NATIONAL ENGINEERING COLLEGE, K.R.NAGAR, KOVILPATTI.
(An Autonomous Institution, Affiliated to Anna University, Chennai)
23EC31C / PROBABILITY, RANDOM PROCESS AND QUEUEING THEORY
Branch/ Semester: ECE / III SEM
Regulation: 2023
CO3- ALL POSSIBLE QUESTIONS
Part A- 2 Marks
Q.No
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Questions
Consider the random process {X(t)}, X (t )  sin( t   ) , where  is uniform in
  
  ,  . Check whether the process is stationary.
 2 2
Define wide sense stationary process
Write down the classification of random processes
Distinguish between wide sense stationary and strict sense stationary of the random
process X(t).
Prove that a first order stationary random process has a constant mean.
Check whether the random process X(t) = A cos2  t is first order stationary, where A
is a constant
What do you mean by jointly WSS process.
A random process has simple functions of the form X (t )  A cos(t   ) in which A and
 are constants and  is a random variable. Prove that this process is not stationary if
 is uniformly distributed over a range of  .
Examine whether the Poisson process X (t ) given by the law
e  t t 
r  0,1,2.... is covariance stationary.
r!
Write any two property of Poisson Process.
The one-step transition probability matrix of a Markov chain with stages {0, 1} is
0 1
 . Draw a transition diagram
given as P  
 1 0
Find the mean of the Poisson process.
What is meant by steady-state distribution of Markov chain?
Define continuous time random process Give an example of it.
1 
 0
 . Find the
The transition probability matrix of a Markov chain is P  
1 / 2 1 / 3 
steady state distributions of the chain.
Define a Markov process and give an example.
PX (t )  r  
r
Define a Markov process and a Markov chains
Consider a Markov chain {Xn; n  1} with state space S = {0, 1, 2} and one-step
Transition probability matrix P = 
0 1 0
 . Is the chain periodic?
0 0 1 
1 0 0
19
20
Define a Markov chain and give an example.
The transition probability matrix of a Markov chain X n  n = 1, 2, 3… having 3 states
 0.1 0.5 0.4 


1, 2 and 3 is P   0.6 0.2 0.2  .Verify that which is irreducible .
 0.3 0.4 0.3 


 0.4 0.6
0
0 

0
0  .Is it
 0.2 0.4 0.1 0.3 


 0
0
0
1 

. Consider the Markov chain with transition probability matrix  0.3 0.7
21
irreducible?
1 / 4 1 / 4 1 / 4 1 / 4 

1 / 3 2 / 3 0  . Find the
 0 3 / 4 1/ 4 0 


 0
0
0
1 

The transition probability matrix of a Markov chain is  0
22
23
24
25
essential states.
A drilling process is considered to be 2 state Markov chain . If process is there in the
production company it is the state ‘0’ and if the process is not there is in state 1. Th
 0.1 0.2 
 . Find the probability that the drilling process will be
TPM is given by P  
 0.3 0.4 
then for 3 days from 1 day?
If two random variables are independent, then prove that their covariance is zero
What is a stochastic matrix? When it is said to be regular?
Part B- 8 Marks
Q.No
1
2
3
Questions
PX (t )  n 

4
5
 | |
If {X(t)} is a WSS process with autocorrelation R( ) = A e
, determine the second
order moment of the RV {X(8) – X(5)}.
Consider the random process x(t) = cos(  0 t +  ) where  is uniformly distributed in
the interval -  to  . Check whether X(t) is stationary or not.
Check whether the process X (t ) whose probability distribution is given by
at n1 , n  1,2,....
1  at n1
is stationary
at
1  at
n0
Show that the process 𝑋 𝑡 = 𝐴𝑐𝑜𝑠 𝜔0 𝑡 + 𝜃 is stationary process in wide sense if 𝐴 and 𝜔0
are constants and 𝜃 is uniformly distributed in 0, 2𝜋
Let 𝑋 𝑡 = 𝐵𝑐𝑜𝑠 100𝑡 + 𝜃 where 𝐵 and 𝜃 are independent random variables such that 𝜃 is
uniform in (−𝜋, 𝜋) and 𝐵 has mean zero and variance one. Find mean and auto correlation
function of 𝑋(𝑡) .
6
7
8
9
10
11
Verify the sine wave process defined by X(t) = Y cos t , where Y is uniformly
distributed over (0, 1) is a SSS.
. For a random process X(t) = Y sin t , Y is a uniform random variable in the interval
(-1, 1). Check whether the process is WSS or not
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 2 consecutive arrivals is
(1) more than 1 minute
(2) between 1 minute and 2inute and
(3) 4 min or less.
Suppose that customers arrive at a bank according to Poisson process with mean rate of
3 per minute. Find the probability that during a time interval of two minutes (i) exactly 4
customers arrive (ii) greater than 4 customers arrive (iii) fewer than 4 customers arrive.
A machine goes out of order, whenever a component fails. The failure of this part
follows a Poisson process with a mean rate of 1 per week. Find the probability that 2
weeks have elapsed since last failure. If there are 5 spare parts of this component in an
inventory and that the next supply is not due in 10 weeks, find the probability that the
machine will not be out of order in the next 10 weeks.
. If {N1(t)} and{N2(t)} are two independent Poisson process with parameters 1 and 2
n
respectively, show that P[N1(t) = k / {N1(t) + N2(t) = n}] =   pk qn-k, where
k 
 
p
1 and
2 .
q
1  2
1  2
The transition probability matrix of a Markov chain X n  n = 1, 2, 3… having 3 states 1,
12
13
14
15
 0.1 0.5 0.4 


2 and 3 is P   0.6 0.2 0.2  and the initial distribution is P (0)  0.7,0.2,0.1 . Find
 0.3 0.4 0.3 


P ( X 2  3) and P( X 2  3, X 1  3, X 0  2)
Consider a Markov chain with state space {0, 1} and transition probability matrix
0 
 1
 .
P  
1 / 2 1 / 2 
(i)Draw a transition diagram (ii) Is the state 0 recurrent? Explain. (ii) Is the state 1
recurrent? Explain.
Show that the process X (t )  A cos t  B sin t , where A and B are random variables, is
wide sense stationary, if (i) E (A) = E (B) = 0 (ii) E (A2) = E (B2) and E (AB) = 0
The TPM of a Markov chain with three states 0,1,2 is 𝑃 =
3
1
4
1
4
1
4
2
3
4
1
4
4
0
0
1
and the initial
1
16
17
state distribution of the chain is 𝑃 𝑋0 = 𝑖 = 3 , 𝑖 = 0,1,2 Find (i) 𝑃 𝑋2 = 2 (ii)
𝑃 𝑋3 = 1, 𝑋2 = 2, 𝑋1 = 1, 𝑋0 = 2 .
Two boys A and B and two girls C and D are throwing a ball from one to the other.
1
Each boys throws the ball to the other boy with probability
and to each girl with
2
1
probability . On the other hand each girl throws the ball to each boy with probability
4
1
and never to the other girl. In the long run, how often does each receive the ball?
2
A man either drives a car or catches a train to go to office each day. He never goes 2
days in a row by train but if he drives one day then the next day he is just as likely to
drive again he is to travel by train. Now suppose that on the first day of the week the
man tossed a fair die and drove to work if and only if a 6 appeared. Find (i) the
probability that he takes a train on the third day (ii) the probability that he drives to work
in the long run
18
Three boys 𝐴, 𝐵 and 𝐶 are throwing a ball to each other. A always throws the ball to 𝐵 and
𝐵always throws the ball to 𝐶. 𝐶 is just likely to throw the ball to B as to 𝐴. Show that the process
is Markovian and classify the states of the Markov chain
19
A salesman territory consists of three cities A,B, and C. He never sells in the same city
on successive days. If he sells in city A ,then the next day ,he sells in city B .However ,if
he sell in either B or C, the next day he is twice as likely to sell in city A as in the other
city .In the long run ,how often does he sell in each of the cities.
20
An average of 10 cars per hour arrive at a single-server drive-in teller. Assume that the average
service time for each customer is 4 minutes, and both inter arrival times and services are
exponential (1) what is the probability that the teller is idle (2) what is the average number of
cars waiting in the line for the teller?
(3) what is the average amount of time a drive-in customer spends in the bank parking (4) on
the average, how many customer per hour will be served by the teller?
21
A psychologist makes the following assumptions concerning the behaviour of mice
subjected to a particular feeding schedule. For any particular trial, 80 % of the mice that
went right on the previous experiment will go right on this trial and 60 % of those mice
that went left in the previous experiment will go right on this trial. If 50 % went right on
the first trial, what would he predict for (i) the second trial (ii) the third trial (iii) the
thousandth trial?