a)

b)

a)

b)

a)

b)

a)

D1002 Pages:

PART ‏تا‎
‎Answer any two full questions, each carries 15 marks
The life time of a certain type of electric bulbs may be considered to follow
exponential distribution with mean 50 hrs. Use central limit theorem to find the
approximate probability that 100 of these electric bulbs will provide a total of more

than 6000 hrs of burning time.

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

1 i
K(i-x—y), O102) = (717೫73) ‏وک 7> ۱۲و‎

0 otherwise
Find (i) the value of K (11) (2 < न्‌ ೫ 3 (iii) the marginal distributions of

X,Y (iv) check whether X, ¢ are independent.
Let X(t) = Acoswt — Bsinwt, where A and B are independent random variables
following 4/(0, ச). Then show that ( X(t)} is WSS.

Find the power spectral density function of the WSS process whose autocorrelation

⋅⋅−≖
നി

2x+3y
54

x = 1,2; y=1,2,3. Find (i) the marginal distributions of ¥ and ¢ (ti) The

The joint probability distribution of ¥ and ४ is given by f (x, y) मः णि

conditional distribution of X for ¥ = क.

w749
The power spectral density of a WSS process is - ‏عل‎ . Find the
P P y ۲ w*+5w7+4

autocorrelation function and power of the process.
PART C
Answer any two full questions, each carries 20 marks
The tpm of a Markov chain with 4 states 0, 1, 2, 3 is given by

02 08 0 0

0 02 0.8 0

0 0 0.2 0.8
0 0 0 1

P=

⋅ 1

with initial distribution மி்‌ = i} = ത i = 0,1,23.

Find (02, = 2/ X, = 1} (ii) P{X, = 3/X, = 1}
(111) P(X, = 3, X, = 2,೫ = 2) (iv) P{X, = 3}

Page 2 of 3

(7)

(8)

(7)

(8)

(7)

(8)

(10)