4. a.

6.

PART B

Find the characteristic function of N( g,a2) and hence find mean and variance

using it. (3) 0. Find the correlation coefficient of (3)

c.

९.

a.

fey ( _ re ം ௫07020, 11
xy OGY) = 0 , otherwise

The two independent random variables X and Y have the variances 36 and

16, find the correlation coefficient between X+Y and X-Y (2)
2-1
T . .
A random vector [20102201 has a covariance matrix K = —1 20
Design ೩ non -trivial transform that will generate from > a new random vector ‏,لا‎
‎whose components are uncorrelated. http:/‘www.ktuonline.com (3)

The transition probability matrix of ೩ markov chain with 3 states 0, I is

3/4 1/4 0
P=1/4 1/2 1/4
0 3/4 1/4

Initial state distribution of the chain is P -i] 1/3 for i =0,1,2.Fi

180೦೦ = 2) 20051052)

#. 2001, ௭23௨-1, 0-2) 20൨5൧൧0512) (3)

Find the autocorrelation function of the Poisson Process and hence find the covariance.

If 216) and X2(t) represent two independent Poisson Process with parameter Aitand

121, then show that +X2(t) is also Poisson Process with parameter (11+ R2)t while
Xi(0)-X2(t) is not a Poisson Process. (3)

b.

الك
‎X(t) is a Gaussian process with p(t) —10 and c(tl,t2) = 16e-t21 (3) find probability‏
‎that X(IO) 58 and also find the probability that X(IO) (654‏

What is random walk and how it can be approximated to a wiener process. Find mean
and variance of random walk. (2)

PART ^

State and Prove Chernoff Bound and Schwarz inequality (5)

A post office handles 10,000 letters per day with a variance of 2000 letters. What
can be said about the probability that this post office handles between 8000 and

12000 letters tomorrow. What about the probability that more than 15000 letters come
in. (3)