Download maesp 102 probability and random

M.TECH DEGREE EXAMINATION
Model Question paper
Specialization: 1. Advanced Communication and Information Systems
2. Advanced Electronics and Communication Engineering
3. Signal Processing
First Semester
MECCI 102 / MECEC 102/ MAESP 102 PROBABILITY AND RANDOM
PROCESSES
(Regular – 2013 Admission Onwards)
Answer all Questions .All questions carry equal marks
Time 3hrs
Max Marks:100
1.a) For a certain binary communication channel, the probability that a transmitted ‘0’ is received as
a ‘0’ is .95 and the probability that a transmitted ‘1’ is received as ‘1’ is .9. If the probability that a ‘0’
is transmitted is .4, find the probability that (i) a ‘1’ is received and (ii)a ‘1’ was transmitted given
that a 1 was received.
(10)
b) If the random variable X takes the values 1,2,3 and 4 such that 2P(X=1)=3P(X=2)=P(X=3)=5P(X=4)
find the probability distribution and cumulative distribution function of X.
(10)
c) Find the mgf of the discrete random variable X with the probability mass function f(x)=1/n for x =
1,2,3,......n
(5)
OR
2.a)If the density function of a continuous random variable X is given by,
f(x)= ax 0≤x≤1
=a
1≤x≤2
= 3a – ax 2≤x≤3
= 0 elsewhere
(i) Find the value of a
(ii) Find the cdf of X
(iii) If x1, x2 and x3 are 3 independent observations of X what is the probability that exactly one
of these 3 is greater than 1.5.
(12)
(b) The joint pdf of a 2 dimensional random variable (X,Y) is given by f(x,y)=xy2 + x2/8 if 0≤x≤2,
0≤y≤1 compute
(i)P(X > 1) (ii) P(X>1/Y<
) (iii) P((X+Y)≤1)
3.(a) State and prove Tchebycheff inequality.
(13)
(12)
(b) Verify central limit theorem for the independent randomvariables Xk, where for each k,
P(Xk=±1)=
(13)
OR
4.a) The joint pdf of (X,Y) is given by f(x,y) = 24xy, x>0,Y>0,x+y≤1 and f(x,y)=0 elsewhere find the
conditional mean and variance of Y given X.
(15)
b)Explain strong law of large numbers.
5.a) Show that the random process X(t) = A cos(
(10)
0t
+ ) is a wss, if A and
0
are constant and
uniformaly distributed random variable in (0,2 )
is a
(7)
b) Given that the autocorrelation function for a stationary ergodic process with no periodic
components is RXX( ) = 25 + (4 / 1+6 2). Find the mean value and variance of the process{X(t)}
(8)
c) Prove that the random process {X(t)} with constant mType equation here.ean is mean ergodic if
t1,t2) dt1 dt2 = 0
OR
6.a) Verify wheather the sine wave process {X(t)}, where X(t) = Y cos t Where Y is uniformaly
distributed in (0,1) is a sss process.
(9)
b) Express the autocorrelation function of the process {XI(t)} in terms of the autocorrelation function
of the process {X(t)}
(8)
c) Find the power spectral density of a wss process with autocorrelation function R( ) =
7.a) Find the mean and autocorrelation function of the Poisson process.
(8)
(8)
b) A man either drives a car or catches a train to go to office each day. He never goes 2 days in a raw
by train but if he drives one day, then the next day he is just as likely to drive again as he is to travel
by train . Now suppose that on the first day of the week, the man tossed a fair dice and drove to
work if and only if a 6 appeared. Find(i) the probability that he takes a train on the third day and (ii)
the probability that he drives to work in the long run.
(17)
OR
8.a)Suppose that a customers arrive at a bank according to a Poisson process with a mean rate of 3
per minute. Find the probability that during a time interval of 2 minute (i) exactly 4 customers arrive
and (ii) more than 4 customers arrive.
(6)
b)Classify different states of Markov Chain.
(9)
c) State and prove Chapman Kolmogorov theorem.
(10)