MA6453-Probability and Queueing theoryUNIT – 1

RANDOM VARIABLESPART A

Random variable1. If A and B are independent events with P(A) = 0.4 and P(B) = 0.5 Find P(A∩B). 2. What is the probability that a leap year selected at random contains 53 Sundays?3. Define Random Variable .

Discrete Random variable4. Define Discrete Random Variable with an example.5. Define Mean or Expected value of a discrete random variable.6. Consider the experiment of tossing a coin 3 times. Let X be the random variable

giving the number of heads obtained. We assume that the tosses are independent and the probability of heads is p, then find P[X=0]

7. Balls are tossed at random into 50 boxes. Find the expected number of tosses required to get the first ball in the fourth box

8. Define Probability mass function.

9. Find C, if P[X=n]= C(23¿¿n ,n=1,2,3….

10. If X is a discrete random variable taking the values 1,2,3…… with probability mass

function defined by P[X=x] = Cx

x ! for x=1,2,3…. Then find the value of c.

11. If random variable X assumes three values -1. 0,1 with probabilities 13, 12, 16

respectively , then find the probability distribution of Y= 3X+1.

12. The probability mass of a random variable X is given by P x ( x )={12x=−1

14x=0

18x=1

18 x=2

Determine the p.m.f of Y if Y=2X2+1.13. The number of hardware failures of a computer system is a week of operations has

the following p.m.f

No. of failures 0 1 2 3 4 5 6

Probability

0.18 0.28

0.25 0.18

0.06 .04

0.01

Find the mean of the number of failures in a week.

14. If the range 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.

15. Let X be a random variable with probability mass function P(X=x)=

{ x10

, x=1,2,3,4

0 , oterwise } compute P(X<3) and E( 12X )

Continuous Random variable16. Define Continuous Random Variable with an example.17. Define Mean or Expected value of a continuous random variable.18. Define expectation of a random variable.19. Determine the expectation of X.20. If Var(X) = 4,find Var(3X+8),where X is a random variable.21. Define Probability density function.22. If X and Y are independent randon variable with variance 2 and 3. Find the variance

of 3X+4Y.23. A continuous random variable X follows the probability law f(x)=Ax2 ,0 ≤ x ≤ 1.

Find A.24. A continuous random variable X that can assume any value between x=2 and x=5

has a density function given by f ( x )=k (1+x). Find P(X<4).25. Check whether the given function f(x) = 6x (1-x) , 0 ≤ x ≤ 1 is density function or

not.

26. Test Whether f ( x )={|x|:−1≤ x≤10 :ot h erwise

can be the probability function of a

continuous random variable.27. A continuous random variable X has probability density function as

f ( x )=3 x2 ,0≤ x≤1. Find K such that P(X>K)=0.05.28. Find the value of (i)C and (ii) mean of the following distribution given by

f ( x )={ C (x−x2)0ot herwise

for 0<x<1

29. A continuous random variable X has the probability density function f ( x )=¿ find a

and P (X<4 ) .

30. Given the random variable X with density function f ( x )={ 2x ,0<x<10 , elsew here

. Find

the probability density function of Y=8 X3.

31. Let X be a random variable with pdf f ( x )={ 2x ,0<x<10 , elsew here

. Find the pdf of

Y= (3 X+1 ).32. Define Cumulative Distribution Function of a discrete random variable X.33. Define Cumulative Distribution Function of a continuous random variable X

Moments and M.G.F34. Define n th moment about origin for a random variable.35. Define central moment of a random variable?36. What do you mean by MGF? Why it called so?37. Define moment generating function

38. If a R.V X has the moment generating function Mx(t)= 3

3−t compute E(X2)

39. Find the moment generating function of the random variable X having the p.d.f. f(x) = 1/4; -2 ¿ x ¿2

Binomial Distribution40. Define Binomial distribution.41. Define Binomial frequency distribution.42. For a binomial distribution mean is 6 and S.D. is √2. Find the first two terms of a

distribution.43. If a fair coin is tossed twice, find P(X≤1), where X denotes the number of

heads in each experimentPoisson Distribution

44. Define Poisson Distribution and state two instances where Poisson Distribution may be successfully employed.

45. Define Poisson frequency distribution.46. Define poisson distribution.47. If X is a poisson variate such that P(X=2)=9P(X=4)+90P(X=6),Find the variance.48. If X and Y are poisson variate then show that X-Y is not poisson variate.

Geometric Distribution49. Define Geometric distribution.

Exponential Distribution50. Define Exponentialdistribution.51. The time (in hours) required to repaira machine is exponentially distributed with

parameter λ =1/252. What is the probability that a repair takes atleast 10 hours given that its duration

exceeds 9 hours.

Gamma Distribution53. Define Gamma distribution.54. Define general Gamma distribution

Normal Distribution55. X is a normal variate with mean 1 and variance 4. Y is another normal variate

independent of X with mean 2 and variance 3. What is the distribution of X+2Y.Part B(8 Marks)Random variable

1. Let X be a R.V with E (X )=1∧E [X (X−1 ) ]=4.Find Var(X2

¿ and Var

(2−3 X ) .Discrete Random Variable

2. A discrete random variable X has the following probability mass function:

X 0 1 2 3 4 5 6 7P(X)

0 k 2k 2k

3k K2 2k2 7k2+k

Find i) k ii)P(X ≥ 6 ), P( X ¿ 6 ) iii) P( 0¿X ¿ 5) iv) the distribution function of X v) If P(X ≤ x )¿1/2, find the minimum value of X.

3. The monthly demand for Allwyn watches is known to have the following probability distribution

Demand 1 2 3 4 5 6 7 8probability 0.08 0.1

20.19 0.24 0.16 0.1

00.07 0.04

Determine the expected demand for watches. Also compute the variance.

4. A fair coin is tossed three times. Let X be the number of tails appearing. Find the probability distribution of X. And also calculate E(X).

5. A random variable X takes the values -2, -1, 0 and 1 with probabilities

18, 18,

14∧1

2, respectively. Find and draw the probability distribution function.

6. The probability mass function of random variable X is defined as P (X=0 )=3c2 ,P (X=1 )=4c−10c2 ,P ( x=2 )=5c−1 where C>0 and P (X=r )=0 if r≠0,1,2 Find i) the value of C ii) P(0<X<2/X>0) iii) the

distribution function of X iv) the largest value of X for which F ( x )< 12

7. A random variable X has the following probability distribution

X -2 -1 0 1 2 3P(x) .1 K .2 2k .3 3k(i) Find the value of k (ii) P(X<1) (iii)P(-1<X≤2) (iv) E(X)

8. A random variable X has the following probability distribution.

x 0 1 2 3 4 5 6 7P(x) 0 K 2K 2K 3K K 2 2K 2 7K 2+

K Find: 1) The value of K

2) P (1.5<X<4.5/⎸X>2 )3) P (X<6 ) , P (X ≥6 ) and

4) t h e smallest value of n for which P (X ≤n )> 12

.

5) find P (X<2 ) ,P (X>3 ) ,P (1<X<5 ) .9. A random variable X has the following probability distribution.

x 0 1 2 3 4 5 6 7 8P(x) a 3a 5a 7a 9a 11a 13a 15a 17a

i) Find the value of aii) Find P(X<3), P(0<X<3), P(X>3).iii) Find the distribution function of X.10. A random variable X has the following probability distribution.

x 0 1 2 3 4P(x) k 3k 5k 7k 9k

Find k, P(X>3) and P(0<X<4).

11. If P(X=x)={Kx ,x=1,2,3,4,5repre sents a p .m. f0 , ot herwise

(i)Find ‘K’(ii)Find P(X being a prime number)(iii)Find P{1/2<X<5/2/X>1}(iv)Find the distribution function.

12. Show that for the probability function P[X=x] =1

x(x+1), x=1,2,3… .. E[x]

does not exist.13. Consider the experiment of tossing a coin 4 times. Define X=0, if 0 or 1 head

appearsX=1, if 2 heads appear; X=2 if 3 or 4 heads appear. Find P[X ¿. Also find the mean and variance of X .

14. If X has the following distribution F ( x )={0 x<1

13 f∨1≤x<4

12for 4≤x<6

56for 6≤x<10

1 for x≥10

Find

a. The probability distribution of X.b. P(2<X<6)c. Mean of Xd. Variance of X.

Continuous Random Variable15. Find k such that the probability density function of a continuous random variable X

is given by f(X) =kx(x-1) ,0¿x ¿ 3.16. A continuous random variable X has probability density function given by f(x) =3x2,

0 ≤ x ≤ 1. Find k such that P(X>k) = 0.0517. If ‘X’ has the probability density f ( x )=K e−3 x , x>0,find K,

P(0.5≤ X ≤1) and the mean of X.

18. The p.d.f of a continuous r.v X is f ( x )=K e−|x|. Find K and F [x]19. A continuous random variable X that can assume any value between X=2 & X=5

has a density function given by f(x)= k(1+x). Find P[x<4].20. If X is a continuous random variable with pdf f(x)=kx2 ,-1<x<1 then find i) the

value of k ii) the mean and variance of X

iii) P(13<X<4 ¿

21. Prove that X with f(x) =12

( x−1 ) ,−1<x<1 is a continuous random variable also

find the mean and variance of X.22. A continuous r.v has the pdf of f ( x )=K x4 ;−1<x<0.F ind the value of K and

also P(X>−12

/X<−14 )

23. If f ( x )={x e−x2

2 , x≥00 , x<0

(i) Show that f(x) is a pd.f of a continuous random variable X.

(ii) Find its distribution function. F(x).

24. If the probability density function of X is given by f ( x )={2 (1−x ) for 0< x<10 ot herwise

(i) Show that E (X r )= 2(r+1 )(r+2)

(ii) Use this result to evaluate E ¿]

25. In a continuous distribution,the probability density is given by f ( x )=Kx (2−x ) ,0< x<2. Find K,mean,variance and the distribution function.Find the pdf of X and evaluate P(⎹ X⎸≤1) using both the pdf and cdf.

26. The sales of a convenience store on a randomly selected day are X thousand dollars, where X is a RV with distribution function of the following form,

F ( x )={0 x<0

x2

20≤ x<1

k ( 4 x−x2 )1≤ x<21x ≥2

. Suppose that this convenience store’s total sales on

any given day are less than $2000 1) Find the value of k 2) Let A and B be the events that tomorrow the store’s total sales are between 500 and 1500 dollars, and over 1000 dollars , respectively . Find p(A) and p(B) 3) Are A and B independent events.

27. Experience has shown that while walking in a certain park, the time X (in mins), between seeing two people smoking has a density function of the form

f ( x )={ λxe− x , x>00elsew here

i) calculate the value of λ.

ii) find the distribution function of Xiii) What is the probability that Jeff, who has just seen a person smoking, will see another person smoking 2 to 5 minutes? In atleast 7 minutes?

28. If the density function of a continuous random variable X is given by

f ( x )={ ax ,0≤x ≤1a ,1≤ x≤2

3a−ax ,2≤ x≤30 , ot herwise .

i) Find the value of a .ii) The cumulative distribution 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?

29. A random variable X has the following probability distribution

X -2 -1 0 1 2 3P(x) .1 K .2 2k .3 3k

i)Find the value of k ii) find the mean of X iii) find P(−2<X<2), P(X<2),P(-2<X<3) iv) find the cumulative distribution of X.

30. The cumulative distribution function (cdf) of a random variable X is given by

F ( x )={0 , x<0

x2 ,0≤ x<1 /2

1− 325

(3−x )2 , 12≤x<3

1 , x ≥331. The cumulative distribution function of a random variable X is

F ( x )=1−(1+x )e− x, x>0. Find the probability density function of X, mean and

variance of X.32. Let X be a discrete RV whose cumulative distribution function is

F ( x )={0 for x<−3

16 for−3≤x<6

12for 6≤x<10

1 for x≥10

i. Find P(X≤4¿ ,P (−5<X ≤4 ) ,ii. Find the probability distribution of X.

Moments & M.G.F33. Find the moment generating function of the RV X whose probability function

P(X=x)=12x , x=1,2 ,….Hence find its mean.

34. A random variable X has density function given by f ( x )={ 1Kfor 0<x<K

0otherwise Find

a. M.g.fb. r th momentc. Mean

d. Variance35. A continuous random variable X has the pdf f ( x )=K x2 e−x , x ≥0. Find the rth

moment of X about the origin. Hence find mean and variance of X.

36. Let the random variable X has the pdf f ( x )={ 12e

−x2 , x>0

0 , ot herwise Find the moment

generating function,mean and variance of X.37. Find the moment generating function of a random variable X whose rt hraw moment

is (r+1)! 2r

38. Find the MGF of the two parameter exponential distribution whose density function is given by f(x) = λ e− λ(x−a) , x>a and hence find the mean and variance.

39. If the pdf of X is given by f(x)= 2(1-x) for 0<x<1 find its r t h Moment . Hence evaluate E ¿].

40. Find MGF corresponding to the distributionf (θ )=12e

−θ2 ,θ>0 and hence find its

mean and variance

41. Let X be a random variable with pdf f ( x )=13e

−x3 , x ≥0.Find the moment

generating function of X and hence find its mean and variance.42. The pdf of a random variable X is given by ( x )=kx (2−x ) ,0≤x ≤1 . Find k ,

mean , variance and rth moment .43. Find the M.G.F. of the random variable X having the probability density function

f ( x )={ x4 e−x2 , x>0

0elsew hereAlso deduce the first four moments about the origin.

44. Find the M.G.F. of the random variable X having the probability density function

f ( x )={ x

4 ex2

, x>0

0elsew here

Also deduce the first four moments about the origin.

45. Find the moment generating function and rth moment for the distribution whose pdf is f ( x )=K e− x ,0≤ x≤∞. Hence find the mean and variance.

46. If X is a discrete RV with probability function P ( x )= 1K x , K is constant , X=

1,2,3,…. Find i) the MGF of X ii) Mean and Variance of X47. A random variable X has pdff ( x )=k x2 e−x , x>0 Find the r t h moment of X

about origin. Hence , find the mean and variance.48. A continuous random variable has the pdf f(x) given by f ( x )=ce−¿ x∨¿−∞<x <∞.¿

find the value of c and moment generating function of X.

49. A rectangular distribution f ( x )= 12a

,−a< x<a. Show that the moments of odd

order are zero and μ2 r=a2 r

2 r+150. Show that all the odd order central moment vanish and the even order central

moments are given by μ2n=3

2n+3 , n=0,1,2,3,….

51. Find the MGF of the random variable ‘X’ having the pdf

f ( x )={ x for0<x<12−x for1< x<2

0ot herwise and hence find the mean and variance.

52. Find the probability distribution of the total number of heads obtained in four tosses of a balanced coin. Hence obtain the MGF of X, mean of X and variance of X.

53. Prove that the moment generating function of the sum of a number of independent random variables is equal ti the product of their respective moment generating functions.

54. Let X be a continuous R.V with probability density function f ( x )={ x e−x x>00ote rwise

find (i) the cumulative distribution function of X (ii) Moment generating function Mx(t) of X (iii) P(X<2) (iv) E(X)

Binomial Distribution55. Determine the binomial distribution for the mean is 4 and the variance is 3.56. Find m.g.f of the Binomial Distribution. Also find the mean and variance.57. 6 dice are thrown 729 times.How many times do you expect atleast three dice to

show 5 or 6?58. An irregular 6 faced dice is such that the probability that it gives 3 odd numbers in7

throws is twice the probability that it gives 4 odd numbers in 7 throws.How many sets of exactly 7 trials can be expected to give no odd number out of 5000 sets?

59. Out of 800 families with 4 children each, how many families would be expected to have (i) 2 boys and 2 girl; (ii)atleast 1 boy;(iii) atmost 2 girls(iv)children of both genders. Assume equal probabilities for boys and girls.

60. The mean and variance of a binomial variate are8 and 6.Find P(X≥2¿

61. A discrete R.V X has moment generating function M X (t )=( 14+ 3

4et)

5

Find

E (X ) ,Var (X )∧P(X=2)62. A coin is biased so that a head is twice as likely to appear as a tail. If the coin is

tossed 6 times. Find the probabilities of getting i) Exactly 2 heads ii) Atleast 3 heads iii) Atmost 4 heads.

63. If m things are distributed among a men and b women, show that the probability

that the number of things received by men is odd is 12 [ (b+a)m−(b−a )m

(b+a)m ]64. Show that the largest value of a binomial distribution is

n4

.

Poisson Distribution65. A machine manufacturing screws is known to produce 5% defective. In a random

sample of 15 screws, what is the probability that there are (1)exactly 3 defectives (2)not more than 3 defectives.

66. In a book of 520 pages, 390 typographical errors occur. Assuming Poisson’s law for the number of errors per page, find the probability that a random sample of 5 pages will contain no error.

67. A manufacturer of television sets known that of an average 5% of his product is defective. He sales television in consignment of 100 and guarentees that not more than 4 sets will be defective. What is the probability that a television set will fail to meet the guaranteed quality?

68. Determine the mean, variance and moment generating function of a random variable X following Poisson distribution with parameter λ.

69. Derive Poisson distribution from Binomial distribution.70. In a large consignment of electric bulbs, 10 percent are defective. A random sample

of 20 is taken for inspection. Find the probability the i) all are good bulbs ii) at most there are 3 defective bulbs iii) exactly there are 3 defective bulbs.

71. A car hire firm has 2 cars which it hires out day by day. The number of demands for a car on each day follows a Poisson distribution with mean 1.5. calculate the proportion of days on which

(1) Neither car is used(2) Some demand is refused.

72. A manufacturer of pins knows that 2% of his products are defective. If he sells pins in boxes of 100 and guarantees that nor more than 4 pins will be defective, what is the probability that a box fail to meet the guaranteed quality?

73. If X is a Poisson variate such that P (X=2 )=9 P (X=4 )+90P (X=6 ) Find i)

mean andE [X 2] ii) P (X ≥2 )74. If X and Y are independent Poisson variates , show that the conditional distribution

of X given X+ Y is binomial.Geometric Distribution

75. A die is tossed until 6 appears . What is the probility that it must be tossed more than 5 times?

76. Suppose that a trainee solider shoots a target in an independent fashion. If the probability that the target is shot in any one shot is 0.7,

(1) What is the probability that the target would be hit on tenth attempt?(2) What is the probability that it takes him less than 4 shots?(3) What is the probability that it takes him an even number of shots?

77. Find the m.g.f of a geometric distribution. Also find the mean and variance.78. State and prove memoryless property of geometric distribution.79. Suppose that a trainee soldier shoots a target in an independent fashion . If the

probability that the target is shot on any one shot is 0.7, i) What is the probability that the target would be hit on tenth attempt? ii)What is the probability that it takes him less than 4 shots? iii) what is the probability that it takes him an even number of shots?

80. If X is a geometric random variable then find the probability that X is divisible by 3.81. If the probability that an applicant for a driver’s license will pass the road test on

any given trial is 0.8. what is the probability that he will finally pass the test i) on the fourth trial and ii) in less than 4 trials?

82. An office has four phone lines. Each is busy about 10% of the time. Assume that the phone lines act independently. i) what is the probability that all four phones are busy? ii) What is the probability that at least two of them are busy?

83. Let P(X = x) = ( 34 )( 1

4 )x−1

, x=1,2,3 ,…. be the probability mass function of a

R.V X compute (i) P(X > 4) (ii)P(X>4/X>2) (iii) E(X) (iv) Var(X)

Uniform Distribution84. Let ‘X’ be a uniform random variable over (0,1).Determine the moment generating

function of X and hence find variance of X.85. Determine the moment generating function of uniform distribution and hence find

mean and variance.86. A random variable ‘X’ has the uniform distribution over (-3,3)

compute(i)P(X<2).P(|X|<2),P(|X-2|<2),(ii)Find K for which P(X>K)=1/387. Starting at 5.00a.m every half hour there is a flight from San Francsisco airport to

Los Angels International Airport.Suppose that none of these planes is completed

sold out and that they always have room for passengers.A person who wants to fly to L.A arrives at the airport at a random time between 8.45 a.m and 9.45 a.m. Find the probability that she waits (1)at m ost 10 min (2)atleast 15 min.

88. Trains arrive at a station at 15 minutes intervals starting at 4 a.m. If passenger arrives at the stop at a random time that is uniformly distributed between 7 and 7 :30 a.m, find the probability that he waits i) Less than 5 minutes ii) At least 12 minutes

89. Trains arrive at a station at 15 minutes intervals starting at 4 a.m. If passenger arrive at the station at a time that is uniformly distributed between 9.00 and 9 :30 a.m, find the probability that he has to wait for the train for

1) Less than 6 minutes2) More than 10 minutes

90. If X1, X2 be independent random variables each having geometric distribution show that the conditional distribution of X1 given X1+X2 is uniform.

91. Let X be uniformly distributed R.V over [-5,5]. Determine (i) P(X ≤ 2) (ii) P(|X|>2) (iii) cumulative distributive function of X (iv) Var(X)

Exponential Distribution92. Determine the moment generating function of an exponential random variable and

hence find its mean and variance.93. State and prove the memoryless property of exponential distribution.94. Find the moment generating function of the exponential distribution

f ( x )= 1

c e−xc

,0≤ x<∞,c>0Hence find its mean and S.D

95. The daily consumption of milk in excess of 20,000 gallons is approximately exponentially distributed with θ=3000.The city has the daily stock of 35,000 gallons.What is the probability that of two days selected at random,the stock is insufficient for both days.

96. The time(in hours) required to repairs a machine is exponential,distributed with parameter λ=1/2.

a) What is the probability that the repair time exceeds 2 hours?b) What is the conditional probability that a repair takes atleast 10h given

that its duration exceeds 9h?97. The mileage which car owner get with a certain kind of radial tyre is a random

variable having an exponential distribution with mean 40,000km. Find the probabilities that one of these tires will last (i) atleast 20,000 km (ii) atmost 30,000 km

Gamma Distribution

98. Describe Gamma distribution.Obtain moment generating function[MGF].Hence or otherwise compute the first four moments.

99. Find the mean and variance of gamma distribution and hence find mean and variance of exponential distribution.

100.Prove that the sum of the independent gamma variates is a gamma variate.101.The daily consumption of a milk in a city,inexcess of 20,000 L is exponentially

distributed as a gamma variate with the parameters α=2∧λ=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?

102.The daily consumption of milk in a city in excess of 20,000 Lis approximately distributed as a gamma variate with the parameters ∝ =2 and λ =1 /10,000 .The city has daily stock of 30,000 litres . What is the probability that the stock is insufficient on a particular day ?

Normal Distribution103.A RVX has a normal distribution with 10. If the probability that the RV will take

on a vlue less than 82.5 is 0.8212. what is the probability that it will take on a value greater than 58.3?

104.The weakly wages of 1000 workmen are normally distributed around a mean of Rs. 70 and with standard deviation of Rs. 5. Estimate the number of workers whose weakly wages will be between Rs. 69 and Rs. 72; more than Rs. 72.

105.Assume that the reduction of a person’s oxygen consumption during a period of Transcendental Meditation (T.M) is a continuous random variable X normally distributed with mean 37.6 cc/mm and S.D 4.6 cc/mm. Determine the probability that during a period of T.M a person’s oxygen consumption will be reduced by i) at least 44.5 cc/mm.

Ii) at most 35.0 cc/mm iii) anywhere from 30.0 to 40.0 cc/mm.106.The marks obtained by a number of students in a certain subject are assumed to be

normally distributed with mean 65 and standard deviation 5. If 3 students are selected at random from this group, what is the probability that two of them will have marks over 70?

107.If X is a random variable following normal distribution with μ=1, σ=2. Find P ¿ / X>0¿

UNIT – 2TWO DIMENSIONAL RANDOM VARIABLES

PART –A 2- Marks2-D RANDOM VARIABLE

56. Define two dimensional random variables57. Define joint probability distribution58.Define joint probability distribution of two random variables of X and Y

59.Define Marginal probability distribution60. Define conditional probability distribution61. Write any two properties of joint cumulative distributive function.62. X and Y are independent random variable with variance 2 and 3 . Find the Var

(3X+4Y)63. Let X and Y be two independent R.Vs with Var(X) = 9 and Var(Y) = 3. Find

Var(4X-2Y+6).DISCRETE RANDOM VARIABLE

64.Define two dimensional discrete random variables65.Define the term joint probability mass function66.Define the term joint probability mass function67.Define marginal probability mass function

CONTINUOUS RANDOM VARIABLE68.Define two dimensional continuous random variables69.Define the term joint probability density function

70. Define conditional probability density function71. If joint pdf of a random variable (X,Y) is given by f(X,Y) =ke−(x2+ y2) ,X > 0 ; Y >

0 . Find the value of K.72. The joint probability density function of a random variable (X,Y) is f(x,y) =

k e−(2 x+3 y) , x≥0 , y≥0. find the value of k.73. If X and Y have the joint density function f ( X , Y ) =

{34+xy ,0<¿ x<1 ,0< y<1

0 , ot herwise Find f (y/x )

74. The joint pdf of R.V (X,Y) is given as f(x,y) = {1x,0< y<x ≤1

0 , oterwise } find the

marginal pdf of YCOVARIANCE

75. Define covariance76. Show that Cov 2 (X,Y) ≤ Var (X) Var (Y)

CORRELATION77. Give any two properties of correlation coefficient78. What is meant by the term correlation 79. The two equations of the variables X and Y ARE X = 19.13 -0.87Y and Y = 13.64 -

0.54X . Find the correlation coefficient between X and Y REGRESSION

80. Distinguish between correlation and regression

81. Explain the term regression 82. Define regression of Y on X83. What are regression lines84. What are the types of regression analysis85. Find the acute angle between the two lines of regression

TRANSFORMATION OF VARIABLE86. If X is uniformly distributed in (-π/2 , π/2 ) find the probability distribution

function Of Y =tan XPART- B 8 MARKS

DISCRETE RANDOM VARIABLE108.The joint probability mass function (PMF) of X and Y is

X\Y

0 1 2

0 0.1 0.04 0.021 0.08 0.20 0.062 0.06 0.14 0.30

Compute the marginal PMF X and of Y , P[ X ≤ 1 , Y ≤ 1 ] and check if X and Y are independent.

109.The joint distribution of ( X , Y ) where X and Y are discrete is given in the following table

Verify whether X, Y are independent.

110.The two dimensional random variable (X ,Y) has the joint density function f(x,y) = x+2y/27 X = 0,1,2 ; y = 0,1,2 . Find the conditional distribution of Y given X =x . Also find the conditional distribution of X given Y = 1.

111.Three balls are drawn at random without replacement from a box containing 2 white , 3 red and 4 black balls . If X denotes the number of white balls drawn and Y denotes the number of red balls drawn. Find the joint probability distribution of (X ,Y).

112.The joint probability mass function of (X ,Y) is given by P (x,y ) = k (2x+3y) , x= 0,1,2 ; y=1,2,3 .Find all the marginal and conditional probability distributions. Also find the probability distribution of (X+Y) and P[X+Y > 3 ]

113.Given is the joint distribution X and Y

Y\ 0 1 2

Y\X 0 1 20 0.1 0.4 0.11 0.2 0.2 0

X0 0.02 0.08 0.101 0.05 0.20 0.252 0.03 0.12 0.15

Obtain i) Marginal distribution and ii) The conditional distribution of X given Y = 0

114.Let X and Y have the joint p.d.f

Find i) P( X + Y > 1 )ii) the probability mass function P ( X = x) of the R.V. X iii) P (Y =1 / X =1 )iv) E [XY]

CONTINUOUS RANDOM VARIABLE

115.Find the costant k such that f(x,y) ={k ( x+1 )e− y ,0<x<1 , y>00ot herwise

is a joint pdf of

the continuous R.V(X,Y). are X and Y independent R.Vs

116.If the joint pdf of a 2D R.V (X, Y) IS given by f(x,y) ={ 2 ,0<Y <¿ x<10 ,OTHERWISE

Find

the marginal density function of X and Y are independent.

117.Given the joint probability density f(x,y) ={ 23 ( x+2 y )

;0<¿ x<1 ,0< y<1

0 ,ot herwise Find

the marginal densities , conditional density of X given Y =y and P (X≤ ½ / Y =1/2 )118.The joint probability density function of a two dimensional random variable (X ,Y )

is given by F(x,y) = xy2 + x2/8 , 0 ≤ x ≤2 , 0≤ y ≤ 1.Compute P(X > 1) ,P( Y< 1/2), P( X>1 / Y <1/2) ,P( Y <½ / X>1), P( X< Y) and P (X+Y≤1).

119.If X and Y have the joint p.d.f. f ( x , y )={34+xy ,0< x<1 ,0< y<1

0 , ot herwise Find f

(y/x) and P( Y > ½ / X = ½ )120.The joint p.d.f.of R.Vs X and Y is given by

a. Find P[ X > 1 / Y =y]

121.The joint density function of the random variable (X,Y) is given by f(x,y) =

{8 xy ;0<¿ x<1 ,0< y<x0 , ot herwise Find i) Marginal density of Y

ii)Conditional density of X/Y = y iii)P(X <1/2 )122.If the joint p.d.f. of random variables of X and Y is

f(x,y) ={xe− x(1+ y) ;∧x>0 , y>00 ,ot h erwise

find f ( y/x) and E [Y/X =x]

123.Find the marginal and conditional densities if f(x,y) = k(x3 y + x y3 ) , 0≤x≤2,0≤y≤2.

124.Let the joint pdf of R.V (X,Y) be given as f(x,y) ={Cx y2 ,0≤ x≤ y≤10 , ot h erwise

determine

(i) the value of C (ii) the marginal pdf of X and Y (iii)the conditional pdf f(x/y) of X given by Y=y.

COVARIANCE125.Find the covariance of the two RV’S whose p.d.f. is given by

f(x,y) ={2;∧x>0 , y>0 , x+ y<10 , ot herwise

126.A joint probability mass function of the discrete R.Vs X and Y is given as

P(X=x,Y=y)={x+ y32

, x=1,2 , y=1,2,3,4

0 , otherwise compute the covariance of X and Y.

127.The random variable [X,Y] has the following joint p.d.f. f(x,y) =

{1/2(x+ y);0≤∧x≤2 ,0≤ y ≤20 , ot herwise

1. Obtain the marginal distributions of X .ii)E[X] ,E[X2]iii)Compute co –variance [X,Y].

CORRELATION AND REGRESSION128..If X,Y,Z are uncorrected 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 .129.Two random variables X and Y are defined as Y = 4X + 9. Find the correlation

coefficient between X and Y.

X\Y 0 1 20 0.1 0.04 0.061 0.2 0.08 0.122 0.2 0.08 0.12

Y\X -1 0 0-1 1/6 1/3 1/60 0 0 01 1/6 0 1/6

130.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).

131. Calculate the coefficient for the following heights of fathers X their sons Y.

132.. Find the

correlation coefficient for the following data

133.Find the coefficient of correlation between industrial production and export using the following data

134.Suppose the joint probability mass function of a bivariate RV (X,Y) is given by

PXY(x,y) ={13; for (0,1 ) , (1,0 ) ,(2,1)

0 , otherwise i) Are X and Y independent ii) Are X

and Y uncorrelated135.Two independent random variables X and Y are defined by f(x) =

{4 ax ;0≤∧x ≤10 , ot herwise F(y)= { 4 ;0≤Y ≤1

0 , ot h erwi se Show that U=X+Y and V=X-Y are

correlated.

136.The joint p.d.f of a bivariate RV (X,Y) is given by f(x,y) ={kxy ;0<¿ x , y<10 , ot herwise

where k is constant .i)find k ii)find P[X+Y<1] iii)are X and Y independent random variables explain ?

137.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.

138.Find the coefficient of correlation and obtain the lines of regression from then data given below

X 50 55 50

60 65 65

65 60 60

50

Y 11 14 13

16 16 15

15 14 13

13

139.The regression relation of two random variables of X and Y are x-(5/4)y +(33/5)=0 and Y – (20/9)x +(107/9) =0. The std. deviation of X is 3 . Find the std. deviation of Y.

140.A statistical investigator 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 Yiii)σY =S.D. of Y if σX S.D. of X =4.

141.the equations of the two lines of regression of y on x and x on y are respectively , 7x-16y +9=0; 5y-4x-3=0, calculate the coefficient of correlation , X and Y .

142.If y=2x-3 and y=5x+7 are the two regression lines find the mean values of x and y . Find the correlation coefficient between x and y . Find an estimate of x when y=1.

143.Can Y = 5+2.8 X and X=3 -0.5Y be the estimated regression equations of Y on X and X on Y respectively ? Explain your answer with suitable theoretical arguments.

144.The two lines regression are 8x-10y+66=0 ; 40x-18y-214=0 .The variance of x is 9. Find i)the mean values of x and y ii)Correlation coefficient between x and y

145.For 10 observations in price X and supply Y the following data were obtained .

∑ X =130, ∑Y =220, ∑ X2 =2288 , ∑Y 2=5506 ,∑ XY=3467. Obtain the line of regression of Y on X And estimate the upply when the price is 16 units.

146.Following table gives the data on rainfall and discharge in a certain river.Obtain the line of regression of y on x.

X 5 6 7 7 8 9 70 2Y 7 8 5 8 2 72 9 1

x: 10 14 8 2 26 30y: 18 12 4 6 0 6

Productioin(X): 55 56 58 59 60 60 62Export(y): 35 38 37 39 44 43 44

Marks inECO. X

25 28 35 32 31 36 29 38 34 32

Statistics 43 46 49 41 36 32 31 30 33 39

Rainfall(inches)(X): 1.53

.78 2.60 2.95 42

Discharge (1000c.c)(y):

33.5

36.3 40.0 45.8 53.5

147.Find the coefficient of correlation and obtain the lines of regression from the

data given below

148.If 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 BombayAverage Price 65 67S.D.of Price 0.5 3.5

S.D. of the difference between the price at Madras & Bombay is 3.1

149.Suppose that the 2D RVS X and Y has the joint p.d.f f(x,y)=

{x+ y ;0<¿ x<1 ,0< y<10 , ot herwise Obtain the correlation coefficient between X and Y.

TRANSFORMATION OF VARIABLE150.Let Y=X2 ,find the pdf of Y if X is a uniform random variable over (-1,2).

151.If X is a uniform random variable in trhe interval (-2,2).Find the pdf of Y=X2

152.If X is a random variable uniformly distributed in (0,1),find the pdf of Y=sin X.Also find the mean and variance of Y.

153.If the joint p.d.f. of (X,Y) is givenby fXY(X,Y)=X+Y,0≤X,Y≤1 Find the p.d.f. of U =XY.

154.If X and Y are independent random variables each following N(0,2) find the probability density function of Z=2X+3Y.

155.If X and Y are independent RVS with density function fX(x)=e− x U(X) and fY(y)=2

e−2 y U(y). Find the density function of Z=X+Y.156.If X has exponential distribution with α, find the probability function of Y=logX .

157.The joint pdf of the continuous R.V (X,Y) is given as f(x,y) = {e−( x+ y) , x>0 , y>00 , oterwise

find the pdf of the R.V U=XY

UNIT -3

MARKOV PROCESSES AND MARKOV CHAINSPART –A

Random process87. Classify the random processes.88. If {X(s,t)} is a random process , what is the nature of X (s,t) when i)s is fixed ii)t

is fixed?Stationary process

89. Find the mean and variance of the stationary random process whose auto correlation function is given by RXX(τ ) =18 +2/6+τ 2

90. A random process has the autocorrelation function R xx ( τ )=4 τ2+6τ2+1

, find

the mean square value of the process.91. Consider the random process {X(t),X(t)=cos (t+ф)} where ф is uniform in (-π/2,

π/2). Check whether the process is stationary.92. Define strict sense and wide sense stationary process.93. Define auto correlation function of the stochastic process . What does it measure ?94. Check whether the random process X(t) =A cos 2πt is first order stationary .95. The random process X(t) is given by X(t) = Y cos(2πt), t > 0, where Y is a

R.V with E(Y) = 1. Is the process X(t) stationary?96. Let X(t) =B Sin wt , where B is a random variable with zero mean and w is a

constant .Then find the mean of X(t).97. Prove that the auto correlation function is an even function 98. Prove that the first order stationary random process has a constant mean

Markov process99. What is stochastic matrix ? when it is said to be regular?100.Define MARKOV PROCESS and MARKOV CHAIN.

101.If the transition probability matrix of a Markov chain is [ 0 11/2 1/2] find the

limiting distribution of the chain.102.At an intersection a working traffic light will be out of order the next day with

probability 0.07 and an out of order traffic light will be working the next day with probability 0.88. Let Xn=1 if on day ‘n’ the traffic light will work ; Xn=0 if on day ‘n’ the traffic light will not work.Is { X n ; n=0,1,2,..} a Markov chain ? If so , write the transition probability matrix.

x: 2 4 65 9 0 1 2 74y): 26 5 39 45 5 2 80 208

103.The one step transition probability matrix of a Markov chain with states (0,1) is

given by P=[0 11 0] . Is it irreducible Markov chain?

104.The transition probability matrix of a Markov chain {Xn}, n =1,2,3,… with three

states 0,1,2 is P =3/ 4 1/4 01/4 1/2 1/40 3/4 1/4

with intial distribution P(0) =(1/3 1/3

1/3 ).Find P (X3=1 ,X2=2,X1=1,X0=2).105.Define irreducible markov chain.106.State Chapman-kolmogorov theorem.

Poisson process107.Examine whether the poisson process {X(t)} given by the law P[x(t)=r] =e− λt

(λt)r/r! ,r= 0,1,2,…is covariance stationary.108.Show that the sum of two independent poisson process is a poisson process.109.Define ergodic process.110.Suppose that a customers arrive at a bank according to poisson process with a

mean rate of 3 per minute , find the probability that during a time interval of two minutes more than four customers arrive ?

111.Obtain the moment generating function of the poisson process.112.Derive the autocorrelation function for a Poisson process with rate λ.

113.State any four properties of poisson process.114.State the Postulates of the poisson process.115.Prove that sum of two poisson process is also a poisson process.116.Prove that the interarrival time of a poisson process with parameter λ has an

exponential distribution with parameter 1λ

.

117.Is the difference of the two poisson process is a poisson process. Explain.118.If {X(t)} and Y{(t)} are two independent poisson process, show that the

conditional distribution {X(t)} given {X(t)+Y(t)} is binomial.PART –B

Stationary process158.Given a random variable Y with characteristic function ф(w)=E [e iwy] and a random

process defined by X (t) =cos [λt+y] show that [X(t)] is stationary in the wide sense of ф (1) =ф(2)=0.

159.Consider a random process y(t) =X(t) cos (ω0+θ) where X(t) is wide sense stationary random process .θis a random variable independent of X(t) and is distributed uniformly in (-π, π ) and ω0is constant. It is assumed that X(t) and θ are independent. Show that Y(t) is a wide-sense stationary.

160.If X(t) =Y cos t +Z sin t for all t where Y and Z are independent binary random variables ,each of which assumes the values -1 and 2 with probabilities 2/3 and 1/3 respectively ,prove that {X(t)} is wide sense stationary.

161.Show that then process X(t) = A COS λt +B sin λt (where A and B are RVS )is WSS if i)E(A) =E(B)=0 ii) E(A2) =E(B2) and iii)E(AB)=0.

162.A stochastic process is described by X(t) =A sin t +B cos t where A and B are independent random variables with zero means and equal standard deviations show that the process is stationary of the second order.

163.Consider a random process X(t) defined by X(t) = U cos t + (V+1) sin t where U and V are independent random variables for which E(U) = E(V) = 0 ; E(U2 ) = E(V2) =1 Find i) Auto covariance function of X(t) ii) Is X(t) wide – sense stationary ? Explain your answer.

164.If X(t) = sin (ω t +Y ) where Y is uniformly distributed in ( 0,2π ) Show that X(t) is wide –sense stationary.

165.A random process X(t) is defined as X(t) =A cos (ωt+θ) where ω and θ are constants and A is a random variable . Determine whether X(t) is a wide –sense stationary process or not.

166.Show that the random process X(t) = cos (t+ф) where ф is a random variable uniformly distributed in ( 0, 2π ) is i)first order stationary ii) stationary in the wide sense iii)Ergodic (based on first order or second order averages)

Markov process167.Let { Xn ; n = 1,2,3,…} be a Markov chain on the space S={1,2,3} with one step

transition probability matrix P=0 1 0

1/2 0 1/21 1 0

i) Sketch the transition diagram

ii) Is the chain irreducible ? Explain 168.Consider a Markov chain {Xn : n = 0, 1, 2,…} having state space S = {1, 2} and

one-step TPM P=[ 410

610

810

210 ] (i) draw a transition diagram. (ii)is a chain

irreducible? (iii) is the state-1 ergodic? Explain. (iv) Is the chain ergodic? Explain169. Let { Xn ; n = 1,2,3,…} be a Markov chain on the space S={1,2,3} with

one step transition probability matrix P=0 1 0

1/2 0 1/21 1 0

and initial probability

distribution P(X0 = i) =13

, i = 1,2,3,… compute (i) P(X3 = 2,X2 = 1, X1=2/X0 =1) (ii)

P(X3 = 2,X2 = 1/ X1=2,X0 =1) (iii) P(X2=2/X0 =2) (iv) Invariant probabilities of the Markov chain.

170.Consider the Markov chain with 2 states and transition probability matrix P =

[3/4 1/41/2 1/2] .Find the stationary probabilities of the chain.

171.Find P(n) for the homogeneous markov chain with the following transition

probability matrix P(1−a ab 1−b) where 0<a<1 ,0<b<1.

172.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 as 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 iff a 6 appeared . Find i) the probability that he takes a train on the 3rd day. ii) the probability that he drives to work in a long run.

173.Three boys A,B,C are throwing a ball each other .A always throw the ball to B and B always throw the ball to C ,but C is just likely to throw the ball to B as to A . S.T. the process is Markovian. Find the transition matrix and classify the states.

174.A raining process is considered as two state Markov chain , If it rains, it is considered to be state 0 and if it does not rain, the chain is in state is 1. The

transition probability of the Markov chain is defined as P[0.6 0.40.2 0.8] Find the

probability that it will rain for 3 days from today assuming that it will rain after 3 days. Assume the initial probabilities of state 0 and state 1 as 0.4 and 0.6 respctively.

175.A person owning a scooter has the option to switch over to scooter ,bike or a car next time with the probability of ( 0.3,0.5,0.2) If the transition probability matrix is 0.4 0 .3 0.30.2 0.5 0 .3

0.25 0.25 0.5 what are the probabilities vehicles related to his fourth

purchase ?176.Assume that a computer system is in any one of the three states busy, idle and under

repair respectively denoted by 0,1,2.Observing its state at 2pm each day, we get the

transition probability matrix as P=0.6 0 .2 0.20.1 0.8 0 .10.6 0 0.4

Find out the 3rd step transition

probability matrix . Determine the limiting probabilities.177.An Engineer analyzing a series of digital signals generated by a testing system

observes that only 1 out of 15 highly distorted signals followed a highly distorted signal with no recognizable signal , whereas 20 out of 23 recognizable signals

follow recognizable signals with no highly distorted signals between .Given that only highly distorted signals are not recognizable. Find the fraction of signals that are highly distorted .

Poisson process178.Suppose that 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 min i) exactly 4 customers arrive and ii) more than 4 customers arrive.

179.In a village road , buses cross a particular place at a Poisson rate of 4 per hour. If a boy starts counting at 9.00a.m. i) What is the probability that his count is 1 by 9.30.a.m.? ii) What is the probability that his count is 3 by 11.00.a.m.?

180.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.

181.Derive Poission process with rate λ and hence find its mean . It Poisson process stationary ? Explain.

182. Let X(t) and Y(t) be two independent poisson processes with parameters λ1 and λ2 respectively. Find (i) P(X(t) + Y(t)) = n, n= 0,1,2,3,…, (ii) P(X(t) –Y(t) = n , n =0,±1,±2,…

183.If the process { N (t) ; t ≥ 0 } is a poisson process with parameter λ obtain P[N (t) = n ] and E[ N (t)].

184. Explain the properties of Binomial process.185.Derive mean of the poisson process.186.Derive autocorrelation of the poisson process.187.State the postulates of a poisson process. State its properties and establish the

additive property for the poisson process.

UNIT -4QUEUEING THEORY

Part A119.What are the basic characteristics of a queuing system?120.Average number of customers in the queue.121.Average waiting time in the system 122.Define (i) balking and (ii) reneging of the customers in the queueing system 123.Consider an M/M/1 queuing system. If λ=6 and μ=8, find the probability of at

least 10 customers in the system.124.For an (M/M/1) :( ∞/FIFO) queue with λ=4/hr, μ=6 / hr, find the average queue

length.

125.In an (M/M/1):(∞/FIFO) queue with inter-arrival time 12 minutes and service time 4 minutes , find the average length of the queue that forms from time to time .

126.For (M/M/1) :( ∞/FIFO) model, write down Little’s formula.127.What is the probability that a customer has to wait for more than 15 minutes to

get his service completed in an (M/M/1) queuing system if λ = 6 per hour and μ = 10 per hour.

128.Consider an (M/M/c) queuing system. Find the probability that an arriving customer is forced to join the queue.

129.What is the probability that an arrival to an infinite capacity 3 server Poisson

queuing system with and Po = 1/9 enters the service without waiting?130.Find the traffic intensity for an (M/M/c) :( ∞/FIFO) queue with λ=10 / hr, μ =

15 / hr and 2 servers.131.For an (M/M/2) :( ∞/FIFO) queue with λ = 1/6, μ = ¼, find the probability of an

empty system.132.What are Little’s formulas for an idle system in an (M/M/c) :( ∞/FIFO) queue?133.Define effective arrival rate for an (M/M/1) :( k/FIFO) queue.134.In an (M/M/1) :( k/FIFO) queue with k = 5, λ = 5 / hr, service time = 10

min/person, find the percentage of time the system is idle.135.For an (M/M/c):(N/FIFO) queue write down the formula for 136.Write down the formula for an idle system in an (M/M/c) :( k/FIFO) queue.137.For an M/M/C/N FCFS (C<N) queueing system, write the expressions for P0 and

PN.

Part B 188.Explain KENDALL’S notation . What are the assumptions are made for simplest

queuing model?189.Draw the state diagram of a birth –death process and obtain the balance

equations hence find the limiting distribution of the process.190.Customers arrive at a watch repair shop according to a Poisson process at a rate

of one per every 10 minutes and the service time is an exponential random variable with mean 8 minutes.

i. Find the average number of customers Ls in the shopii. Find the average time a customer spends in the shop Ws

iii. Find the average number of customers in the queue Lq

iv. What is the probability that the server is idle?191.A T.V. repairman finds that the time spent on his jobs has an exponential

distribution with mean 30 minutes. If he repairs sets in the order in which they come in and if the arrival of sets is approximately Poisson, with an average rate

of 10 per 8 hour day, what is the repairmen’s idle time each day? How many jobs are ahead of the average set brought in?

192.Consider a single-server queue where the arrivals are Poisson with rate λ = 10/hour. The service distribution is exponential with rate μ = 5/hour. Suppose that customers balk at joining the queue when it is too long. Specifically, when there are ’n’ in the system, an arriving customer joins the queue with probability

1n+1 . Determine the steady-state probability that there are ‘n’ customers in the

system.193.Self service system is followed in a super market at a metropolis. The customer

arrivals occur according to a Poisson distribution with mean 40 per hour. Service time per customer is exponentially distributed with mean 6 minutes.

i. Find the expected number of customers in the system.ii. What is the percentage of time that the facility is idle

194.People arrive to purchase cinema tickets at the average rate of per minute, and It takes an average of 7.5 seconds to purchase a ticket. If a person arrives 2 mins before the picture starts and it takes exactly 1.5 minutes to reach the correct seat after purchasing the ticket, a)can he expect to be seated for the start of the picture? b)What is the probability that he will be seated for the start of the picture?

195.Customers arrive at a one man barbershop according to a Poisson process with a mean inter arrival time of 20 minutes.Customers spend an average of 15 minutes in the barber chair.If an hour is used as a unit of time ,then

i. What is the probability that a customer need not wait for a hair cut?

ii. What is the expected number of customers in the barbershop and in the queue?

iii. How much time can a customer expect to spend in the barbershop?

iv. Find the average time that the customer spend in the queue?v. What is the probability that there will be 6 or more customers

waiting for service?196.Arrival rate of telephone calls at a telephone booth is according to Poisson

distribution with an average time of 9 minutes between two consecutive arrivals. The length of Telephone call is assumed to be exponentially distributed with mean 3 minutes.

i. Determine the probability that a person arriving at the booth will have no wait.

ii. Find the average Queue length that forms from time to time

iii. What is the probability that an arrival will have to wait for more than 15 minutes before the phone is free?

197.A duplicating machine maintained for office use is operated by an office Assistant who earns Rs.5/- per hour .The time to complete each job varies according to an exponential distribution with mean 6 minutes. Assume a Poisson input with an average arrival rate of 5 jobs per hour. If an 8-hour day is used as a base, determine a) the percentage idle time in the system b) the average time a job is in the system c) the average earning per day of the assistant.

198.A one man barber shop takes exactly 25 minutes to complete one haircut. If customers arrive at the barber shop in a Poisson fashion at an average rate of one every 40 minutes, how long on the average a customer spends in the shop? Also find the average time a customer must wait for service

199.A supermarket has two girls ringing up sales at the counters . If the service time for each customer is exponential with mean 4 minutes and if people arrive in a poisson fashion at the counter at the rate of 10 per hour then calculate i)The probability of having to wait for service ii)The expected percentage of idle time for each girl iii)The waiting time of a customer in a system

200.On every sunday morning a dental hospital renders free dental service to the patients.As per the hospital rules 3 dentists who are equally qualified and experienced will be onduty then ittakes on an average 10 minutes for a patient to get treatment and the actual time taken as known to vary approximately exponentially around theis average . The patients arrive acoording to the Poisson distribution with an average of 12 per hour.The hospital management wants to investigate the following i)The expected number of patients waiting in the queue.ii)The average time that a patient spends at the hospital

201.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) What fraction of time will the entire typist be busy? b) What is the average number of letters waiting to be typed?

202.A general insurance company has three claim adjusters in its branch office. People with claims against the company are found to arrive Poisson fashion, at an average rate of 20 per 8 hour day. The amount of time that an adjuster spends with a claimant is found to have exponential distribution with a mean service time 40 minutes. Claimants are processed in the order of their appearance. Assuming that the company works 5 days a week .How many hours a week can an adjuster expect to spend with Claimants? How much time on an average does a claimant spend in the branch office?

203.A petrol pump station has 4 petrol pumps. The service time follows an exponential distribution with mean of 6 minutes and cars arrive for service in a poisson process at the rate of 30 cars per hour. Find the probability that no car is

in the system. What is the probability that an arrival will have to wait in the queue? Find the mean waiting time in the system.

204.A petrol pump station has 4 pumps. The service time follows the exponential distribution with mean of 6 min and cars arrive for service in a Poisson process at the rate of 30 cars per hour .what is the probability that an arrival would have to wait in line ?For what percentage of time would a pump be idle on an average?

205.Patients arrive at a clinic according to Poisson distribution at a rate of 30 patients per hour.The waiting room does not accommodate more than 14 patients. Examination time per patient is exponential with mean rate of 20 per hour. i)Find the effective arrival rate at the clinic. ii)What is the probability that an arriving patient will not wait? iii)What is the expected waiting time until a patient is discharge from the clinic?

206.A barber shop has 2 barbers and 3 chairs (totally 5 chairs) for waiting customer’s .Assume that customers arrive in Poisson fashion at a rate of 5 per hour and that each barber services customers according to exponential distribution with mean of 15 minutes .Further, if a customer arrives and there are no empty chairs in the shop he will leave .Find the steady state probabilities.

207.A one person barber shop has 6 chairs to accommodate people waiting for a haircut. Assume that customers who arrive when all the 6 chairs are full leave without entering the barber shop. Customers arrive at the rate of 3 per hour and spend an average of 15 minutes in the barber’s chair. Compute (i)P0 (ii) Lq (iii) P7 (iv) Ws

UNIT VNon – Markovian Queues and Queue Networks

PART A138.Write Pollaczek – Khintchine formula and explain the notations.139. M/G/1 queueing system is Markovian. Comment on this statement.140.Consider a service facility with two sequential stations with respective service

rate of 3/min and 4/min. The arrival rate is 2/min. What is the average service time of the system, if the system could be approximated by a two stage Tandem queue?

141.Define Series queues.142. Define Queue networks.143.State arrival theorem.144.What do you mean by bottleneck of a network?145.When we say a node is bottle neck of a queueing system?146.Define a open Jackson network.147.Define Open networks.

148.Define Closed networks.149.Distinguish between open and closed networks.

PART – B

208.Derive Pollaczek – Khinchine formula.209.Derive the Pollaczek – Khinchine formula for M/G/1 queueing model210.Discuss an M/G/∞ FCFS queueing system and hence pollaczek-Khintchine(p-k)

mean value formula. Deduce also the mean system size for the M/M/1/∞:FCFS queueing system from the P-K formula.

211. A car manufacturing plant uses one big crane for loading cars into a truck. Cars arrive for loading by the crane according to a Poisson distribution with a mean of 5 cars per hour. Given that the service time for all cars is constant and equal to 6 minutes determine LS , Lq, WS and Wq.

212.An automatic car wash facility operates with only one bay. Cars arrive according to a Poisson distribution with a mean of 4 cars/hr. and may wait in the facility’s parking lot if the bay is busy. Find LS , Lq, WS and Wq if the service time is constant and equal to 10 minutes

i. follows uniform distribution between 8 and 12 minutes.

i. follows normal distribution with mean 12 minutes and S.D 3 minutes.

ii. follows a discrete distribution with vales 4,8 and 15 minutes with corresponding probabilities 0.2, 0.6 and 0.2.

213.A police department has 5 patrol cars. A patrol car breaks down and repairs service one every 30 days. The police department has two repair workers, each of whom takes an average of 3 says to repair a car. Breakdown times and repair time are exponential. Determine the average number of patrol cars in good condition. Also find the average down time for a patrol car that needs repairs.

214.A repair facility is shared by a large by a large number of machines for repair. The facility has two sequential stations with respective rates of service 1 per hour and 3 per hour. The cumulative failure rate of all the machines is 0.5 per hour. Assuming that the system behavior may be approximated by a two-station tandem queue. Findi) The average number of customers in both stationii) The average repair timeiii) The probability that both service stations are idle.

215.In a heavy machine shop the overhead crane is 75% utilized. Time study observations gave the average singing time as 10.5 minutes with a standard deviation of 8.8 minutes . What is the average calling rate for the services of the

crane and What is the average delay in getting service ?If the average service time is cut to 8 minutes with a standard deviation of 6 minutes how much recduction will occur on average in the delay of getting served?

216.An average of 120 students arrive each hour(inter-arrival times are exponential) at the controller office to get their hall tickets. To complete this process, a candidate must pass through three counters. Each counter consists of a single server. Service times at each counter are exponential with the following mean times: counter1, 20 seconds: counter2, 15 seconds and counter 3, 12 seconds. On the average, how many students will be present in the controller’s office.

217.Consider a queueing system where arrivals are according to a poisson distribution with mean 5/ hour.Find expected waiting time in the system if the service time distribution is i)Uniform time t=5 min to t=15 min ii) Normal with mean 3 min and variance 4 min 2

218.Consider a system of two servers where customers from outside the system arrive at server arrive at server 1 at a Poisson rate 4 and at server 2 at a Poisson rate 5. The service rates in 1 and 2 are respectively 8 and 10. A customer upon completion of service at server 1 is equally likely to go to server 2 or to leave the system; whereas a departure from server 2 will go 25% of the time to server 1 and will depart the system otherwise. Determine the limiting probabilities, average number of customers and average waiting time of a customers and average waiting time of a customer in the system.

219.Explain the open and closed Jackson networks.220.Write short notes on the following: Queue networks ,Series queues ,Open

networks ,Closed networks ,Bottleneck of a network

221.Consider a open queueing network with parameter values shown below:Facility j sj uj α j i=1 i=2 i=3

j=1 1 10 1 0 0.1 0.4j=2 2 10 4 0.6 0 0.4j=3 1 10 3 0.3 0.3 0

i) find the steady state distribution of the number of customers at facility 1, facility 2 and facility 3.ii) find the expected total number of customers in the systemiii) find the expected total waiting time for a customer.

222.Explain how queueing theory could be used to study computer networks.223.Jackson network with three facilities that have the parameters given below

P11=0,P12=0.6,P13=0.3,P21=0.1,P22=0,P23=0.3 , P31=0.5,P32=0.4,P33=0,µ1=10,µ2=10,µ3=10,C1=1,C2=2,C3=1,r1=1,r2=4,r3=3. Find i. The total arrival rate at each facility

ii. Expected number of customers in the entire system

iii. Expected time a customer spends in the system.