RANDOM PROCESSES QUESTION BANK UNIT II PART A 1) Suppose the
duration X in minutes of long distance calls from your (1/ 8e - ( x
/ 8) ) forx > 0 home,follows exponential law with PDF f(x)= .
Find 0otherwise P[x>5], P[3 X 8], P [X / X 4], mean and variance
of X. 1 for0 < x < 1 2) Let X be a RV with pdf f(x)= . If Y =
-2 log X find the PDF 0otherwise of Y and E(Y). 3) If X is
binomially distributed RV with E(X) = 2 and var(X) = 4/3. Find P[X
=5]. 4) A RV is uniformly distributed over [0,2 p ] . If Y = cos X
then 1. Find PDF of Y and X 2. E(X) and E(Y) 5) The life time of a
component measured in hours is Weibull distribution with parameter
a = 0.2 and b = 0.5. Find the mean life time of the component. 6)
If X is a Poisson variate such that P[X=2] = 9 P[X=4] + 90 P[X=6]
find the variance. 2 . Find the first two terms of the 7) For a BD
mean is 6 and SD is distribution. 8) If the RV X is uniformly
distributed over (-1,1). Find the density function of Y = sin (p x
/ 2). 9) The life length [in months] X of an electronic component
follows an exponential distribution with parameter l = (1/2). What
is the probability that the components survives atleast 10 months
Given that already it had survived for more than 9 months. 10) If M
things are distributed among a men and b women find the number of
things received by men is odd.
PART B 1. Let X be the length in minutes of a long distance
telephone (1/10)e - ( x /10) forx > 0 conversation. The PDF of X
is given by f(x)= . 0otherwise 2. The life time X in hours of a
component is modeled by Weilbull distribution with b = 2 starting
with a large number of components . It is observed that 15% of the
components that have lasted 90 hours fail before 100 hours. Find
the parameter a . 3. State central limit theorem in Lindberg Levy
form. A random sample of size 100 is taken from a population whose
mean is 60 and variance is 400. Using CLT with what probability can
we assert that the mean of the sample will not differ from m = 60
by more than 4? Area under the standard normal curve from z =0 to
z=2 is 0.4772. 4. If the life X [in years] of a certain type of car
has a Weibull distribution with parameter b = 2. Find the value of
a given that the probability that the life of the car exceeds 5 is
e -0.25 for these values of a and b find the mean and variance of
X. 5. It is known that the probability of an item produced by a
certain machine will be defective is 0.05. If the produced items
are sent to the market in packets of 20, find the number of packets
containing at least exactly and at most 2 defective items in a
consignment of 100 0 packets using Poisson approximation to
Binomial distribution. 6. State and prove the memoryless property
of the exponential distribution and geometric distribution. 7. The
daily consumption of milk in a city in excess of 20,000 litres is
approximately distributed as an gamma variate with the parameters
k=2 and l = (1 /10,000).the city has a daily stock of 30,000
litres. What is the probability that the stock is insufficient on a
particular day? 8. Derive the mgf of negative binomial
distribution. Also obtain its mean and SD. 9. Prove that poisson
distribution is the limiting case of binomial distribution. 10.
Find the MGF of a poisson distribution and hence find its mean and
variance.
RANDOM PROCESSES UNIT IV TWO DIMENTIONAL RANDOM VARIABLES PART A
1 1. If X and Y are random variables having density function f(x,y)
= (6 - x - y ) , 8 0 < x < 2, 2 < y < 4, Find P( X + Y
< 3 ). 2. State the equation of two regression lines.what is the
angle between them? 3. The following table gives the joint
probability distribution of X and Y. Find (i) marginal density
function of X. (ii) marginal density function of Y. Y X 1 2 3 1 0.1
0.1 0.2 2 0.2 0.3 0.1 4. If the joint pdf of the random variable is
given by f(x,y) = kxy e-( x + y ) , x > 0 , y > 0 , find the
value of k. 5. The tangent of the angle between the lines of
regression y on x and x on y is 0,6 and 1 sx = sy , find the
correlation coefficient between X and Y. 2 1 ,0 x < 2 6. If the
joint pdf of (X,Y) is f(x,y) = 4 . 0,otherwise Find P(x+y 1) if
P(y=1) = 0.4 and P(Y = 2 ) = 0.6 x + y ,0 < x < 1,0 < y
< 1 7. If X and Y have joint pdf f(x,y) = ,check whether X and Y
0, otherwise are independent. 8. Find the marginal density
functions of X and Y if 1 (2 x + 5),0 x 1,0 y 1 . f(x,y) = 4 0,
otherwise 9. Find the marginal density functions of X and Y from
the joint density function 2 (2 x + 3 y ),0 x 1,0 y 1 . f(x,y) = 5
0, otherwise 10. X and Y are two random variables having the joint
density function 1 ( x + 2 y) , where x and y assumes the integer
values 0,1 and 2. Find the f(x,y) = 27 marginal probability
distribution of X. 11. Find the value of k if f(x,y) = k(1-x)(1-y)
for 0 < x,y < 1is to be a joint density function. 12. Find k
if the joint probability density function of a bivariate random
variable (X,Y) is2 2
given by f(x,y) = k(1-x)(1-y) if 0 < x < 4, 1 < y <
5 and 0 otherwise.
PART B 1.The joint density finction of a random variable X and Y
is f(x,y) = 2, 0 < x < Y < . Find marginal and conditional
probability density functions. Are X and y independent? 4ax,0 x 1
2. Two independent random variables X and Y are defined by f(x) =
and 0,oterwise 4by,0 y 1 f(y) = . Show that U = X + Y and V = X +Y
are uncorrelated. 0,otherwise 3.(X,Y) is a two dimensional random
variable uniformly distributed over the triangular 4 region R
bounded by y = 0, x = 3 and y = x . Find the correlation
coefficient rxy. 3 1 4. X and Y are two random variables having
density function f(x,y) = (6 - x - y ) , 8 0 < x < 2, 2 <
y < 4. Find (i) P ( X < 1 Y < 3 ) (ii) P( X + Y < 3 )
(iii) P( X < 1 / Y < 3 ). 5. Given the joint distribution of
X and Y. Y/X 0 1 2 0 0.02 0.08 0.10 1 0.05 0.20 0.25 2 0.03 0.12
0.15 Obtain (i) marginal distribution and (ii) the conditional
distribution of X given Y = 0 . 6. A statistical investigation
obtains the following regression equations in a survey. X Y 6 = 0
and 0.64X + 4.08 = 0. Here X = age of husband and Y = age of wife.
Find (i) Mean of X and Y (ii) Correlation coefficient between X and
Y and (iii) sy = S.D. of Y if sx = S.D. of X = 4. 7. Given the
joint density function f(x,y) = cx(x-y), 0 < x 0 , evaluate k. e
- l lx p xq x - y , 14. The joint pmf of the random variables X and
Y is p(x,y) = y!(x - y )! y = 0,1,2, x, x = 0,1,2, .. where l >
0 , 0 P 1, p+q = 1 are constants.Find the marginal and conditional
distributions. 15. Two dimentional random variables aX and y have
joint pdf f(x,y) = 8xy, 0 < x < y < 1: 0 otherwise. Find
(i) marginal and conditional distributions (ii) Whether X and Y are
independent? 16. The joint pdf of the random variable (X,Y) is
given bykxy,0 < x < 1,0 < y < 1 f(x,y) = where
0,otherwise
k is a constant. (i) Find the value of k. (ii) Find P(X+Y 0, y
> 0 f(x,y) = , find f(y/x) and f(Y/X = x). 0, otherwise 18. If y
= 2x -3 and y = 5x + 7 are the two regression lines, find the
values of x and y . Find the correlation coefficient between x and
y. Find an estimate of x when y = 1. 19. If the independent random
variables X and Y have the variances 36 and 16 respectively , find
the correlation coefficient between ( X + Y ) and ( X Y ). 20. From
the following data , find (i) the two regression equations (ii) the
coefficient of correlation between the marks in Economics and
Statistics (iii)the most likely marks in statistics when marks in
Economics are 30 Marks in 25 28 35 32 31 36 29 38 34 32 Economics
Marks in 43 46 49 41 36 32 31 30 33 39 Statistics 21. Two random
variables X and Y have joint density function
xy ,0 < x < 4,1 < y < 5 .Find E(X) , E(Y), E(XY), E(
2X + 3Y ), V(X), f(x,y) = 96 0, elsewhere V(Y), Cov(X,Y).What can
you infer from Cov(X,Y). 22. The joint probability density function
of the two dimentional random variable is 8 xy,1 < x < y <
2 f(x,y) = 9 0, otherwise (i)Find the marginal density functions of
X and Y. (ii)Find the conditional density function of Y given X =
x. kx,0 x 2 2k,2 x 4 23. X is a continuous random variable with pdf
given by f(x) = 6k - kx,4 x 6 0, elsewhere Find the value of k and
also the cdf 0f f(x). 24. If the joint pdf of a random variable
(X,Y) is given by xy f(x,y) = x 2 + ,0 x 1,0 y 2 , find the
conditional densities of X given Y and 3 Y given X.
25. Let X and Y be random variables having joint density
functions 3 2 2 (x + y ),0 x 1,0 y 1 f(x,y) = 2 .Find the
correlation coefficient rxy. 0, elsewhere
RANDOMPROCESSES 1. Define Stochastic process. 2. Classify Random
Process. 3. What is a continuous random sequence? Give an example.
4. Give an example of stationary random process and justify your
claim. 5. Distinguish between wide sense stationary and strict
sense stationary processes. 6. Prove that the first order
stationary process has a constant mean. 7. What is a Markov
processes. 8. Define Markov chain and one step transition
probability 9. Give an example of Markov Processes. } 10. Find the
invariant probabilities for Markov chain {X n ; n 1 with state
space S = [0,1]0 1 and one-step TPM P = . 1/ 2 1/ 2 11. What is
stochastic matrix? When is said to be regular? 12. Define
irreducible Markov chain and state Chapman Kolmorgow theorem. 13.
What is meant by Steady state distribution of Markov chain? 14.
State the Postulates of Poisson process. 15. State any two
properties of Poisson process. 16. What will be the super position
of n independent Poisson processes with respective average rates
l1,l2, ln ? 17. Define auto correlation function and state any two
of its properties. 18. Define autocorrelation function and prove
that for a WSS process RXX (-t) = RXX (t). 19. Define cross
correlation function and state any two of its properties. 20. Given
the autocorrelation function for a stationary ergodic process with
no periodic 4 . Find the mean and variance of the process.
components is R(t) = 25 + 1 + 6t 2 4 , find the 21. If the
autocorrelation function of a stationary process is RXX(t) = 36 + 1
+ 3t 2 mean and variance of the process.
22. Define birth and death process. PART- B 1. Define a random
(stochastic) processes. Explain the classification of random
process. Give an example to each class. 2. Consider a random
process y(t) = x(t) cos(w0t + q) where x(t) is distributed
uniformly in (-p,p) and w0 is a constant. Prove that y(t) is wide
sense stationary. 3. Two random processes X(t) and Y(t) are defined
by X(t) = Acoswt + Bsinwt and Y(t) = Bcoswt - Asinwt. Show that
X(t) and Y(t) are jointly Wide-Sense stationary if a and B are
uncorrelated random variables with zero means and same variances
and w is constant. 4. Show that the process X(t) = A coslt + B
sinlt where A and B are random variables is wide sense stationary
if (i) E(A) = E(B) = 0 (ii) E(A2) = E(B2) and (iii) E(AB) = 0. 5.
Show that the random process X(t) = A sin(wt+j) where j is a random
variable uniformly distributed in (0,2p) is (i) first order
stationary (ii) stationary in the wide sense. 6. For a random
process X(t) = Y sinwt, Y is an uniform random variable in ( -1, 1
). Check whether the process is wide sense stationary or not.
7. If X(t) is a wide sense stationary process with
autocorrelation function RXX(t) and if Y(t) = X(t+a) X(t-a) , show
that RYY(t) = 2RXX(T+2a) - RXX( T-2a). 8. If X(t) = 5 sin(wt+j) and
Y(t) = 2cos(wt+j) where w is a constant q+j = p/2 and is a random
variable uniformly distributed in (0,2p), find RXX(t) , RYY(t) ,
RXY(t) and RYX(t). Verify two properties of autocorrelation
function and cross correlation function. 9. If the process {N (t);t
0 is a Poisson process with parameter lt, obtain P[N(t) = n and } ]
E[N(t) .] 10. Find the mean and autocorrelation of of Poisson
process. 11. State the Postulates of Poisson process. Discuss any
two properties of Poisson processes. 12. Prove that the sum of two
independent Poisson process is also a Poisson Process. 13. Let X be
a random variable which gives the interval between two successive
occurrences of a Poisson process with parameter l. Find out the
distribution of X. 14. Given a random variable Y with
characteristic function j(w) = E(eiwy) and a random process defined
by X(t) = cos(lt + y). Show that {X (t )} is stationary in the wide
sense if j(1) = j(2) = 0. 15. 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 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 train
on the third day (ii) the probability that he drives to work in the
long run. 16. Three boys A, B and C are throwing a ball to 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. 17. The transition probability matrix of a
Markov Chain {X n } , n = 1, 2, 3, having 3 states
0.1 0.5 0.5 1, 2 and 3 is P = 0.6 0.2 0.2 and the initial
distribution is 0.3 0.4 0 .3
P(0) = ( 0.7, 0.2, 0.1
) . Find (i) P(X 2= 3) (ii) P(X 3= 2, X 2= 3, X 1= 3, X 0= 2). }
18. The one step TPM of a Markov chain {X n ; n = 0,1,2 having
state space
S = [1,2,3 ]is
0.1 0.5 0.5 P = 0.6 0.2 0. 2 and the initial distribution is
P(0) = ( 0.7, 0.2, 0.1 ) . Find (i) P(X2 0.3 0. 4 0 .3 = 3) (ii)
P(X3 = 2, X2 = 3, X1 = 3, X0 = 0) P(X2 = 3/ X0 = 1) . 19. Let {X n
} , n = 1, 2, 3, be a Markov chain with state space S = 0,1,2 and
one step 1 0 0 1/ 4 1/ 2 1/ 4 Transition Probability Matrix P = 0 1
0 (i) Is the chain ergodic? (ii) Find the invariant
probabilities.20. Show that random process X(t) = A cos(wt+q) is
wide sense stationary if a and w are constants and q is uniformly
distributed random variable in ( 0,2p). 21. For a random process
X(t) = Y , Y is uniformly distributed random variable in ( -1, 1 ).
Check whether the process is wide sense stationary or not.
(at ) n-1 , n = 1,2,... n+1 22. The process P{X (t) = n =}(1 +
at ) at ,n = 0 1 + at Show that X(t) is stationary. 23. If X(t) = A
cosly + B sinlt ; t 0 is a random process where A abd B are
independent N(0, s2) random variables, examine the stationary of
X(t). 24. Let X(t) = A cos(wt+Y) be a random process where Y and w
are independent random variables. Further the characteristic
function j of Y satisfies j(1) = 0 and j(2) = 0 , while the density
function f(w) of w satisfies f(w) = f(-w). Show that X(t) is wide
sense stationary . 25. The autocorrelation function of a stationary
random process is RXX(t) = 9 16 + . Find the variance of the
process. 1 + 6t 2 } 26. Let {X(t) ;t 0 be a random process where
X(t) = total number of points in the interval (0,t) = k , say 1if k
is even = Find the autocorrelation function of X(t). -1if k is
odd
QUEUEING THEORY PART-A 1. For ( M/M/1 : ( / FIFO) model, write
down the littles formula. 2. For ( M/M/c) : ( N/ FIFO) model, write
down the formula for (a) average number of customers in the
queue.(b) Average waiting time in the system. 3. In a given M/M/1,
queue, the arrival rate =7 customers/ hour and service rate h = 10
customers/ hour. Find P ( X > 5) where X is the number of
customers in the system. 4. What is the effect arrival rate for
M/M/1/N queuing system 5. In the usual notation of an M/M/1 queuing
system if =12 per hour and =12 per hour system. 6. Write pollaczck
khintchinine formula and explain the notations. 7. What are the
basic characteristics of queuing process? 8. Obtain the steady
state probabilities of on ( M/M/1); ( N/ FIFO) queuing System. 9.
In a given ( M/M/1 : ( / FCFS) =0.6 what is the probabilities that
the queue contain 5 or more customer 10. What is the effective
arrival rate for ( M/M/1 : ( A/ FCFS) queuing model when =2and =5
=24 per hour .find average the number of customers in the
PARTB1. Obtain the
steady state probabilities for ( M/M/1) : ( N/ FCFS) queuing
Model. 2. A petrol pump station has 2 pumps . The service times
follow the Exponential distribution with mean of 4 min and cars
arrive for service is Poisson process at the rate of cars per hour
. Find the probabilities that a customer has to wait for service .
What is the probabilities that pumps remain ideal.
3. In a given ( M/M/1) queuing System the ag arrival is 4
customer per minute =0.7 what are (i) ) mean number of customer Lq
in the queue (ii) mean number of customer standing in the
queue(iii) Probabilities that the server is ideal (iv) mean waiting
time W6 in the system. 4. There are three typists in an office
.Each typist type on any of 6 letter per hr. If letters arrive for
being typed at the rate of 15 letters per hr. (i)What fraction of
time all the typist will be busy? ( ii)What is the average number
of letters waiting to be typed? (iii) What is the average time of
letter has to spend waiting and for being typed? 5. A 2person
barber shop has 5 chair to accommodate the waiting customer
potential customer who arrive when all 5 chairs are foll Leave
without entering the barber shop customers arrive at the average
rate 4 per hr. and spend on average of 12 min in the barbers chair.
compute P0, P1, P7 and Lg 6. In the railway marshalling yard goods
trains arrive at a rate of 30 trains per day. Assume that the int
distribution er arrival time follows the Exponential distribution
and the service time distribution is also Exponential with an
average 36 minutes . Calculate the following a. The mean square
size b. The probabilities that the queue size exceeds 10 if the
input of trains increase to an average of 33 per day , what will be
the change in the above quantities? 7. Arrival rate of telephone
calls at telephone booth are according to Poisson distribution with
an average time of 12 min. between two consecutive calls arrival.
The length of telephone call is assumed to be exponentially
distributed with mean.4 min a. Determine the probabilities that
person arriving at booth will have to wait. b. Find the average
queue length that is formed from time to time c. The telephone
company will install second booth when convinced that an arrival
would expect to have wait at least 5 min for the phone find their
increase in follows of arrival which will justify second booth.
d. What is the probabilities that an arrival will be wait for
more then 15 min before thje phone is free. 8. Patients arrive at
clinic according to Poisson distribution at a rate of 30 patients
per hr. The waiting room does not accommodate more than 14patients
.Examine time per patient is exponential with mean rate of 20 per
hr. a. what is the probabilities that an arriving patient will not
wait? b. What is the effective arrival rate 9. Automatic car W has
facility operator with only one boy .Cars arrive according to
Poisson distribution with mean of 4 cars per hr and may wait in the
facilities parking Lot if the boy is busy . If the service time for
all cars is content and equal to 10 min .Determine Ls Lq, Ws and
Wq. 10. Derive pollaccekkhinchine formula for the average number of
customer in the M/M/I queuing system