INTERNAL EXAMINATION – I

SUBJECT : PROBABILITY AND QUEUEING THEORY

BRANCH : II YEAR B.E./B.TECH., - CSE/IT

DATE : 18/01/2011 TIME : 3 HRS. MAX. MARKS : 100

PART – A

ANSWER ALL THE QUESTIONS (20 x 2 = 40)

1. Define discrete random variable and give an example.

2. Write down the properties of probability mass function..

3. Let X be a random variable with E(X) = 1, E[X(X – 1)] = 4. Find Var(X).

4. If X denotes the outcome when a die is tossed , find the mean and variance of X.

x / 3 / 6 / 9
P(x) / 1/6 / 1/2 / 1/3

5. Find E[(1+2X)2] for the following distribution

6. 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).

7. Find the moment generating function for the following distribution where f(x) = 2/3 at x = 1

1/3 at x = 2

0 otherwise

8. A continuous random variable X has pdf f(x) = 3x2 , 0 ≤ x ≤ 1. Find k such that P(X > k) = 0.05.

9. If the MGF of a random variable X is , find the standard deviation of X.

10. A binomial random variable X has mean 2 and variance 4/3. Find P(X > 1).

11. If X is a Poisson variate such that P(X = 2) = 9P(X = 4) + 90P(X = 6), find the variance.

12. The MGF of a random variable X is given by . Find P(X = 1).

13. Derive the recurrence formula for geometric distribution.

14. What do you mean by negative binomial random variable?

15. If the mean and variance of a uniform random variable X are 2 and 4/3 respectively, find the pdf of X.

16. An applicant for a driver’s license will pass on any given trial is 0.8. What is the probability that he will pass the test in fewer than four trials?

17. Derive the mean of exponential random variable.

18. Find the MGF of uniform random variable.

19. If a random variable X is uniformly distributed over (0 , 10), find the probability that (i) X < 3 (ii)

20. State the MGF, mean and variance of Gamma distribution.

PART – B

ANSWER ANY FIVE QUESTIONS (5 x 12 = 60)

21. (a)If a random variable X takes the values 1,2,3 and 4 such that 2P(X = 1) = 3P(X = 2) = P(X = 3) = 5P(X = 4), find the probability distribution and the cumulative distribution function of X.

(b) A random variable X has the following distribution

x / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7
p(x) / 0 / k / 2k / 2k / 3k / k2 / 2k2 / 7 k2+k

Find (1) the value of k (2) P(1.5 < X< 4.5 / X > 2) (3) the smallest value of λ for which P(X ≤ λ) > ½

22. (a) If the probability density function of X is given by f(x) = kx2e –x , x ≥ 0

find the value of k, mean, variance and rth moment about origin.

(b)The cdf of a rv X is given by

F(x) = 0 , x < 0

x2 , 0 ≤ x < ½

1 – (3-x)2 , ½ ≤ x < 3

1 , x ≥ 3

Find the pdf of X and evaluate P(⃓ x⃓ < 1), P(1/3 ≤ X < 4) using both pdf and cdf.

23. (a) If a random variable X has the density function f(x) = x , 0 ≤ x ≤ 1

2 – x , 1 ≤ x ≤ 2

0, otherwise

Find MGF, mean and variance.

(b) A random variable X has the probability distribution given by P(X = j) = , j= 1, 2, 3, ….

(i) Prove that total probability is 1 (ii) Also find P( X is even) and P(X is divisible by 3) and Mean of X.

24. (a) A continuous random variable X has pdf f(x) = kx (1 – x) , 0 ≤ x ≤ 1.

Find (1) the value of k (2) the cdf of X (3) P( X ≤ 1/2 / 1/3 < X < 2/3).

(b) Derive the MGF of Binomial distribution and hence find its mean and variance.

25. (a)6 dice are thrown 729 times. How many times do you expect at least three dice to show a 5 or 6?

(b) A and B shoot independently until each has hit his own target. The probabilities of their hitting the target at each shot are 3/5 and 5/7 respectively. Find the probability that B will require more shots than A.

26. (a) A manufacturer of cotter pins knows that 5% of the product is defective. If he sells cotter pins in boxes of 100 and guarantees that not more than 4 pins will be defective. What is the approximate probability that a box will fail to meet the guaranteed quality?

(b) State and prove the memoryless property of Geometric distribution.

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

(b) The time (in hours) required to repair a machine is exponentially distributed with parameter λ = ½.

(i)What is the probability that the repair time exceeds 2 hours?

(ii)What is the conditional probability that a repair takes at least 10 hours given that its duration exceeds 9 hours?

28. (a)Buses arrive at a specified stop at 15 minutes intervals starting at 7 A.M., that is they arrive at 7, 7:15,7:30, 7:45 and so on. If a passenger arrives at the stop at a random time between 7 and 7:30 A.M., find the probability that he waits (a) less than 5 minutes for a bus (b) at least 12 minutes for a bus.

(b)In a certain city, the daily consumption of electric power in millions of kilowatt hours can be treated as a random variable having Erlang distribution with parameters (1/2, 3). If the power plant of this city has a daily capacity of 12 million kilowatt hours, what is the probability that this power supply will be inadequate on any given day?

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INTERNAL TEST-II

PROBABILITY AND QUEUEING THEORY

Date: 22.02.2011 Marks: 100 Branch :B.E./B.TECH.-CSE/IT Time: 3 hrs

Section A:Answer all the Questions: (20 x 2 = 40)

  1. Define Marginal probability density functions.
  2. If the Jpdf of (X,Y) is f(x,y) = .
  3. Determine the value of ‘C’ such that the function f( x , y ) = C (1-x)(1-y) and 0 < x < 1 and 0 < y < 1 satisfies the properties of Jpdf.
  4. The Joint pdf of (X, Y) is .Examine X and Y are independent.
  5. The joint distribution of X and Y is given by ,x = 1, 2, 3; y = 1, 2.

Find the marginal distributions.

  1. The conditional pdf of X and Y=y is given by f(x/y)= Find P(X<1/Y=2).
  2. Show that Cov2(x ,y) Var(X) Var(Y)
  3. If X and Y are independent rvs with Variance 2 and 3. Find the variance of 3X+4Y.
  4. State the equations of two regression lines. Write down the angle between them.
  5. The tangent of the angle between the lines of regression of y on x and x on y is 0.6 and

. Find the Correlation coefficient.

  1. State Central Limit Theorem.
  2. Two rv’s X and Y are defined as Y = 4X+9.Find the correlation coefficient between X and Y.
  3. Write the properties of correlation coefficient.
  4. Let X and Y be random variables having Joint pdf .
  5. Find the Marginal densities of X and Y Whose Joint pdf is given by
  1. If X has mean 4, Variance 9,while has mean -2 and Variance 5 and the rvs are independent. Find E(XY2).
  2. Define random process and its classification.
  3. Define Strict sense stationary process and given example..
  4. If the Transition probability matrix of a Markov chain is .Find the limiting distribution of the chain.
  5. Examine whether the Poisson process {X(t)},given by the probability law

P{X (t) = r} =, r = 0, 1, 2, 3…. is Covariance stationary.

Section B: Answer any five Questions: (5 x 12 = 60)

  1. a) The joint probability mass function of X and Y is

Y X / 0 / 1 / 2
0 / 0.10 / 0.04 / 0.02
1 / 0.08 / 0.20 / 0.06
2 / 0.06 / 0.14 / 0.30

Find i) The Marginal density functions of X and Y ii) Conditional distribution of X

given Y=2. iii) and iv) check if X and Y are independent.

b) Let the joint pdf of the r.v (X,Y) be given by f( x , y) = 4xy,x > 0 and y > 0.

Are X and Y independent? Why or why not?

  1. . a) The Joint pdf of a two dimensional r.v (X, Y) is given by

. Compute (i) P(X > 1) (ii) P(Y< ½)

(iii)P(X>1/ Y<1/2) (iv) P(Y< ½ /X>1) (v) P(X<Y) (vi) P(X+Y).

. b) Given f(x,y) = c x(x-y) , 0 < x < 2, -x < y < x and 0 otherwise. Evaluate c and find

Marginal densities. Also find f(y/x).

  1. . a) The joint pdf of X and Y is given by

f( x , y ) = 3( x + y ) ,Find Cov(X,Y)

b) Find the coefficient of correlation between X and Y from the following data.

X: / 25 / 28 / 35 / 32 / 31 / 36 / 29 / 38 / 34 / 32
Y: / 43 / 46 / 49 / 42 / 36 / 32 / 31 / 30 / 33 / 38
  1. a) The two lines of 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.

b). If the Joint pdf of X1 and X2 is given by

.Find the pdf of V = .

  1. a) If X and Y each follow an exponential distribution with parameter 1 and are

independent, find the pdf of U = X-Y

b) If the Joint pdf X and Y is given by f( x , y) =

Find (i) P(X < 1) (ii) P(X < 1 / Y < 3) (iii) P(X + Y < 3)

  1. a) Let the joint pdf of the r.v (X,Y) be given by

Compute the correlation coefficient between X and Y.

b) A coin is tossed 300 times. Find the probability that number of heads obtained is between 140 and 150.

  1. a) Show that the random process X(t) = A sin() is wide-sense stationary process

where A and are constants and is uniformly distributed in (0, 2).

b) The transition probability matrix of a Markov chain {Xn},n = 1,2 3,……having 3

states 1,2 and 3 is and the initial distribution is p(0) = (0.7,0.2,0.1).

Find (i) P{X2 = 3} and (ii) P{X 3 = 2,X2 = 3,X1 = 3,X0 = 2 }.

  1. a) A fair die is tossed repeatedly. If Xn denotes the maximum of the numbers occurring in the first n tosses,

find the transition probability matrix P of the Markov chain {Xn}.Find also P2 and P(X2 = 6).

b) The process {X (t)} whose probability distribution under certain conditions is given by

P{X(t) = n} =

Show that {X(t)} is not stationary.