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Posted By: CONFIDENCE IS THE COMPANION OF SUCCESS Member Level: Diamond Posted Date: 16 Apr 2008
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2007 Anna University B.E Computer Science Probability and Queuing theory Question paper
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B.E/B.Tech Degree Examination May/June-2006
Fourth Semester CSE - MA 1254-Probability and Queuing theory
PART-A Answer all the Questions
1.A coin is tossed an infinite number of times. If the probability of a head in a single toss is p, find the probability that k th head is obtained at the nth tossing but not earlier, with q=1-p.
2.A continuous random variable X that can assume any value between x=2 and x=5 has density function given by f(x)=k(1+x).Find P[X<4].
3.If X is uniformly distributed in (-p/2,p/2), find the p.d.f of Y=tan X.
4.Find the moment generating function of a geometric distribution.
5.Show that correlation coefficient is independent of change of origin and scale.
6.The two regression lines are 4x-5y+33=0 and 20x-9y=107 and variance of x=25. Find the mean of x and y. Also find the value of r.
7.Examine whether the Poisson process {X(t)} given by the law P[X(t)=r]=(e-?t(?t)r)/r!, r=0,1,2… is covariance stationary.
8.Define a Markov process and a Markov chain.
9.What are the basic characteristics of a queuing system?
10.Derive the average number of customers in the system for (M/M/I):(a/FIFO) model.
PART-B
11.(i).The process {X(t)} where probability distribution is given by
(at)n-1/(1+at)n+1, n=1,2,3….
P[X(t)=n]= (at)/(1+at), n=0 . Show that it is not stationary.
11.(ii).A raining process is considered as a two-state Markov chain. If it rains it is considered to be in 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. Find the probability that will rain for three days from today assuming that it is raining today. Find also the unconditional probability that will rain after three days with the initial probabilities of state 0 and state 1 as 0.4 and 0.6 respectively.
12.(a).(i).Out of (2n+1) tickets consequently numbered three are drawn at random. Find the probability that the numbers on them are in arithmetic progression.
12.(a)..(ii).If A and E are independent events, than show that A and E are also independent events. Also show that A and E are also independent events.
Or
12.(b).(i).The content of urns I,II and III are as follows
1 white, 2 black, and 3 red balls
2 white, 1 black, and 1 red ball
4 white, 5 black, and 3 red balls
One urn is chosen at random and two balls are drawn from it. They happen to be white and red. What is the probability that they come from urns I, II and III?
12.(b).(ii).Let the random variable X assume the value r with probability law
P(X=r)=qr-1p, r=1,2,3…. Find the moment generating function and hence its mean and variance.
13.(a).(i).If m things are distributed among ‘a’ men and ‘b’ women, find the probability that the number the number of things received by men is odd.
13.(a).(ii).If X and Y are independent Poisson variates, find the conditional distribution of X given X+Y.
or
13.(b).(i).If X1 and X2 are independent uniform variates on[0,1], find the distribution of X1/X2 and X1X2.
13.(b).(ii).Find the moment generating function of a normal distribution.
14.(a).(i).Two random variables X and Y have the following joint p.d.f
2-x-y, 0=x=1,0=y=1
f(x,y)= 0, otherwise
14.(a).(ii).Let (X,Y) be a two-dimensional non-negative continuous random variable having the joint density
4xye-(x2+y2),x=0,y=0
f(x,y)= 0, otherwise
Find the density function of U=v(x2+y2).
Or
14.(b).(i).Find the coefficient of correlation and obtain the lines of regression from the data given below:
X: 62 64 65 69 70 71 72 74
Y: 126 125 139 145 165 152 180 208
14.(b).(ii).Let the random variable X have the marginal density
f1(x)=1, -1/2<1/2 and let the conditional density f(y/x) be given by
1,x<0
f(y/x)= 1,-x<1-x,0<1/2
Show that the variables are uncorrelated.
15.(a).Customers arrive at a one-man barber shop according to a Poisson process with a mean inter arrival time of 20 min. Customers spend an average of 15 min 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 barber shop and in the queue?
iii. How much time can a customer expect to spend in the barber shop?
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?
15.(b).Derive the formula for the average number of customers in the queue and the probability that an arrival has to wait for (M/M/C) with infinite capacity. Also derive foe the same model the average waiting time of a customer in the queue as well as in the system.
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