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 PX 0  i   , i = 1,2,3
1
3
Find i) PX 1  2
ii) PX 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)