UNIV4VINU | ENGINEERING ACHIEVERS

B.E./B.Tech. DEGREE EXAMINATION,

Second Semester

Information Technology

MA 039 — PROBABILITY AND STATISTICS

Time : Three hours Maximum : 100 marks

Answer ALL questions.

PART A — (10 ? 2 = 20 marks)

1. and are events with and Find .

2. The time (in hours) required to repair a machine is exponentially distributed with

parameter . What is the probability that the repair time exceeds

3 hours.

3. Find the value of if, for is to be a joint density function.

4. Given the random variable X with density function

find the probability density function if .

5. When are the processes and said to be jointly stationary in the wide sense?

6. Define a Markov Process.

7. Reliability of a component is 0.4. Calculate the number of components to be

connected in parallel to get system reliability 0.8.

8. The following data was collected for an automobile :

Mean time between failures : 500 hr

Mean waiting time for spares : 5 hr

Mean time for repairs : 48 hr

Mean administrative time : 2 hr

Compute the availability of the automobile.

9. Name the basic principles of experimental design.

10. Find the lower and upper control limits for -chart and -chart if and .

PART B — (5 ? 16 = 80 marks)

11. (i) Find the M.G.F. of the random variable with probability density function :

Also find . (8)

(ii) The joint pdf of the two dimensional random variable is given by

Find the marginal density functions of X and Y. Find also the conditional density

function of Y given and the conditional density function

of given . (8)

12. (a) (i) is a continuous random variable with pdf given by

Find the value of K and also the cdf . (8)

(ii) A random sample of size 100 is taken from a population whose mean is 60 and

variance is 400. Using capital Limit Theorem, find with what probability can we

assert that the mean of the sample will not differ from by more than 4? (8)

Or

(b) (i) State Tchebycheff’s inequality. Using the inequality for a r andom variable X

with pdf show that and show also that the actual probability is . (8)

(ii) Let the random variables and have the joint pdf

Compute the correlation coefficient between X and Y. (8)

13. (a) (i) Define Random Process. Specify the four different types of Random

Process and give an example to each type. (8)

(ii) The transition probability matrix of a Markov chain having 3 states 1, 2 and 3 is

and the initial distribution is . Find and . (8)

Or

(b) (i) Prove that the difference of two independent Poisson process is not a Poisson

process. (8)

(ii) A random process has the probability distribution

Show that the process is evolutionary. (8)

14. (a) (i) The density function of the time to failure of an appliance is ( is in years)

(1) Find the reliability function

(2) Find the failure rate

(3) Find the MTTF. (6)

(ii) Calculate the system reliability for the units connected as below : (6)

(iii) If a device has a failure rate of where is in years, calculate the reliability for a 5

year design life, assuming that no maintenance is performed. (4)