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E4802
The autocorrelation function for a stationary process X(t) is given by Ryx(t) =

9 12௨11, Find the mean value of the random variable Y = രമ

and the variance of X(t).

A random process X(t) is defined by X(t) = Y(t) cos(wt + 6) Where Y(t) is a
WSS process, w is a constant and @ is a random variable which is uniformly
distributed in [0,27] and is independent of Y(t). Show that X(t) is WSS.

Consider the random process X(t) = Acos(wt + 6) where A and w are constants
and 6 is a uniformly distributed random variable in (0,27). Check whether or not
the process is WSS.

The joint PDF of two continuous random variables X and Y is

_ (8xy,0fy) = [ 0, otherwise
i) Check whether X and Y are independent ii) Find P(X + Y > 1)

PART C
Answer any two full questions, each carries 20 marks
The number of particles emitted by a radioactive source is Poisson distributed. The source
emits particles at a rate of 6 per minute. Each emitted particle has a probability of 0.7 of
being counted. Find the probability that 11 particles are counted in 4 minutes.
Assume that a computer system is in any one of the three states: busy, idle and under
repair, respectively, denoted by 0,1,2. Observing its state at 2 P. M. each day, the transition

0.6 0.2 0.2
probability matrixis P=]0.1 0.8 0.1
0.6 0 0.4

Find out the third step transition probability matrix and determine the limiting probabilities.
If customers arrive at a counter in accordance with a Poisson process with a mean rate of 2
per minute, find the probability that the interval between two consecutive arrivals is:

i) More than | minute ii) Between | minute and 2 minutes

111) Less than or equal to 4minutes.
Use Trapezoidal rule to evaluate 18 x? dx considering five subintervals

Using Newton’s forward interpolation formula, find y at x = 8 from the following
table: x: 0 5 10 15 20 25

y: 7 11 14 18 24 32
Using Euler’s method, solve for y atx ಎ 0.1 from = =x+ytxy, y(0)=1
taking step size 1 ಎ 0.025.
The transition probability matrix of a Markov chain {Xn 10} having three states

| 0.2 0.3 0.5
1,2and3is 2-101 0.6 0.31 and the initial probability distribution is
0.4 0.3 0.3
ற(0)-[0.5 0.3 0.2]. Find the following:
1) 2( = 2} ii) P{X, = 3,X, =2,X, =1,X) = 3}.

Using Newton-Raphson method, compute the real root of f(x) =x? - 22-5
correct to 5 decimal places.

Using Lagrange’s interpolation formula, find the values of y when x = 10 from
the following table :

: 5 6 9 11
y: 12 13 14 16

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