chapter3and4extra yousef

8/18/2019 Chapter3and4extra Yousef

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The Islamic University of Gaza

Faculty of Engineering

Electrical Engineering Department

Probability and Stochastic Processes

(EELE 3340)

Fall semester 2014

Review Problems (ch3+ch4)

Engineer: Yousef Awad Shaban

2014


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Problem (1)

The discrete random variables X and Y have joint PMF PX,Y(x,y) given by

X,Y (x,y) x=0 x=1 x=3 x=5

y=-1 1/12 1/6 1/12 0 y=3 1/6 1/12 0 1/12

y=4 0 1/12 1/6 1/12

(a) Find the marginal PMF P X (x) and P y(y).

(b) Find the conditional PMF PX|Y(x|3) and PY|X(Y|3).

(c) Find P{X ≤ Y} and P{ X + Y ≤ 8 }

Problem (2)

Is the following function

0 F X ,Y ( x, y)

1

x y 1

x y 1

a valid joint CDF. Why or why not? Prove your answer and show your work.

Problem (3)

The random variables X and Y have joint probability density function

(a) Sketch the xy plane with an indication of the region where f X,Y (x,y) is nonzero.(b) Find the joint CDF of X and Y. Specify the value of F X,Y (x,y) for all x and y.(c) Find the marginal CDFs of X and Y .

(d) Find the marginal pdfs of X and Y .

i. integrating the joint pdfii. differentiating the marginal CDFs found in part (c)

(e) Find the conditional pdf of X given Y=y , where y > 0.

(f) Find P{Y > 3X}.

(g) For β > 0, find P{ X + Y ≤ β }.


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c

Problem (4)

Suppose that the distribution of X is uniform on [−1, 1]. Compute E[Xn ] for each n≥1.

Problem (5)

Which of the following are valid cumulative probability distribution functions? For

those that are not valid cdfs, state at least one property of the cdf which is not

satisfied.

0 , x 1

(a) F ( x ) 2 x x 2 , 1 x 2

1 x 2

(b ) F ( x ) 0.5e2 x

, x

x 0 1 0.25e 3 , x 0

(c ) F ( x )

0.5e2 x

,

x

x 0

1 0.25e 3 , x 0

Problem (6)

X is a random variable with probability density function

c x ,

f ( x)

(6 x) ,

0 ,

0 x 3

3 x 6

otherwise

(a) Find the value of the constant c.

(b) Compute the cdf F X (x) of X.

(c) Plot F X (x) .

(d) Show that the function F X (x) computed in part (b) satisfies all properties of cdfs.

(e) What is P[A]=P[ X > 3] , P[B]=P[1.5≤ X ≤

9]?(f) Are events A and B independent?

Problem (7)

X is a continuous random variable with probability density function

If E[x]=2/3 find P[X


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Problem (8)

Two fair dice are thrown .Let X 1 be the number shown in the first die and X 2 be the number

shown on the second die. Define the random variable Z as follows:

0 : Z

X 2 is not evenly divisible by X 1

1 : X 2 is evenly divisible by X 1

Note: X 2 is evenly divisibly by X 1 if and only if X 2/ X 1 is an integer.

(a) Compute the PMF for Z.

(b) Compute the conditional probability P({Z=1} | { X 2 = 4}).

(c) Compute the conditional probability P({ X 1 = 3} | { Z = 1}).

Problem (9)

Let X denote a random variable uniformly distributed on the interval [ – 1, 4] , Find F X (x).

Problem (10)

f X (x) denotes the probability density function (pdf) of a continuous random variable X.

Which of the following properties are satisfied by all pdfs?

1 f X (x) 1 for all x, – < x < . 2 lim x – f X (x) = 0.

3 lim x f X (x) = 1. 4 P{a < X < b} = P{a X b}

Problem (11)

Which of the following four statements are NOT properties of all CDFs?

1 P{X > b} = 1 – FX(b). 2 If FX(a) < FX(b), then a 0 X < 1/2}(b) Find the expected value of X.

(c) Find the expected value of X

otherwise


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Problem (13)

C, D, and E are events of nonzero probability .Which of the following statements are true?

(a) P(EC D) = P(EC) + P(ED) – P(EC D)

(b) If P(C) = P(D), then P(CD) = P(DC) (c) If P(ED) = P(DE), then P(D) = P(E)

Problem (14)

Let X denote a geometric random variable with parameter 1/3 .

(a) What is the average value of (X – 2)2?

(b) What is the probability that X = 2 given that X < 4 ?

Problem (15)

Which of the following four statements are true for all random variables X and Y with

identical finite variance 2.

(a) E[X2] = E[Y

2]

(b) var(4X – 5Y) = var(4Y – 5X)

(c) |cov(X, Y)| 2

(d) X + Y and X – Y are uncorrelated R.Vs

Problem (16)

Problem (17)

X is exponential random variable with expected value of 1/ for x>0 and Y = exp(X).Find f Y (y), the PDF of Y.


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Problem (18)

The joint probability density function f X,Y (x,y) for the continuous random variables X and Y

has constant value on the region {(x, y) : 0 < x < 2, 0 < y < 2, 1 < x + y < 2}.

(a) Find the joint PDF f X,Y (x,y).

(b) Find f X (x), the marginal PDF for X.

Problem (19)

Continuous random variables X and Y each take on experimental values between zero and

one, with the joint PDF indicated below (the cutoff between probability density 0.8and 1.6

occurs at x=0.5and y=0.5):

(a) Find f X (x) and f Y (y) and plot them.

(b) Are X and Y independent? Present a convincing argument for your answer.

Problem (20)

Random variables X and Y have the joint PDF of

6 x , x 0 , y 0 , f

X ,Y( x , y )

0, otherwise

x y 1

(a) Obtain an expression for and sketch the PDF of f Y (y)

(b) Determine the conditional expectation and variance for X given Y =0.5.

Problem (21)

Random variables X and Y have the joint PDF of

1 / 10, 0 x 4 , 0 y 4 f X ,Y ( x , y )

0, otherwise

Find the conditional PDF of X and Y given the event B=[ {X+Y} ≥ 4 ]


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Problem (22)

X is a R.V has a density function f X (x)

x / 2 f X ( x)

0

0 x 2

otherwise

Let Y=4x-2

(a) Compute CDF and PDF of Y ?

(b) Compute var[y]?

(c) Compute var[y2] ?

Problem (23)

Let the random variable X denote the time until a computer server connects to your machine

(in milliseconds), and let Y denote the time until the server authorizes you as a valid user (in

milliseconds). Each of these random variables measures the wait from a common starting

time and X < Y. Assume that the joint probability density function for X and Y is

f X ,Y ( x , y ) 6 10 -6

e (0.001 x 0.002 y )

for x


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( x , y ) 3

Problem (25)

Let X and Y be continuous random variables with joint density function

8 xy

X ,Y for 0 x 1, x y 2 x

0 otherwise.

Calculate the covariance of X and Y.

Problem (26)

Let X and Y be continuous random variables with joint density function

What is the conditional variance of Y given that X = x ?

Problem (27)

The joint PDF of random variables Y1,Y2 ,Y3 is:

(a) Find the marginal PDFs f Y1(y1 ), f Y2(y2 ), f Y3(y3 ) .

(b) Determine whether the random variables are Y1, Y2, Y3 are independents. (c) Event A has { Y1≤ 1/2 , Y2≤ 1/2 , Y3≤ 1/2 } find P[A].

chapter3and4extra yousef

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