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B4A0102

PART 8

Answer any two questions. Each carries 15 marks

A factory has two outlets to sell its products. The daily sales from the first outlet
is uniformly distributed between Rs. 50,000 and 60,000 and from the second
outlet is uniformly distributed between 40,000 and 60,000. The sales of the
outlets are independent.
(i) What is the probability that the total sales from both the outlets combined is
more than Rs.100000.
(ii) If 20% of the amount from the sales is profit, find the expected daily profit
from both the outlets combined, and the variance = of the profit.
A computer generates 100 random numbers uniformly distributed between 0 and
1. Use central limit theorem to find the probability that
i) their sum is 60 or more,
11) their average is 0.7 or less.
A random process X(t) is defined by X(t) = sin(t + ©)where ಅ is a random

variable taking values 0 or 7 with equal probability. Find the mean, autocorrelation
and autocovariance of X(t). Is it a wide sense stationary process?

Find the power spectral density of a wide sense stationary process X(t) with
autocorrelation function Ry(t) = 3.

The joint probability distribution of two discrete random variables X and Y is given
by
1
p(x,y) = उठ + 1), = 0,1,2 ى٣‎ = 0,1,2,3
Find the correlation coefficient between Xand Y.

Find the autocorrelation function and average power of a wide sense stationary
process X(t) with power spectral density given by

1- ८, |o#|<1
5 =
x(@) { 0 otherwise.
PART ^

Answer any two questions. Each carries 20 marks

The number of enquiries arriving at a call centre is a Poisson process with rate 5 per
hour.

i) Find the probability that there would be 3 calls between 10 AM and 11
AM and 4 calls between 2 PM and 4 PM.

ii) A call is categorized as ‘long’ if it lasts more than 10 minutes. The
probability that an arriving call is long is 0.2. Find the probability that
the time between two consecutive long calls is less than 1 hour.

2,,7 = 0,1,2, ...15 a Markov chain on state space (1,2,3) with initial probability
distribution P(X 9 = 1) = ‏ئ[)م‎ = 2) = P(X) = 3) = 1/3 and transition
probability matrix given by

0.2 06 0.2
0.5 0.2 0.3
0.3 0.4 0.3

Find i) P(X, = 3) ii) P(X, = 3,X> =2,X3 = 2) iii) P(X, = 1|Xp = 2)

The table gives the area under the normal probability curve from 0 to certain values
x.

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