Anna University Coimbatore 2009 B.E Electronics & Communication Engineering Probability and Random Process-Model - Question Paper
MODEL EXAMINATION
PROBABILITY & RANDOM PROCESSES
Year/Semester & Branch: II / IV / ECE
Max. Marks: 100 Time: 180 min
PART-A Answer ALL Questions (20 X 2 = 40)
- If the range set of X is the set {0,1,2,3,4} and P( X = x ) = 0.2, determine the mean and variance of the random variable.
- Given the random variable X with
density function
.
Find the probability density function of 
. - The mean of a Binomial distribution is 20 and standard deviation is 4. Find the parameters of the distribution.
- Is the function defined as
follows a density function?
. - If the joint probability density
function is given by
,
. Find E[XY]. - If the joint probability density
function of (X,Y) is
,
find 
. - If Y=-2X+3, find the cov(X,Y).
- State the central limit theorem for independence and identically distributed random variables.
- Define Wide sense stationary random process.
- The one-step transition
probability matrix of a Markov chain with states (0,1) is given by
. Is it irreducible
Markov chain? - Define Sine wave process.
- Define Markov process and a Markov chain.
- Define auto correlation function
and prove that for a WSS process
. - The power spectral density of a
random process
is given
by 
Find
its autocorrelation function. - Find the power spectral density
of a WSS process with auto correlation function
. - Find the variance of the
stationary process {X(t)} whose auto correlation is given by
. - Define thermal noise and white noise.
- Describe linear system with random input.
- Write a note on noise in communication system.
- Examine whether the following
system is linear:
.
PART-B Answer ANY 5 Questions (5 X 12 = 60)
- a) In a continuous distribution, the probability
density is given by
.
Find k, mean, variance and the distribution function. (6 Marks)
b) Out of 800 families with 4 children each, how many families would be expected to have
(i) 2 boys and 2 girls ; (ii) atleast 1 boy ; (iii) atmost 2 girls and (iv) children of both the genders.
Assume equal probabilities for boys and girls. (6 Marks)
- The joint probability density function of a two
dimensional random variable (X,Y) is given by
. Compute 
, 
and 
. - Show that the random process
where 
is a random variable
uniformly distributed on 
is
a) first order stationary b) stationary in wide sense c) ergodic(based
on first order and second order averages). - a) The power spectral density of a WSS process
is given by
.
Find the auto correlation function of the process. (6 Marks)
b) The auto correlation function of a wide sense stationary random process is given by
. Determine the power
spectral density of the process. (6 Marks)
- a) A white noise signal of zero mean and power
spectral density
is applied
to an ideal low pass filter whose
bandwidth is B. Find the auto correlation of the output noise signal. (6 Marks)
b)
If {X(t)} is a Gaussian process with 
and
find the probability that
(i) 
and (ii) 
. (6 Marks)
- a) The number of personal computer(PC) sold in a computer world is uniformly distributed
with a minimum of 2000 PC and a maximum of 5000 PC. Find (i) The probability that daily
sales will fall between 2500 and 3000 PC. (ii) What is the probability that the computer world
will sell atleast 4000 PC ? (iii) What is the probability that the computer world will exactly
sell 2500 PC? (6 Marks)
|
Production (X): |
55 |
56 |
58 |
59 |
60 |
60 |
62 |
|
|
Export(Y): |
35 |
38 |
37 |
39 |
44 |
43 |
44 |
(6 Marks) |
b) Find the coefficient of correlation between industrial production and export using the following
data
- a) If
where
Y is uniformly distributed in 
.
Show that X(t) is wide
sense stationary. (6 Marks)
b) Three boys A,B,C are throwing a ball each other. A always throws the ball to B and B always
throws the ball to C, but C is just as likely to throw the ball to B as to A. Show that the process
is Markovian. Find the transition matrix and classify the states. (6 Marks)
- a) Find the power spectral density of the random
process {X(t)} if
and
(6 Marks)
b) Consider a system with transfer function 
. An input signal with
auto correlation function
is fed as input to the
system. Find the mean and mean square value of the output.
(6 Marks)
| Earning: Approval pending. |
