean = 5x 106 hours and standarddeviation of 5x 105 hours. A mainframe manufacturer requires that at least95% of a batch should have a lifetime greater than 4x106 hours. Will the dealbe made? [8+8]


= -2-x, 1 ≤ x ≤ 2

= 0, else where







1 of 2




jw+2Find mean









⋆ ⋆ ⋆ ⋆ ⋆

2 of 2





⋆ ⋆ ⋆ ⋆ ⋆








A and B,−



B1 B2 B3A1 3/36 * 5/36A2 5/36 4/36 5/36A3 * 6/36 *

P(Bj) 12/36 14/36 *




3. (a) If X and Y are two random variables which are Gaussian. If a random variableZ is defined as Z=X+Y, Find fZ(Z).

(b) Prove that the characteristic function and probability density function form afourier transform pair. [8+8]

1 of 2


4. (a) Prove that PSD and Auto correlation function of Random process form afourier transform pair.

(b) A random process has the power density spectrum Sxx() = 6ω2

1+ω4

Find the average power in the process. [8+8]








H (P1, P2, .........Pn) = λn∑

i=1

pi log2 pi

(b) A man is informed that when a pair of dice was rolled, the result was a seven.How much information is there in this message? [8+8]

⋆ ⋆ ⋆ ⋆ ⋆

2 of 2





⋆ ⋆ ⋆ ⋆ ⋆

1. (a) The number of times that an electric switch operate before having to be dis-carded is found to be a random variable with probability mass function (pmf):

p(x) = A(1/3)x for x = 0, 1, 2,......= 0 otherwise.

i. Find the value of A that makes p(.) a p.m.f.

ii. Sketch the p.m.f.

iii. What is the probability that the number of times the switch will operatebefore having to be discarded is greater than 5, an even number (regard0 as even.), and an odd number.

(b) A continuous random variable has the p.d.f. given byfx(x) = 5 − Cx; 0 ≤ x ≤ 5.If Y = ax2+b, find the p.d.f.of Y. [8 + 8]


p(x) = Ax x = 1, 2, 3, ..................., 50

= A(100 − x) x = 51, 52, ..............100

= 0 otherwise



A. more than 50

B. less than 50

C. equal to 50


E. an odd number.


3. (a) The joint probability density function of random variables X and Y is f (x, y) =14exp (|x − y|)−

′

α < x <′

α , and −′

α < y <′

α If another random variable ‘Z’ isdefined such that, Z = X + Y Find fz(Z).

1 of 2


(b) Two random variables x and y have the following joint probability densityfunction f(x,y) = 2-x-y, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1

= 0, other wise Find

i. Marginal probability density functions of x and y

ii. Var (x) and Var (y) [8+8]


i. Acos0τ

ii. AΠ(τ/τ0) where Π(x) is a unit area rectangular function


Sxx() = a + ((jb/w));−W < < W

= 0 : ElsewhereWhere W > 0, a and b are real constants. Find the cross correlation function.

[8+8]






(b) Discuss the role of temperature variation in the operation of electronic devicesin the point of view of noise generation. [5+5+6]


(b) Evaluate the equivalent noise temperature of a two port device with a matchedsource and a matched load. [8+8]



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⋆ ⋆ ⋆ ⋆ ⋆





f(x) =x

α2exp

−1

2

x

α

3. (a) Find the rth moment of the random variable X in terms of its characteristicfunction φx().

(b) Prove that σ2x+y = σ2

x + σ2y + 2[E[XY]]-E[X]E[Y]] where X and Y are two

random variables. [8+8]








jw+2Find mean



1 of 2


(b) Explain the components of noise power spectral density. [8+8]


(b) Describe the quadrature representation of narrowband noise. [8+8]





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II B.Tech II Semester Supplimentary Examinations, Apr/May 2009PROBABILITY THEORY AND STOCHASTIC PROCESS



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f(x) =x

α2exp

−1

2

x

α



φ (t) = 1 − |t|, |t| <= 1




1 of 2


Figure 4b



jw+2Find mean



(b) Describe the behavior of Zero mean stationary Gaussian bandlimited Whitenoise [8+8]



(b) A card is drawn at random from ordinary deck of 52 playing cards. Find theinformation in bits that you receive when you are told that the card is a heart,a face card, and a heart face card. [8+8]

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(c) Consider the experiment of tossing four fair coins. The random variable Xis associated with the number of tails showing. Compute and sketch theCumulative distribution function of X [4+6+6]



f(x) =x

α2exp

−1

2

x

α

3. (a) A Gaussian distribution random variable X of zero mean & variance is σ2

transformed by rectifiers Characterized by input-output relationY = ax2, X.+0

= 0, X<0Determine the Probability of Y.

(b) Find the nth moment of uniform random variable & hence its mean [8+8]



Rxx(τ) = 1 − |τ |, |τ | ≤ 1


Sxx (ω) =

[

Sinω/2

ω/2

]2

1 of 2




jw+2Find mean





(b) Discuss the role of temperature variation in the operation of electronic devicesin the point of view of noise generation. [5+5+6]



when Ti = T0 . Find Te and noise figure. [10+6]




Figure 8b

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(b) Let X and Y be jointly continuous random variables with joint density function

f(x, y) = xy exp

h−

12(x2+y2)

i; x > 0, y > 0

= 0 otherwiseCheck whether X and Y are independent. Find

i. P(X ≤ 1, Y ≤ 1) and

ii. P(X + Y ≤ 1) [4+12]



∅(t) =e−−t

2σ2

2

[10+6]



Rxx(τ) = 1 − |τ |, |τ | ≤ 1


Sxx (ω) =

[

Sinω/2

ω/2

]2

1 of 2




(a) Flicker noise

(b) Partition noise



8. (a) Find the channel capacity of BSC as shown in figure8a.

Figure 8a

(b) Show that in general H (x 1, x 2, ..............., x n) ≤n∑

i=1

H(xi)

When does the equality hold? [8+8]

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3. (a) Given the following table

X 1 2 3 4 5 6 7P(x) 0.05 0.l 0.3 0 0.3 0.15 0.1

Find

i. E[X]

ii. E[X2]

iii. V[X]

iv. V[2x ± 3]

(b) Prove that cov(ax,by) = ab cov(x,y) [8+8]


(b) Consider a train of Rectangular pulses having as amplitude of 2 volts andwidths which are either 1 µs or 2µs with equal probability. The meantimebetween pulses is 5 µs. Find the PSD Gn(f) of the pulse train. [8+8]

1 of 2




jw+2Find mean






8. (a) Distinguish between discrete memory less source and discrete memory source.

(b) Explain the importance of coding for efficient transmission.

(c) A certain data source has 16 equiprobable symbols, each 1ms long. The sym-bols are produced in blocks of 15, separated by 5 ms space. Find the sourceinformation rate. [6+4+6]

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II B.Tech. II Semester Regular Examinations, April/May -2005PROBABILITY THEORY & STOCHASTIC PROCESS



? ? ? ? ?

1. (a) Define the following and give one example for each.

i. Sample space.

ii. Event.

iii. Mutually exclusive events

iv. Collectively exhaustive events.

(b) In three boxes, there are Capacitors as shown in the following table:

Value in mf Number in box↓ →

1 2 31-0 70 80 1450.1 55 35 750.01 20 95 25

An experiment consists of first randomly selecting a box (assume that each boxhas the same probability of selection) and then randomly selecting a capacitorfrom the chosen box.

i. What is the probability of selecting 0.01uf capacitor, given that the box2is chosen?

ii. If a 0.01 mf capacitor is chosen, what is the probability that it came fromthe second box?

2. (a) Define Conditional Probability mass function.

(b) If two random variables have the joint probability density

f (x1, x2) = 23(x1 + 2x2) for 0 < x1 < 1, 0 < x2 < 1

= 0 elsewhere. Find

i. the marginal density of x2

ii. Conditional density of the first given that the second takes on the valuex2.

(c) A pair of dice is tossed. Define a random variable X to be the difference ofthe face values turned up. Determine the probability mass function of X.

3. (a) State and prove any four properties of mean of a random variable X.

1 of 2


(b) Prove that the density function of sum of two statistically independent randomvariables is the convolution of their individual density functions.

4. Let the Random process be given as = Z(t) = x(t) cos [ω0t + θ ] where x(t) instationary Random process with E[x(t)]=0 and E[x2(t)] = σ2

x



the interval (−Π, Π) ) show that E[Z(t)] = 0 and E[Z2(t)] = σ2x

2.

5. White noise n(t) with G(f) = η/2 is passed through a low pass RC network with a3dB frequency fc.


(b) Sketch PR(τ) = R(τ)/R(0)

(c) Find $c(t) such that P(τ)£ ≤ 0.1.


(b) How the autocorrelation function of the White noise represented? What is itssignificance?

7. (a) Show that the effective noise temperature of ‘n’ networks in cascade is givenby,Te= Te1+Te2/g1+Te3/g1g2+............+Ten/g1g2.....gn−1)

(b) A low noise receiver for satellite ground station consists of the following stages

Antenna with Ti = 125K

Waveguide with a loss of 0.5dB

Power amplifier with ga = 30dB, Te = 60K, BN = 20MHz

TWT amplifier with ga = 16dB, F= 6dB, BN = 20 MHz

Calculate the effective noise temperature of the system.

8. (a) Consider an AWGN channel with SN= 104 . Find the maximum rate for reliable

information transmission when, B = 1 KHz, 10 KHz and 100 KHz.

(b) The Binary Erasure Channel (BEC) has two source symbols 0 and 1, and threedestination symbols 0, 1 and E, where E denotes a detected but uncorrectableerror. The forward transition probabilities are,

P( 00) = 1−α P(E

0) =α P(1

0) = 0

P( 01) = 0 P(E

1) =α P(1

1) = 1−α

I (x, y) is maximum when source symbols are equiprobable. Find Cs (channelcapacity) in terms of α.

? ? ? ? ?

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? ? ? ? ?

1. (a) Explain the concept of random variable.

(b) What is the probability of picking an ace and a king from a deck of 52 cards?

(c) A box contains 4-point contact diodes and 6 alloy junction diodes. What is theProbability that 3 diodes picked at random contain at least two point contactDiodes?


(b) Let X and Y be jointly continuous random variables with joint density functionf(x, y) = xy exp

[

−12(x2 + y2)

]

; x > 0, y > 0

=0 otherwise

Check whether X and Y are independent. Find

i. P(X ≤ 1, Y ≤ 1)and

ii. P(X + Y ≤ 1)

3. (a) Find the moment generating function of the random variable having proba-bility density function fx(x) = x, 0 ≤ x ≤ 1

= 2 − x, 1 ≤ x ≤ 2

= 0, else where

(b) Find the moment generating function of the random variable whose momentsare mr = (r + 1)!2r


(b) Consider the Random process x(t)= A cos(ω0t + θ) where A and ω0 are realconstants and θ is a random variable uniformly distributed on the interval(

0, Π2

)

find the average power Pxx in x(t).


(b) X(t) is a stationary random process with zero mean and auto correlationRXX (τ) e−2|τ | is applied to a system of function H (w) = 1

jw+2. Find mean


6. (a) What are the sources of flicker noise and how can it be reduced?

(b) How noise equivalent bandwidth of a electronic circuit can be estimated?

1 of 2


7. (a) An amplifier has input and output impedances of 75 ohm, 60dB power gain,and a noise equivalent bandwidth of 15KHz. When a 75 Ω resistor at 2900K isconnected to the input, the output rms noise voltage is 75microvolt. Determinethe effective noise temperature of the amplifier assuming that the meter isimpedance matched to the amplifier.


8. (a) A source is transmitting six messages with probabilities 0.3, 0.25, 0.15, 0.12,0.10 and 0.08 respectively. Find the binary ”HUFFMAN” Source code for theabove messages and code efficiency.

(b) Consider an AWGN Channel with 4 MHz bandwidth and noise PSD is η2= 10−12 w

Hz..

The signal power is required at the receiver is 0.1 mw. Find the capacity ofthe channel.

? ? ? ? ?

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? ? ? ? ?

1. (a) Explain why there must be a mistake in each of the following statements:

i. If the probability that an ore contains Uranium is 0.28, the probabilitythat it does not contain Uranium is 0.62.

ii. A Company is working on the construction of two shopping centers. Theprobability that the larger of the two shopping centers will be completedis 0.35 and the probability that both shopping centers will be completedon time is 0.42.

iii. The probability that a student gets A in a particular course is 0.32 andthe probability that he will get either an A or a B is 0.27.

(b) A jar contains 52 badges numbered 1 to 52.Suppose that the numbers 1 thro 13are considered lucky. A sample of size 2 is drawn from the jar with replacement.What is the probability that

i. both badges drawn will be lucky?

ii. Neither badge will be lucky?

iii. Exactly one of the badges drawn will be lucky.

iv. at least one of the badges will be lucky.


(b) Find the mean value, the mean squared value and the cumulative distributionfunctio for the Rayleigh distribution with parameter α> 0, , specified by thepdf f(x) = x

α2 exp

−12

xα

3. (a) Find the density function whose characteristic function is exp (− |t|).

(b) Let X be a continuous random variable with pdf fx (x)=8/ x3, x > 2. FindE[W] where W = X/3


i. Acosω0τ

ii. AΠ( ιτ0

)whereΠ(x) is a unit area rectangular function


Sxx ( ω ) = a+(jb ω /w ) ; -W < ω < W

= 0 : Elsewhere

Where W > 0, a and b are real constants. Find the cross correlation function.

5. (a) Find the PSD of a random process where z(t)= X(t) + y(t) where x(t) andy(t) are zero mean, individual random process.

1 of 2


(b) A wss random process x(t) is applied to the input of an LTI system whoseimpulse response is 5t.e−2t. The mean of x(t) is 3. Find the output of thesystem.


(b) What are the different noise sources that may be present in an electron devices?



8. (a) For a system the bandwidth is 4 KHz and SN

ratio is 14. If the bandwidthis increased to 5 KHz, find the required S

Nratio to have the same channel

capacity and find the percentage change in signal power.

(b) Using the definition of H (x) and H (X/y), Show that, I(x, y) =∑

x,y

P (x, y) log P (x,y)P (x),P (y)

? ? ? ? ?

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? ? ? ? ?


(b) Consider the probability density f(x) = a e −b|x| where x is a random variableWhose allowable values range from x = −∞to∞. Find

i. the CDF F (x)


iii. the probability that the outcome x lies between 1 and 2.

2. The Rayleigh density function is given by f(x) = xe−x2/2 x ≥ 0

= 0 x < 0


i. f(x)≥ for all x and

ii.∫ ∞

∞f(x) dx = 1


(c) Find P (0.5 < x ≤ 2)

(d) Find P(0.5 ≤ x < 2).


(b) Let X and Y be independent random Variables, prove that Var (XY) = Var(X) Var (Y) if E[X] = E[Y] = 0


(b) Consider a train of Rectangular pulses having as amplitude of 2 volts andwidths which are either 1µs or 2µs with equal probability. The meantimebetween pulses is 5 µs . Find the PSD Gn(f) of the pulse train.

5. White noise n(t) with PSD= n2

is passed through a low pass RC network with a 3db frequency fc.

(a) Find the auto correlation R(τ)) of the o/p noise of the network.

(b) Sketch ρ(t) = R(τ)R(o)


1 of 2






whereGo(f), Gin(f),and K are respectively open circuited voltage, spectral density and the voltagegain of n/w.

8. (a) A source is transmitting two symbols A and B with P (A) = 116

and P (B) =1516

Construct a source code to provide a code efficiency of around 50%.

(b) A binary data source has P (0) = 38, P (1) = 5

8. Due to noise in the channel,

P(

10

)

=34, P

(

01

)

= 116

. Find the conditional entropy H yx

and maximum entropyof ‘x’ where ‘x’ is source and ‘y’ is receiver.

? ? ? ? ?

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PROBABILITY THEORY & STOCHASTIC PROCESS(Bio-Medical Engineering)





i. the CDF F (x)


iii. the probability that the out come x lies between 1 and 2.

[7+9]

2. Two discrete random variables X and Y have joint p.m.f. given by the followingtableX ↓ 1 2 3 Y

←

1 1/12 1/6 1/122 1/6 1/4 1/123 1/12 1/12 0

Compute the probability of each of the following events

(a) X ≤ 11/2

(b) XY is even

(c) Y is even given that X is even.

[5+5+6]

3. (a) For a function Y=(X −mx)/σx, prove that mean is zero & variance is 1




[8+8]


1 of 2


(b) Consider the Random process x(t)= A cos(0t + θ) where A and 0 are realconstants and θ is a random variable uniformly distributed on the interval(0, π/2) find the average power Pxx in x(t).

[8+8]



(a) Flicker noise

(b) Partition noise

(c) Johnson’s noise

[5+5+6]



where Go(f),

Gin(f), and K are respectively open circuited voltage, spectral density andthe voltage gain of n/w.

[10+6]

8. (a) Consider an AWGN channel with S/N = 104. Find the maximum rate forreliable information transmission when, B = 1 KHz, 10 KHz and 100 KHz.


P (0/0) = 1− α P (E/0) = α P (1/0) = 0

P (0/1) = 0 P (E/1) = α P (1/1) = 1− α


[6+10]

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[8+8]


←

1 1/12 1/6 1/122 1/6 1/4 1/123 1/12 1/12 0


(a) X ≤ 11/2

(b) XY is even


[5+5+6]

3. (a) For a function Y=(X −mx)/σx, prove that mean is zero & variance is 1




[8+8]






1 of 2





[8+8]


(b) What are the causes of shot noise?

[8+8]


8. (a) Discuss the necessity for “Source coding”.

(b) A source has an alphabet a1, a2, a3, a4, a5, and a6 with correspondingprobabilities 0.1, 0.2, 0.3, 0.05, 0.15, and 0.2. Find the entropy of its source.Compare the entropy with the entropy of a uniformly distributed source withsame alphabet.

[8+8]

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(c) A coin is tossed. If it turns up heads, two balls will be drawn from box A,otherwise, two balls will be drawn from box B. Box A contains three blackand five white balls. Box B contains seven black and one white balls. In bothcases, selections are to be made with replacement. What is the probabilitythat Box A is used, given that both balls drawn are black?

[5+6+5]


f(x) = x e−x2

/2 x ≥ 0

= 0 x < 0



ii.∫ ∞

∞f(x) dx = 1


(c) Find P(0.5 < x ≤ 2)

(d) Find P(0.5 ≤ x < 2).

[2+2+4+4+4]

3. (a) Given the following table

X 1 2 3 4 5 6 7P(x) 0.05 0.l 0.3 0 0.3 0.15 0.1

Find

i. E[X]

1 of 2


ii. E[X2]

iii. V[X]

iv. V[2x ± 3]

(b) Prove that cov(ax,by) = ab cov(x,y)

[8+8]

4. (a) Explain Ergodic random process

(b) State and prove properties of Auto correlation function

[8+8]

5. White noise n(t) with G(f) =η/2 is passed through a low pass RC network with a3dB frequency fc.


(b) Sketch P(τ) =R(τ)/R(0)

(c) Find c(τ)such that P (τ) ≤0.1.

[8+4+4]


(b) What are the causes of shot noise?

[8+8]

7. (a) Show that the effective noise temperature of n networks in cascade is givenby, Te = Te1 + Te2/g1 + Te3/g1g2 + ................... + Ten/g1g2gn−1

(b) A low noise receiver for satellite ground station consists of the following stagesAntenna with Ti = 125KWaveguide with a loss of 0.5dBPower amplifier with ga = 30dB, Te = 6K, BN = 20 MHzTWT amplifier with ga = 16dB, F = 6dB, BN = 20 MHzCalculate the effective noise temperature of the system.

[8+8]




(b) Suppose 100 voltage levels are employed to transmit 100 equally likely mes-sages. Assume the system to be a Gaussian channel with λ = 3.5 and band-width B = 104 Hz. Find S/N.

[8+8]

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[8+8]


←

1 1/12 1/6 1/122 1/6 1/4 1/123 1/12 1/12 0


(a) X ≤ 11/2

(b) XY is even


[5+5+6]

3. (a) Prove that the second moment of binomial distribution is given by E(X2) = (np)2+npq.

(b) From the nth moment of exponential distribution, determine its variance tobe 1/α2, where α is a constant.

[8+8]


1+(ω

k)2


[8+8]



[8+8]

1 of 2




[8+8]



[8+8]

8. (a) Consider an AWGN channel with S/N = 104. Find the maximum rate forreliable information transmission when, B = 1 KHz, 10 KHz and 100 KHz.


P (0/0) = 1− α P (E/0) = α P (1/0) = 0

P (0/1) = 0 P (E/1) = α P (1/1) = 1− α


[6+10]

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