Part-B

4. A two dimensional random vector [X is described by the probability density

function
0 << + <1

(२२) = { 0 , otherwise
Find the correlation coefficient between X and Y. (9)

5. Define the following
1) Wiener process 11)
Poisson process
iii) Birth —Death Markov chain (9)
6. a) Consider the random process X(t)=A cos(oot+0), where A and are
constants, e is a uniformly distributed random variable in the interval (-
Jt,1t). Find the mean and auto correlation ofthe random process X(t). (5)

b) IfX and Y are two discrete random variables, then show that
[ECX/Y)]- E(X) http:/*'mxww.knuonline.com

Part-C
7. a) State and prove Markov Inequality (6)

0) A continuous random variable X has mean R=12 and variance ox 2 = 9
and has an unknown probability distribution. Find the lower bound for the
probability of the event P(6
8. a) Show that the convergence of a random sequence with probability- I
implies the convergence in probability. (6)

b) State and prove weak law of large numbers. (6)

9. a) The transition probability matrix (tpm) of a Markov chain having three
states 1,2 and 3 is