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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
SECOND SEMESTER – APRIL 2008
ST 2902 – PROBABILITY THEORY AND STOCHASTIC PROCESSES
Date : 29-04-08 Dept. No. Max. : 100 Marks
Time : 1:00 – 4:00
Part-A
Answer all the questions: 10 X 2 = 20
1.) Find the constant c such that p(x) = c (2/3)x x=1,2,3,… zero elsewhere, satisfies the
conditions of a probability distribution.
2.) If the events A and B are independent show that A and B are independent.
3.) The MGF of a random variable X is (1-2t)-n/2 .Find the E(X).
4.) Define the period of a state. When do you say that a Markov chain is aperiodic?
5.) Define a process with independent increments. When do you say that it is stationary?
6.) Define a Markov process.
7.) Define covariance between two random variables X and Y. What happens to the Covariance when
they are independent?
8.) Given the MGF of a random variable X as (1/3 + 2/3 et)5 .Find P(X=0)
9.) What is a Renewal process?
10.) State the additive property of Poisson distribution.
Part-B
Answer any 5 questions: 5 X 8 = 40
11.) The transition probability matrix of a Markov chain with three states 0,1,2 is
P =
and the initial distribution is given by P(X0= i) = 1/3 i=0,1,2
Find i.)P[X2 = 2] ii.)P[X3 =1 ; X2=2 ; X1=1 ; X0=2]
12.) Show that discrete queue forms a Markov chain.
13.) Show that the distribution function F(x) is non decreasing and right continuous.
14.) Let X have a pdf f(x)=4x3 , 0<x<1, zero elsewhere .Find the distribution function and
pdf of Y= -2 ln X4
15.) Derive the MGF of Normal distribution.Hence obtain the mean and variance.
16.) Given the joint pdf of the random variables X and Y as
f (x, y) = 8xy, 0<x<y<1, zero elsewhere.
Obtain i.) Marginal pdf of x1
ii.)conditional pdf of Y given X=x
iii.)E[Y/x]
iv.)Var [Y/x]
17) In a Markov chain if i ↔ j then show that
i.) d(i) = d(j)
ii.) If i is recurrent then j is also recurrent .
18.) Derive the expression for Pn (t) in Yule-Furry process
Part-C
Answer any 2 question: 2 X 20 = 40
19.)a.)Let An be increasing sequence of events. show that P(lim An) = lim P( An) .
Deduce the result for decreasing sequence. (8+4)
b.)Five numbers are drawn without replacement from the first 10 positive integers.
Let X represent the next to the smallest of the numbers drawn .Find the probability
distribution of X.Also obtain F(x). (5+3)
20.)a.)State and prove addition theorem for n-events.
b.)Bowl I contains 3 red chips an7 blue chips.Bowl II contains 6 red chips and 4 blue
chips .A bowl is selected at random and then 1 chip is drawn from the bowl.
i.)Compute the probability that this chip is red.
ii.)Given that the drawn chip is red,find the conditional probability that it is drawn
from bowl II.
c.)Show that Binomial distribution tends to Poisson distribution under some
conditions.
21.)a.)State the postulates of a Poisson process and obtain the expression for P(t).
b.)If the Process {Xt} has stationary independent increments with a finite mean, show that
E[Xt}=m0+m1 t where m0 = E (X)
m1= E (X) – m0
22.)a.)Given the Markov chain with transition probability matrix with states 1,2,3,4
and
P =
i.)Show that the chain is irreducible, aperiodic and non-null recurrent.
ii.)Obtain the probability of reaching the various states as n → ∞. (8+6)
b.)A gambler has Rs.2.He bets Rs.1 at a time and wins Rs.1 with probability 1/2.He
stops the playing if he loses Rs.2 or wins Rs.4.
What is the transition probability matrix of the Markov chain? (6)