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D192004 Pages:3

The joint density function of two continuous random variables X,Y is given by

നാ, x >0,y>0
fay) = (£ otherwise

Find (é)the value of K (ii)P(¥ > 1) (iii) the marginal distributions of X,Y and (iv)
check whether X, ¢ are independent.
Find the power spectral density function of the WSS process whose autocorrelation
function is A? e721"!
Let X(t) = 4 ८०5७ (50 + 6), where A and 6 are independent random variables. A is a
random variable with mean 0 and variance 1 and @ is uniformly distributed in (नया, 7).
Show that { X(t)} is WSS.
If {X(t)} is a random process with mean 3 and
(2) = ‏و‎ + 4൦5. Find തി (i) VIX(8)] (iti) Cov [x(5),X(8)]
3 balls drawn at random without replacement from a box containing 2 white, 3 red and 4
black balls. Let ¥ denotes number of white balls drawn and 3” denotes number of red balls
drawn. Find the joint pdf of X and Y, find the marginal pdfs and check whether X and 7

are independent.

PART ட்‌
Answer any two full questions, each carries 20 marks
0.1 0.5 04
The tpm of a Markov chain with 3 states 1, 2, 3 is given by P = [os 0.2 3 with initial
0.3 04 03

distribution P{0} = [0.7,0.2,0.1].

Find (i)P{X, = 2/ ‰ = 1} (ii) P(X, = 3, 2 = 2.21 = 1 X, 2)
(11) P(X, = 3/0 = 1} (iv) P{X, = 3}

The tpm of a Markov Chain is P = . Find the steady state distribution of the chain.

५० | ‏صم‎ pope
५० | ‏عم | ذم يم‎

Suppose that customers arrive at a bank according to a Poisson process with mean rate of 3
per minute. Find the probability that during a time interval of 2 minutes

(i) exactly 4 customers arrive (ii) more than 4 customers arrive.

Use Newton’s forward interpolation formula to find the interpolating polynomial for the

following data. Hence evaluate (1.5).

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