# Anna University MA1252-Probability and Queuing Theory 4th Semester CSE Question paper 2006

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(PART A-10*2=20marks)

1.The odds in favour of A solving mathematical problem are 3 to 4 and the odds against B solving the problems are 5 to 7. Find the probability that the problem will be solved by at least one of them.

2.A die is loaded in such a way that each odd number is twice as likely to occur as even number. Find P(G), where G is the event that a number greater than 3 occurs on a single roll of the die.

3.Define a continuous random variable. Give an example.

4.Find the value of (a). C and (b). mean of the following distribution given

C(x-x2), for 0<1

f(x)= 0, elsewhere

5.If the probability is 0.40 that a child exposed to a certain contagious will catch it, what is the probability that the tenth child exposed to the disease will be the third to catch it?

6.If X is uniformly distributed over (0,10) Calculate the probability that (a).X>6, (b).3<8.

7.Find the moment generating function for the distribution where

2/3, at x=1

f(x)=1/3, at x=2

0, otherwise

8.State central limit theorem.

9.Define random process and its classification.

10.What are the basic characteristics of Queuing process?

(PART B-5*16=80 marks)

11(a).(i).If the probability density of X is given by

2(1-x), for 0<1

f(x)=0, otherwise

1).Show that E[Xr]=2/((r+1)(r+2))

2).Use this result to evaluate e[2X+1)2]

11(a).(ii).Given a binary communication channel, where A is the input and E is the output, let P(a)=0.4, P(E/A)=0.9 and p[E/A]=0.6. Find

1).P(A/E)

2).P(A/E)

11(b).(i).A random variable X has density function given by

1/k, for0

f(x)= 0, elsewhere

Find,(1).m.g.f

(2).r th moment

(3).mean

(4).variance.

11(b).(ii).Given that a student studied, the probability of passing a certain quiz is 0.99. Given that a student did not study. The probability of passing the quiz is 0.05. Assume that the probability of studying is 0.7. A student flunks the quiz. What is the probability that he or she did not study?

12.(a).(i).Let the random variable X following binomial distribution with parameters n and p. Find,

(1).Probability mass function of X.

(2).Moment generating function.

(3).Mean and variance of X.

12.(a).(ii).The number of personal computer (PC) sold daily at a computer World is uniformly distributed with a minimum of 2000 PC and a maximum of 5000 PC. Find

(1).The probability that daily sales will fall between 2500 and 3000PC.

(2).What is the probability that the computer World will sell at least 4000 PC's?

(3).What is the probability that the computer World will sell exactly 2500 PC's?

12.(b).(i).Define the probability density function of normal distribution and standard normal distribution. Write down the important properties of its distribution.

12(b).(ii).An electric firm manufactures light bulbs that have a life, before burnout, that is normally distributed with mean equal to 800 hours and standard deviation of 40 hours. Find

(1).The probability that a bulb burns more than 834 hours

(2).The probability that bulb between 778 and 834 hours

13.(a).(i).In producing gallium-arsenide microchips, it is known that the ratio between gallium and arsenide is independent of producing a high percentage of workable wafer, which are main components of microchips. Let X denote the ratio of gallium to arsenide and Y denote the percentage of workable micro wafers retrieved during a 1 hour period. X and Y are independent random variables with the joint density being known as

(x(1+3y2))/4, 0<2,0<1

f(x)= 0, otherwise

Show that E(XY)=E(X)E(Y).

13(a).(ii).If the joint density of X1 and X2 is given by

6.e-3x1-2x2, for x1>0, x2>0

f(x1,x2)= 0, otherwise

Find the probability density of Y= X1 and X2

13.(b).(i).Two random variables X and Y have joint density function

fXY(x,y)=x2+(xy)/3; 0=x=1, 0=y=2

Find the conditional density functions. Check whether the conditional density functions are valid.

13.(b).(ii).If the joint probability density of X1 and X2 is given by

ex1+ x2, for x1>0, x2>0

f(x1,x2)= 0, otherwise

Find the probability of Y= X1/( X1 + X2)

14.(a).(1). 13(a).(11)Find the correlation coefficient and obtain the lines of regression from the following data:

x: 50 55 50 60 65 65 65 60 60 50

y: 11 14 13 16 16 15 15 14 13 13

14.(a).(ii).Let z be a random variable with probability density f(z)=1/2 in the range -1=z=1. Let the random variable X=z and the random variable Y=z2. Obviously X and Y are not independent since X2=y. Show none the less, that X and Y are uncorrelated.

14.(b).(i).Two random variables X and Y are defined as Y=4X+9.Find the correlation coefficient between X and Y.

14.(b).(ii).A stochastic process is described by X(t)=Asint + Bcost where A and B are independent random variables with zero means and equal standard deviation. Show that the process is stationary of the second order.

15.(a).(i).A raining process is considered as two state Markov chain. If it rains, it is considered to be the state 0 and if it does not rain, the chain is in state 1. The transition probability of the Markov chain is defined as

0.6 0.4

P 0.2 0.8 in matrix form

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

15.(a).(ii).A person owing 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 0.3

0.2 0.5 0.5

0.25 0.25 0.5 . What are the probabilities vehicles related to his fourth purchase?

15.(b).(i).Define Kendall's notation. What are the assumptions are made for simplest queuing model.

15.(b).(ii).Arrival rate of telephone calls at telephone 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 times.

1).Determine the probability that person arriving at the booth will have to wait.

2).Find the average queue length that is formed from time to time.

3).The telephone company will install second booth when convinced that an arrival would expect to have to wait at least 5 min for the phone. Find the increase in flows of arrivals which will justify a second booth.

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