Indian Institute of Technology, Kharagpur

Date of Exam.: .04.18 (FN/AN) Time: 3 Hrs. FulllVlarks: 50 No. of Students: 340End (Spring) Semester Examination (2017-18) Department: MathematicsSubject No. MA20106 Subject Name: Probability & Stochastic Processes

Instructions:(i) Use of calculator and Statistical tables is allowed.(ii) Answers of numerical questions must be in at least two decimal places/ fraction form.(iii) All the notations are standard and no query or doubts will be entertained. If anydata/statement is missing, identify it on your answer script.(iv) Answer All questions. (v) All parts of a question Must Be answered at One Place.

1. For this question write only answers on the first page of your answer script in the giventabular form. Detail working may be carried out on other pages. [1x 10]

a b c d e f h

(a) A town has two doctors X and Y operating independently. If the probability thatdoctor X is available is 0.9 and that for Y is 0.8, what is the probability that atleastone doctor is available when needed?

(b) Two defective tubes get mixed up with two good ones. The tubes are tested one byone without replacement until both defectives are found. What is the probabilitythat the last defective tube is obtained on the third test?

(c) Suppose the rv X takes values in {I, 2,3, ... } with pmf px(n) = ~.~2. Find E(X).

(d) Let the random variable X has pdf f(x) = 3e-3x, x 2: O. Find the median of rv X.(e) Let the random variable X has MGF Mx(t) = e2t+2t2. Find P(X2 - X < 6).

(f) Let X and Y independent random variables and X rv N(l, 2), Y rv N(3,4). FindP(2X + 3Y > 9).

(g) One percent of jobs arriving at a computer system need to wait until weekends forscheduling owing to core size limitations. Find the probability that among a sampleof 200 jobs, there are no jobs that have to wait until weekends.

(h) Compute the probability that the 5th child of a family is the 3rd son.

(i) Let Z is normal variate with mean 0 and variance l. Find the pdf of IZI.(j) What is the expected number of trials in rolling a fair die until all 6 faces occur?

2. For this question write only answers on the first page of your answer script in the giventabular form. Detail working may be carried out on other pages. [1x 10]

a b c d e f h

(a) Consider a Markov chain {Xn} with state space S = {O, I, 2} and transition proba-bility matrix (tpm)

P = (~/O/~~j~ 1~4)1/3 2/3

1

Whether the Markov chain is reducible or irreducible.

(b) With reference to the problem in (a), determine P(X2 = 2, XI = 11Xo = 0).

(c) An urn initially contains a single red ball and a single green ball. A ball is drawn atrandom, removed, and replaced by a ball of opposite color, and this process repeatsso that there are always exactly two ball in the urn. Let Xn be the number of redballs in the urn after n draws, with Xo = l. Specify the transition probability matrixof Markov chain {Xn}.

(d) Consider a Markov chain with state space S = {O, I, 2} and tpm

(

0 0.5 0.5)P = 0.5 0 0.5

0.5 0.5 0

Determine whether the states are recurrent or transient.

(e) Consider a Markov chain with state space S = {O, I, 2} and tpm

(

0.1 0.6 0.3)P = 0.5 0.1 0.4

0.1 0.2 0.7

Determine the stationary distribution (7ro, 7rI, 7r2).

(f) Events occur according to a Poisson process with rate A = 3 per hour. Starting atnoon, what is the expected time at which the third event occurs?

(g) A small full service station is opertated by the owner, Amit, by himself. On Mondaymornings customers (car) arrive randomly at the average rate of 15 per hour. Mr.Amit provides exponential service with mean service time of 2.5 minutes. What isthe mean number of customers waiting (queueing) for service.

(h) A random vector (X, Y) has the joint density function

f(x, y) = 4xy 0::; x ::; 1,0 ::; y ::; 1; 0 otherwise.

Are X and Y independent? Find E(XY).

(i) Let the random variable X has pdf f(x) = 2x, 0 ::; x ::; 1 and let the rv Y = 3 - 2X.Find correlation p between X and Y.

(j) Suppose the r.vs. X and Y have a bivariate normal distribution with parametersfLx = 2, fLy = 5, (Jx = 3, (Jy = 6 and fJ = 2/3. Find j.LYII = E[YIX = 1] and(J~II = V[YIX = 1].

3. (a) A tetrahedron shaped die gives one of the numbers I, 2, 3, 4 with equal probabil-ities. We roll two of these dice and denote the two outcomes by Xl and X2. LetYI=Min(XI, X2) and Y2 = IXI - X21. Find (i) the joint pmf of Xl and X2. (ii) thejoint pmf of Yj and Y2. (iii) The marginal pmf of Yj and Y2. (iv) P(Y1Y2 ::; 2). (v)E(YjY2).

(b) The joint pdf of (X, Y) is given by

{c «=:» 0::; y ::; x < 00,

f(x, y) = 0,' otherwise

Find the constant (i) c and (ii) P (X + Y < 1). [4+3]

2

.1

4. (a) Let X and y" denote the heart rate (in beats per minute) and average power output(in watts) for a 10 min. cycling time trial performed by a professional cyclist. Assumethat X and Y have a bivariate normal distribution with parameters I'-e = 180,IJ,y =400, ax = 10, ay = 50, p = 0.9. Find (i) E(YIX = 200) (ii) V(YIX = 200)(iii) P(Y ~ 450lX = 200).

(b) Let X and Y have joint probability density function

j'(x 7) = {2, x > 0, y > 0, x + Y < 1,, ,I) 0 otherunse,

Let Zl = X + Y and Z2 = Y. Find (i) Joint distribution of Zl and Z2 (ii) marginaldistribution of ZI and Z2 (iii) E(Zd (iv) E(Z2) (v) E(ZIZ2) (vi) CO\'(Zj, Z2)' [4+4]

5. (a) You start with five dice. Roll all the dice and put aside those dice that come up6. Then, roll the remaining dice, putting aside those dice that come up 6. And soon. Let Xn be the number of dice that are sixes after n rolls. For the Markov chain{Xn, n ~ I}, find the following elements of tpm(i) Diagonal elements i.e Pii i = 0,1, ... 5(ii) P02, P12, P23, P35, P55

(iii) PlO, P21, P32

(b) Consider a Markov chain {Xn' n = 0,1,2, ... }, having state space E = {0,1,2},and TPM

[

1/4 3/4 0 j1/4 1/4 1/2 .1/2 1/2 0

Let the initial state probability distribution is P(Xo = i) = 1/3, i = 0, 1,2.

1. Find P(X3 = 2, Xl = 11Xo = 1);II. P(X3 = 1, Xl = 2, Xo = 0);Ill. Find the p.m.f. of X2. [4+3]

6, (a) Bella is fishing and the time it takes to catch a fish is exponentially distributed withmean 20 minutes. Every time she catches a fish, she throws it back in the water, andcontinue fishing. Find the probability that she will catch five fish in the first hour.

(b) Cars arrive at a toll booth, according to Poisson process at a rate of 2 cars perminute. On an average it takes 20 second to collect the toll. If the time taken tocollect the toll is exponentially distributed, then obtain (i) the average number ofcars waiting in the queue (ii) average waiting time of a car in the queue at the toll(iii) find the probability that there are more than 3 cars at the toll booth.

(c) Explain M/ /\11/1 queuing system. Obtain the expression for the distribution of num-ber of customers in the system in steady-state Pn. Also obtain Land Lq. [2+3+3]

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