LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 M.Sc. DEGREE EXAMINATION – MATHEMATICS SECOND SEMESTER – April 2009 ST 2902 - PROBABILITY THEORY AND STOCHASTIC PROCESSES Date & Time: 02/05/2009 / 1:00 - 4:00 Dept. No. YB 39 Max. : 100 Marks PART-A Answer all questions (10 x 2 = 20) 1) Determine C such that f x C 1 x2 , ∞ < x < ∞ is a pdf of a random variable X. - 2) If the events A and B are independent show that AC and B C are independent. 3) Write the MGF of a Binomial distribution with parameters n and p. Hence or otherwise find E X . 4) If two events A and B are such that A B , show that P(A) ≤ P(B) 5) Given the joint pdf of X 1 and X 2 , f( x1 , x 2 ) = 2, 0< x1 < x 2 <1.obtain the conditional pdf of X 1 given X 2 x2 . 6) Define a Markov process. 7) Define transcient state and recurrent state. 8) Suppose the customers arrive at a bank according to Poisson process with mean rate of 3 per minute. Find the probability of getting 4 customers in 2 minutes. 9) If X 1 has normal distribution N(25,4) and X 2 has normal distribution N(30,9) and if X 1 and X 2 are independent find the distribution of 2 X 1 + 3 X 2 . 10) Define a renewal process. PART-B Answer any five questions (5 x 8 = 40) 11) Derive the MGF of normal distribution. 12) Show that F (-∞) = 0, F (∞) = 1and F(x) is right continuous. 13) Show that binomial distribution tends to Poisson distribution under some conditions to be stated. 14) Let X and Y be random variables with joint pdf f(x,y) = x+y, 0<x<1, 0<y<1, zero elsewhere. Find the correlation coefficient between X and Y 15) Let { X n } be a Markov chain with states 1,2,3 and transition probability matrix 3 4 1 4 0 1 4 1 2 3 4 0 1 4 1 4 with PX 0 i , i = 1,2,3 1 3 Find i) PX 1 2 ii) PX 0 2, X 1 1, X 2 2, X 3 2 1 16) Obtain the expression for Pn t in a pure birth process. 17) State and prove Chapman-Kolmogorov equation on transition probability matrix. -x m-1 18) Let X have a pdf f(x) = 1/m e x , 0 < x < ∞ and Y be another independent -y n-1 random variable with pdf g(y) = 1/n e y , 0 < y < ∞ .obtain the pdf of U= X X Y . PART-C Answer any two questions (2 x 20 = 40) 19) . a) State and prove Bayes theorem. b) Suppose all n men at a party throw their hats in the centre of the room. Each man then randomly selects a hat. Find the probability that none of them will get their own hat. (10 + 10) 20) a) let { An } be an increasing sequence of events. Show that P(lim An ) = limP( An ). Deduce the result for decreasing events. b) Each of four persons fires one shot at a target. let Ai , i = 1,2,3,4 denote the event that the target is hit by person i. If Ai are independent and P( A1 ) = P( A2 ) = 0.7, P( A3 ) =0.9, P( A4 ) = 0.4. Compute the probability that 1. All of them hit the target 2. Exactly one hit the target 3. no one hits the target 4. atleast one hits the target. (12 + 8) 21) . a) Derive the expression for Pn t in a Poisson process. b) If X 1 t and X 2 t have independent Poisson process with parameters 1 and 2 . Obtain the distribution of X 1 t = γ given X 1 t + X 2 t = n. c) Explain Yules’s process. (10 + 5 + 5) 22). a) verify whether the following Markov chain is irreducible, aperiodic and recurrent 0 1 P 3 2 3 2 3 0 1 3 1 3 2 3 0 Obtain the stationary transition probabilities. b) State the postulates and derive the Kolmogorov forward differential equations for a birth and death process. ******************** 2 (10 +10)