MA2262 PROBABILITY & QUEUING THEORY UNIT 1 2 Marks 1. If Var(X) = 4, find var(3X+8) , where X is a random variable. (A.U, Model) 2. X and Y are independent random variables with variance 2 and 3 . Find the variance of 3X + 4Y.[A.U. April/May ,2003] 3. Let X be a R,V with E(X) = 1, and E(X(X-1)] = 4. Find Var X and Var(2-3X).[A.U. May,2000] 4. Let X be a R.V taking values –1 , 0 and 1 such that P(X = -1 ) = 2P(X=0) =P(X = 1) . Find the mean of 2X-5. 5. A continuous random variable X has probability density function given by f(x) = 3x2, 0 ≤X≤ 1. Find K such that P(X > K ) = 0.05(A.U. Model ) 6. The p.d.f of a random variable X is f(x) = 2x, 0 < x < 1, find the p.d.f of Y = 3X + 1.[A.U. A/M 2003] 7. A random variable X has the p.d.f f(x) given by f(x) = C X e-x, If X > 0 0, If X ≤ 0 Find the value of C and c.d.f of X 8. The first four moments of a distribution about X 4 are 1, 4 10 and 45 3 =Respectively . Show that the mean is 5 , variance is 3, 4 = 26.[A.U.0 and
Transcript
MA2262 PROBABILITY & QUEUING THEORY
UNIT 1
2 Marks
1. If Var(X) = 4, find var(3X+8) , where X is a random variable. (A.U, Model)
2. X and Y are independent random variables with variance 2 and 3 . Find the variance of 3X + 4Y.[A.U. April/May ,2003]
3. Let X be a R,V with E(X) = 1, and E(X(X-1)] = 4. Find Var X and Var(2-3X).[A.U. May,2000]
4. Let X be a R.V taking values –1 , 0 and 1 such that P(X = -1 ) = 2P(X=0) =P(X = 1) . Find the mean of 2X-5.
5. A continuous random variable X has probability density function given by f(x) = 3x2, 0 ≤X≤ 1. Find K such that P(X > K ) = 0.05(A.U. Model )
6. The p.d.f of a random variable X is f(x) = 2x, 0 < x < 1, find the p.d.f of Y = 3X + 1.[A.U. A/M 2003]
7. A random variable X has the p.d.f f(x) given by
f(x) = C X e-x, If X > 0
0, If X ≤ 0
Find the value of C and c.d.f of X
8. The first four moments of a distribution about X 4 are 1, 4 10 and 45
3 =Respectively . Show that the mean is 5 , variance is 3, 4 = 26.[A.U.0 and
mN/D 2004
8. 2 . Find the first two termsFor a binomial distribution mean is 6 and S.D is
of the distribution.[A.U. A/M 2004]
10. Define poisson distribution [A.U. N/D 2005]
11. If X is a poisson vitiate such that P( X = 2) = 9 P( X =4) + 90 P(X =6), find
the variance. A/M 2003]
12. The moment generating function of a random variable is given by
Mx (t) = e3(e – 1) . Find P( X =1)[AU, Model]
13. Find the moment generating function of the generating
distribution.[A.U.Dec.98]
14. Find the moment generating function of a uniform distribution.[A.U,2000]
15. If the random variable X is uniformly distributed over (-1,1), find the density
x/2)Function of Y = sin(
/2) , find the probability distribution/2,16. If X is uniformly distributed in (-
function of y = tan x.[A.U.N/D 2003]
17. Define Gamma distribution.[A.U.N/D 2004]
18. Let X be a R.V with p.d.f given by
f(x) = 2x, 0 < X < 1
0, elsewhere
Find the p.d.f of Y = (3X + 1)[A.U.2000)
19.The life time of a component measured in hours follows weibull = 0.5. Find mean lift = 0.2, distribution with parameter time of the component.
[A.U. April ‘03]
40).[ A.U. Nov’07 ,May ‘03] X =10. Find P(15 = 20 and S.D . 20. A normal distribution has mean
16 Marks
1.Let X be a discrete random variable whose cumulative distribution function is [A.U May 2000]
0 for x < -3
F(X) = 1/6 for 6 X –3
1/2 for 10 X 6
1 for x 10
(a) 4) , P (-5Find P( X < 4 ),X
P( x = -3 ), P( x= 4 )
(b) Find the prob. Mass function of x
2. A random variable X has the following probability function
X: 0 1 2 3 4
P(X) K 3K 5K 7K 9K
(i) Find the value of K
(ii) Find P( X < 3) , P( 03 ), P( X < X < 4 )
( iii)Find the distribution function of X
2. Letr X be a R. V with p.d.f given by
f(x) = 2x, 0 < x < 1
0, elsewhere
Find the pdf of Y = (3X + 1) [A.U. 2000]
3.Find the moment generating function of an exponential random variable and hence find its mean and variance.[ A.U. N/D 2004]
4. A continuous random variable X has the p.d.f f(x) = Kx2 e-x, X 0. fiind the rth moment of X about the origin. Hence find mean and variance of X. [ A.U. A/M 2003]
5. If ‘X ‘ has the probability density f(x) = Ke-3x , X > 1 ) and the mean of X. [A.U. A/M 2004] X 0 , fiind K , P(0.5
6. Let the random variable X have the p.d.f
½ e-x/2
f(x) = 0, otherwise
Find the moment generating function, mean and variance of X.
7.Define Binomial distribution, obtain its MFG , mean and variance.[A.U.N?D 2003,A/M 2004]
9. The probability of a bomb hitting a target is 1/5 , Two boms are enough to destroy a bridge. If six bombs are aimed at the bridge, find the probability that the bridge is destroyed? [ A.U Dec ‘98]
10. Describe Binomial B(n,p) distribution and obtain the moment generating function. Hence compute (i) The first four moment and (ii) the recursion relation for the central moments.[ A.U.A/M 2005]
11. If X and Y are independent poisson random variables, show that the conditional distribution of X given X+Y is a binomial distribution.[A.U. model]
12. If a random variable X has a negative binomial distribution. Obtain the mean and variance of X.[A.U. A/M 2004]
13. Describe negative binomial distribution X follows NB( k,p) where X = number of failures preceding the kth success in a sequence of Bernoulli trials and p = Probability success. Obtain the MGF of X, mean and variance of X.
14. Let ‘X’ be a uniform random variable over 90,1) . Determine the moment generating function of X and hence find variance of X . Given that ‘X’ is a uniform random variable over (0,1) . Hence its p.d.f is given by f(x) = 1/1-0 =1 , 0 < x < 1.
15. Let Y = X2 , find the p.d.f of Y if X is a uniform random variable over (-1,2).
16. If a poisson variate X is such that P( X =1) = 2P(X = 2). Find P( X = 0) and var(X) . If X is a uniform random variable in [-2,2] , find and E[Y].Xthe p.d.f of Y =
17. If x is a uniform random variable in the interval (-2,2) find the p.d.f of Y = X2 [A.U.A/M 2005]
18. Find the moment generating function of an exponential random variable and hence find its mean and variance.[A.U. N/D 2004]
19. If X and Y are independent exponential distributions with parameter 1, find the pdf of U = X –Y. [A.U. Model]
20. The daily consumption of milk in excess of 20,000 gallons is = 3000 . The city has aapproximately exponentially distributed with daily stock of 35,000 gallons. What is the probability that of two days selected at random, the stock is insufficient for both days.[ A.U.A/M 2003]
21. Obtain the moment generating funtion of a Gamma variable X. Hence or otherwise calculate the mean and variance of X.
UNIT 2
2 Marks
1.The joint pdf of two random variables X and Y is given by fxy(x,y) = 1/8x(x-y) ; 0 < x < 2; -x < y <x and otherwise fin fy/x = (y/x)[A.U.model
2. State the basic properties of joint distribution of (X,Y) when X andY are random variables.[A.U.A/M 2005]
3. If the point pdf of (X,Y) is given by
f( x,y) = e-(x+y) 0 find E [XY].[A.U A/M 2005]0, Y, x
4.If X and Y are random variables having the joint density function f(x,y) = 1/8(6-x-y) , 0< x <2, 2 < y <4, find P(X+Y<3)[A.U.A/M 2003]
5.Find the marginal density function of X, if the joint density function of two continuous random variable X and Y is
1 y x f(x) =2(2-x-y), 0
0 , Otherwise
6.If the joint pdf of a 2D R V (X,Y) is given by
f(x,y) = 2, 0< x < 1; 0< y < x <1.
0, otherwise
Find the marginal density function of X and Y
7.If the joint density function of the two R V s ‘X ; and ‘Y ‘ be
f(x,y) 0 0, Y= e-(x+y) X
0, otherwise
Find (i) P( X < 1) and (ii) P( X+ Y < 1) [A.U. N/D 2003]
8. Find ‘K ‘ if the joint probability density function of a bivariate random variable (X,Y) is given by
f(x,y) = K (1-x) (1-y) if 0< x < 4 ; 1 < y < 5
0, otherwise
Var ( X) . Var(Y) [A.U. N/D 2004]9.Show the Cov 2 (X,Y)
10. The two equations of the variables X and Y are x = 19.13 –0.87 y and y = 11.64 – 0.50 x. Find the correlation co – efficient between X and Y .[AU, May ‘99]
11.Find the acute angle between the two lines of regression [A.U A/M 2003]
12.Can Y = 5 + 2.8 x and X = 3 – 0.5 y be the estimated regression equations of Y on X and X on Y respectively? Explain your answer.
[A.U. Nov 2007]
13.The tangent of the angle between the lines of regression of y on x y , find the correlation coefficientx = ½ and x on y is 0.6 and between x and y.
14. Two random variables X and Y are related as Y = 4X+9 . Find the coefficient between X and Y[ A.U. 2007]
15.State the equations of the two regression lines. What is the angle between them.[ A.U N/D 2003]
16.If X and Y are linearly related find the angle between the two regression lines.[A.U A/M 2004]
17.X and Y are independent random variables with variance 2 and 3 . Find the variance of 3X + 4Y .[A.U.A/M 2003]
18. State the central limit theorem for independent and identify distributed random variables.[ A.U A/M 2005]
19. Write the note on ‘ Central limit theorem’ [A.U. N/ D 2005]
20.Distinguish between correlation and regression.
16 Marks
1. The joint probability mass function (PMF) of X and Y is
P(x,y) 0 1 2
X 0 0.1 .04 .02
1 .08 .20 .06
2 .06 .14 .30
1 ) and check if X and Y are independent . [A.U N/ D 2004] 1, Y Compute the marginal PMF X and Y , P(X
2. If the joint density function of the two R V s ‘X ; and ‘Y ‘ be
f(x,y) 0 0, Y= e-(x+y) X
0, otherwise
Find (i) P( X < 1) and (ii) P( X+ Y < 1) [A.U. N/D 2003
3.If X and Y have the joint p.d.f
¾+ xy, 0 < x < 1, 0 < y < 1
f(x,y) = 0 , otherwise
Find f(y/x ) and P( ( y > 12 / X = 1/2 ) A.U 2000,
4. X and Y are two random variable having joint function
f(x,y) = 1/8 (6-x –y ) 0 < x < 2, 2< y < 4
0 , otherwise
Find the (i) P( X < Y1 < 3 ) (ii) P ( X + Y < 3 ) (iii) P ( X < 1/Y < 3 ) [ A.U A/M 2003
5. The joint p.d.f of a R .V (X,Y ) is given by,
f(x,y) = 4xy , 0 < x <1, 0< y < 1
0, otherwise
Find P( X + Y < 1) [ A.U N/D 2005]
6. Find the marginal density function of X , if the joint density function of two continuous random variable X and Y is
f(x,y) 1 y x = 2 ( 2-x –y ), 0
0 , otherwise
7. If X,Y and Z are uncorrelated RV’s with zero mean and S.D 5, 12 and 9 respectively, and if U = X + Y and V = Y + Z , find the correlation coefficient between U and V .[ A.U Model]
8. Find the covariance of the two random variables whose p.d.f is given by [A.U. May 2000]
2 for x > 0 , y> 0 , x+ y < 1
f(x,y)=
0, otherwise
9.Calculate the correlation co-efficient for the following heights ( in inches) of fathers X their sons Y .[A.U. N/D 2004]
X: 65 66 67 67 68 69 70 72
67 68 65 68 75 72 69 71
10.Suppose that the 2D RVs ( X,Y ) has the joint p.d.f
f(x,y) = x+y, 0 < x < 1, 0 < y < 1
0 , otherwise
Obtain the correlation co-efficient between X and Y .
11. Two independent random variables X and Y are defind by
f( x) 1 x = 4ax , 0
0 , otherwise
Show that U = X + Y and V = X – Y are uncorrelated. [A.U. A/M 2003]
12.A statistical investigator obtains the following regression equations in a survey:
X – Y – 6 = 0 and 0.64 X + 0.48 = 0 .
Here X = are of husband and Y = age of wife. Find
(i) Mean of X and Y
(ii) Correlation coefficient between X and Y and
y(iii) y = S.D of X = 4.= A.D . of Y if
13. The random variable [X,Y ] has the following joint p.d.f
f(x,y) = 2 yn 2 and 0 x ½ (x + y ) , 0
0, otherwise
(1) Obtain the marginal distribution of X.
(2) E(X) and E ( X2)
(3) Compute co-variance (X,Y ) [ A.U A/M 2005]
14 . Find the Cor ( x,y ) for the following discrete bi variate distribution
X 5 15
10 0.2 0.4
20 0.3 0.1
15. Find the coefficient of correlation and obtain the lines of regression from the data given below: [A.U. N/D 2003]
X 62
64 65 69 70 71 72 74
Y 126
125 139 145 165 152 180 208
16.Following table gives the data on rainfall and discharge in a certain river. Obtain the line of regression of y on x . [A.U. May,99]
17. For the following data find the most likely price at Madras corresponding to the price 70 at Bombay and that at Bombay corresponding to the price 68 at Madras.
Madras Bombay
Average price 65 67
S.D of price
0.5 3.5
S.D of the difference between the price at Madras and Bombay is 3.1?
18.If X and Y are independent random variables each normally 2 , find the density functiondistributed with mean zero and variance tan-1 (Y/X ). [A.U. Dec 03] X2 + Y2 and of R =
19.If X and Y are independent random variables each following N(0,2) , find the probability density function of Z = 2X + 3Y . [A.U A/M 2003]
20. A random sample of size 100 is taken from a population whose mean is 60 and the variance is 400 . Using CLT , with what probability can = 60we assert that the mean of the sample will not differ from by more than 4 ?(A.U. A/M 2003)
UNIT 3
2 Marks
1.Give an example of Markov process.[A.U. N/D 2003]
2.Consider the Markov chain with tpm:
0.4 0.6 0 0
0.3 0.7 0 0
0.2 0.4 0.1 0.3
0 0 0 1
is it irreducible ? If not find the class. Find the nature of the states. [A.U. A/M 2004]
3.Define Markov chain and one – step transition probability [A.U A/M 2004]
4. What is a Markov chain? When can you say that a Markov Chain is homogeneous?[A.U .N/D 2004]
5. Find the nature of the states of the Markov chain with the transition probability matrix [A.U N/D 2004]
0 1 0
½ 0 ½
0 1 0
6. Define irreducible Markov chain? And state Chapaman – Kolmogorow theorom.[A.U .N /D 2003]
7. What is a Markov process? [A.U.N/D 2005}
18. Find the invariant probabilities for the Markov chain { Xn ; n } with state space S ={ 0,1} and one – step TPM . P = 1 0
½ ½
9.State any two properties of a poisson process. [A.U. A/M 2003]
10. What will be the superposition of ‘n’ independent poisson n?[A.U N/D 2004]2, …,1, processes with respective average rates
11. Define poisson random process. Is it a stationary process Justify the answer?[A.U May 1999]
12.Let A = 0 1
½ ½ be a stochastic matrix check whether it is regular. [A.U June ‘06]
13. If the customers arrive at a counter in accordance with a Poisson Process with a mean rate of 3 per minute, find the probability that the interval between 2 consecutive arrivals is
(i) More than 1 minute
(ii) Between 1 minute and 2 minutes
(iii) 4 minutes or less
14.A radar emits particles at the rate of 5 per minute according to Poisson distribution. Each particles emitted has probability 0.6. Find the probability that 10 particles are emitted in a 4 minute period.[A.U. June ’05 ,Dec ‘07]
16 Marks
1. Prove that a first order stationary random process has a constant mean
[A.U Model]
2. Give an example of stationary random process and justify your claim.[A.U N/D 2005]
3. ) is widet + Show that the random process X( t) = A cos ( sense stationary if A & ’ is uniformaly are constant and ‘ ).[A.U. N/D , 2003 ,2005]distributed random variable in ( 0, 2
4. Consider a Markov chain with 2 states and transition probability matrix
P = ¾ ¼
½ ½
Find the stationary probabilities of the chain.[A.U Moel]
5. Consider a Markov chain with state space { 0, 1 } and the tpm P = 1 0
½ ½
(i) Draw a transition diagram.
(ii) Show that state 0 is recurrent.
(iii) Show that state 1 is transient.
(iv) Is the state 1 periodic? If so, what is the period?
(v) Is the chain irreducible?
(vi)Is the chain ergodic ? Explain.
5. Find P(n) for the homogenous Markov chain with the following transition
Probability matrix P = 1-a a
b 1-b Where 0< b < 1. [A.U. A/M 2004]
6.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 t 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 (1) the probability that he takes a train on the third day . (2) the probability that he drives to work in the long run.[A.U. A/M 2003]
7.Define a Markov chain (MC) . Explain: (i) How you would clarify the states and identify different classes of a MC. Give an example t each class. [A.U. N/D 2004]
8.State the postulates of a poisson processes. State its properties and establish the additive property for the poisson process.[A.U.N/D 2004]
9.The inter arrival time of a poisson process (i.e., ) the interval between two successive occurrence of a poisson process with parameter has and exponential distribution with mean 1/ [A.U A/M 2004 ,2005]
10.The difference of two independent poisson process is not a poisson process.[A.U.A/M 2004,N/D 2003]
11.If the number of occurrence of an event E is an interval of length t is a poisson process {X (t) } with pararmeter and if each occurrence of E has a constant. Probability P of being recorded and the recordings independent of each other, then the number N9t) of the recorded occurrences in t is also poisson process with parameter .[A.U.A/M 2004]
Find the mean and autocorrelation of the poisson process.[A.U A/M 2003]
12.Derive the distribution of poisson process and find its mean and variance.[A.U model]
13Prove that the difference of two independent poisson processes is not a poisson process.[A.U N/D 2003]
and hence find its mean it poisson process stationary? Explain. [A.U .A/M 2005]14. Derive poisson process with rate
UNIT 4
2 Marks
1. What is the prob that a customer has to unit more than 15 min. to get his service =10 per = 6 per hour and completed in a M/M 1 queueing system, if hour?
2. What is the prob. That a customer has to wait more than 15 minutes = /FIFO) queue system if to get his service completed in ( M/M1) : ( =10 per hour?6 per hour and
3. /FIFO) model, write down the little’s formula.[A.U A/M 2003]For (M/M/1) : (
4. Consider an M/M/1 queueing system. Find the probability that at least ‘n’ customers in the system.[A.U. Model]
5. Consider an M/MC queueing system. Find the probability that an arriving customer is forced to join in the queue.[A.U Model,A.U.A/M 2005]
6. = 8, find the probability of atleast 10 customers in the system. =6 and Consider an M/M/1 queueing system. If
7. ? FCFS queue , P = 0.6 .What is the probability that the queue contains 5 or more customers?[A.U.N/D 2005]In a given M/M/1 /
8. In a telephone booth the arrivals are on the average 15 per hours. A call on the average takes 3 minutes. If there are just one phone ( Poisson and Exponential arrival), find the expected number of customer in the booth and the idle time of the booth.
16 Marks
1. What is the average length of the queue, that forms from time to time [A.U. N/D 2004]
1. There are 3 typists in an office. Each typist can type an average of 6 letters per hour. If letters arrive for being typed at the rate of 15 letters per hour.
[A.U .N/D2004]
(i) What fraction of the time all the typists will be busy?
(ii) What is the average number of letters waiting to be typed?
(iii) What is the average time a letter has to spend for waiting and for being typed?
(iv) What is the probability that a letter will take longer than 20 min. waiting to be typed and being typed?
3.A bank has 2 letters working on savings accounts. The first teller handles withdrawals only. The second teller handles deposits only. It has been found that the service time distributions for both deposits and withdrawals are exponential with mean service time of 3 min. per customer. Depositors are found to arrive in a poisson fashion throughout the day with mean arrival rate of 16 per hour. Withdrawals also arrive in a poisson fashion with mean arrival rate of 14 per hour. What would be the effect on the average waiting tine for the customers if each teller could handle both withdrawals and deposits? What could be the effect, if this could only be accomplished by increasing the service time to 3.5 min? [A.u.A/M 2003]
1. A car servicing station has 2 boys where service can be offered simultaneously. Because of space limitations, only 4 cars are accepted for servicing. The arrival pattern is possion with12 cars per day. The service time in both the boys is exponentially distributed with m =8 cars per day. Find the average number of cars in the service station, the average number of cars waiting for service and the average time a car spends in the system.[A.U.A/M2005]
2. A 2-person barbershop has 5 chairs to accommodate waiting customers. Potential customers, who arrive when all 5 chairs are full leave without entering barber shop [[. Customers arrive at the average at the average rate of 4 per hour and spend an average of 12 min. in the barber’s chain compute P0,b P1,P7,and E(W).
3. Obtain the expressions for steady state probabilities of M/M/C queueing system.[A.U 2004]
4. Derive the formula for:
(1) Average number Lq of customers in the queue
(2) Average waiting time of a customer in the queue for (M/M/1): (¥/FIFO)
Model [A.U 2003]
1. Explain the model (M/M/S): (¥ /FIFO), (ie.,) multiple sever with infinite capacity. Derive the average number of customers in the queue (i.e., average queue length: Lf ).[A.U.A/M 2004]
2. Automatic car wash facility operates with only one buy cars arrive according to a Poisson distribution, with a mean of 4 cars per hour and may wait in the facility’s parking lot if the bay is busy. If the service time for all cars is constant and equal to 10 min, determine Ls, Lf,Ws and Wf.[A.U.A/M 2004]
3. A petrol pump station has 2 pumps. The service times follow the exponential distribution with mean of 4 minutes and cars arrive for service is a Poisson process at the rate of 100 cars per hour. Find the probability that a customer has to wait for service. What is the probability that the pumps remain idle?[A.U 2005]
4. In a given M/M/1 queueing system the average arrivals is 4 customers per minute. r=0.7 What are
1. 1. Mean number of customer Ls in the system,2. 2. Mean number of customers Lq in the system.3. 3. Probability that the server is idle.4. 4. Mean waiting time Ws in the system.[A.U 2005]
UNIT 5
2 Marks
1.What is the probability then an arrival to an infinite capacity 3 server poisson
= 2 and Po =1/9 enters the service without/queueing system with
