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)

MODEL PAPER

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

Second Semester — Information Technology

Fourth Semester — Industrial Bio–Tech

MA 039 — PROBABILITY AND STATISTICS

Time : Three hours Maximum : 100 marks

PART A — (10 ´ 2 = 20 marks)

Answer ALL questions.

1.From an ordinary deck of 52 cards, we draw car ds at random, with replacement and

successively until an ace is drawn, What is the probability that atleast 10 draws are

needed?

2.For a random variable , find .

3.Let the conditional pdf of given be given by . Find .

4.Let be uniformly distributed over and . Check if the random variables and are

correlated?

5.Check for the stationarity of the random process if and are constants and q is a

uniformly distributed in .

7.A salesman’s territory consists of 3 cities and . He never sells in the same city on

successive days. If he sells in city then the next day he sells in . However, if he sells

either in or ,then the next day he is twice as likely to sell in city A as in the other city.

Find the transition probability matrix.

8.An engine is to be designed to have a minimum reliability if 0.8 and a minimum

availability of 0.98 over a period of hours. Determine the mean repair time and

frequency of failure of the engine.

9.Compute the mean time to failure of the component having a failur e rate , is a

constant.

10.Compare and contrast the Latin Square Design with the Randomised Block

Design.

What is meant by process control in industrial statistics?

PART B — (5 x16 = 80 marks)

11.(a)(i) A cost accountant is asked to set up a system for controlling waste in a

certain department, converting rolls of paper into sheets. The pounds of waste are

recorded by shifts for a period of 10 days as shown below; prepare and charts and

indicate whether the process is in satisfactory control. (8)

Shift 1 2 3 4 5 6 7 8 9 10

1 89 112 121 91 75 86 123 98 96 97

2 99 108 106 117 79 105 106 100 83 114

3 115 132 103 98 81 93 105 114 87 124

(ii) The data below gives the results of daily inspection of sewing machine needles for

a particular quality characteristic. Compute the trial control limits and plot as a p–

chart. Assume that the number of defectives follows a binomial distribution. Also

comment on your finding.

No. inspected : 110, 120, 30, 0, 35, 60, 165, 18, 140, 35, 190, 160, 35,

50, 70.

No. of defectives : 5, 8, 1, 0, 2, 3, 15, 2, 10, 0, 16, 20, 5, 5, 5. (8)

12.(a) (i) A father asks his sons to cut their backyard lawn. Since he does not

specify which of the three sons is to do the job, each boy tosses a

coin to determine the odd person, who must then cut the lawn. In

the case that all three get heads or tails, they continue tossing until

they reach a decision. Let p be the probability of heads and

, the probability of tails. Find the probability that they

reach a decision in less than n tosses. If , what is the

minimum number of tosses required to reach a decision with

probability 0.95? (10)

(ii) A woman and her husband want to have a 95% chance for atleast one boy and

atleast one girl. What is the minimum number of children that they should plan to

have? Assume that the events that a child is a girl and a boy are equiprobable and

independent of the gender of other childr en born in the family. (6)

Or

(b) (i) Let the probability density function of X be

for some . Using the method of distribution functions, calculate the probability density

function of . (8)

(ii) Suppose that, on average, a post office handles 10,000 letters a day with a

variance of 2000. What can be said about the probability that this post office will

handle between 8,000 and 12,000 letters tomorrow? (8)

13.(a) There are 2 white marbles in urn A and 3 red marbles in urn B. At each step of

interchanged. Let the state of the system be the number of red marbles in A after i

changes. What is the probability that there are 2 r ed marbles in A after 3 steps? In the

long run, what is the probability that there are 2 red marbles in urn A?

Or

(b) (i) Let be a Poisson process with rate l. For , show that

.

(ii) Suppose customers arrive at a store according to a Poisson process at a rate 10 per

hour. Calculate the conditional probability that in

5 hours 20 customers arrived given that in 10 hours 30 customers arrived.

14.(a) Obtain the steady–state availability for a 2–unit parallel system with repair.

Or

(b) (i) Estimate the reliability and MTTF of the following system by assuming that the

system are identical with constant hazard rate l.

(4)

(ii) Determine the failure rate of a 2–unit system subject to preventive maintenance at

every 1000 hours. A unit failure rate is 0.01 per

100 hour. (6)

(iii) Let be the failure rate of a component. The component has only two states : state

0 : the component is good and state 1 : the component is failed. Obtain the reliability

of the component. (6)

15.(a) A laboratory technician measures the breaking strength of each of 5 kinds of

linen threads by using four different measur ing instruments, and obtains the following

results, in ounces :

Thread 1 20.9 20.4 19.9 21.9

Thread 2 25.0 26.2 27.0 24.8

Thread 3 25.5 23.1 21.5 24.4

Thread 4 24.8 21.2 23.5 25.7

Thread 5 19.6 21.2 22.1 22.1

Analyse the data using the .05 level of significance.

Or

(b) An experiment was designed to study the performance of 4 different detergents for

cleaning fuel injectors. The following ‘‘cleanness’’ readings were obtained with