PART A — (10 ? 2 = 20 marks)
1. If a random variable X takes the values 1, 2, 3, 4 such that . Find the probability distribution of X.
2. If X is a Poisson variate such that find the variance.
3. X and Y are independent random variables with variance 2 and 3. Find the variance of .
4. If the random variable X is uniformly distributed over , find the density function of .
5. State any four properties of Poisson process.
6. What is meant by steady–state distribution of Markov chain?
7. Define preventive maintenance downtime.
8. A system consists of 3 identical units connected in parallel. The unit reliability factor is 0.9. If the unit failures are independent of one another and the successful operation of the system depends on the satisfactory performance of any one unit, determine the system reliability.
9. For model, write down the Little’s formula.
10. For model, write down the formula for
(a) Average number of customers in the queue
(b) Average waiting time in the system.
PART B — (5 ? 16 = 80 marks)
11. (i) A bank has two tellers working on savings account. 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 minutes per customer. Depositors are found to arrive in a Poisson fashion throughout the day with mean arrival rate of 16 per hour. Withdrawers also arrive in a Poisson fashion with mean arrival rate of 14 per hour. What would be the effect on the average waiting time for the customers if each teller could handle both withdrawals and deposits. (8)
(ii) In a heavy machine shop, the overhead crane is 75% utilised. Time study observations gave the average slinging 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 8.0 minutes, with a standard deviation of 6.0 minutes, how much reduction will occur, on average, in the delay of getting served? (8)
12. (a) (i) An urn contains 5 balls. Two balls are drawn and are found to be
white. What is the probability of all the balls being white? (8)
(ii) The daily consumption of milk in excess of 20,000 gallons is approximately exponentially distributed with . 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) (i) A man draws 3 balls from an urn containing 5 white and 7 black balls. He gets Rs. 10 for each white ball and Rs. 5 for each black ball. Find his expectation. (8)
(ii) Two unbiased dice are thrown. If X is the sum of the numbers showing up, prove that using Chebyshev’s inequality. (8)
13. (a) (i) is a two–dimensional random variable uniformly distributed over the triangular region R bounded by and . Find the correlation coefficient . (8)
(ii) If X and Y are independent random variables each following , find the probability density function of . (8)
(b) (i) X and Y are two random variables having joint density function
Find (1) (2) (3) .
(ii) A distribution with unknown mean has variance equal to 1.5. Use central limit theorem, to find how large a sample should be taken from the distribution in order that the probability will be atleast 0.95 that the sample mean will be within 0.5 of the population mean. (8)
14. (a) (i) Two random processes and are defined by
and . Show
that and are jointly wide–sense stationary if A and B
are uncorrelated random variables with zero means and the same
variances and w is constant. (8)
(ii) Find the mean and autocorrelation of the Poisson process. (8)
(b) (i) Given a random variable Y with characteristic function and a random process defined by , show that is stationary in the wide sense if . (8)
(ii) If is a Gaussian process with and , find the probability that (1)
(2) . (8)
15. (a) (i) 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 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. (10)
(ii) Find the hazard rate function corresponding to the Weibull distribution given by . (6)
(b) (i) Five elements are connected as shown below. Their reliabilities are also given. Calculate the system reliability. (8)
(ii) Three boys A, B and C are throwing a ball to each other. A always throws the ball to B and B always throws the ball to C, but C is just as likely to throw the ball to B as to A. Show that the process is Markovian. Find the transition matrix and classify the states. (8)