Anna University Model Question Paper BE II sem IT PROBABILITY AND STATISTICS

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 cards 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 .
  6. 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.
  7. 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.
  8. Compute the mean time to failure of the component having a failure rate ,  is a constant.
  9. Compare and contrast the Latin Square Design with the Randomised Block Design.
  10. What is meant by process control in industrial statistics?

PART B — (5 ´ 16 = 80 marks)

  1. (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)

Days

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)

  1. (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 children 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)

  1. (a)     There are 2 white marbles in urn A and 3 red marbles in urn B. At each step of the process, a marble is selected from each urn and the 2 marbles selected are 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 red 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.

  1. (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)

A

B

C

 

 

 

 

 

                       (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)

  1. (a)     A laboratory technician measures the breaking strength of each of 5 kinds of linen threads by using four different measuring 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 specially designed equipment for 12 tanks of gas distributed over 3 different models of engines :

Engine 1

Engine 2

Engine 3

Totals

Detergent A

45

43

51

139

Detergent B

47

46

52

145

Detergent C

48

50

55

153

Detergent D

42

37

49

128

182

176

207

565

                       Perform the ANOVA and test at .01 level of significance whether there are differences in the detergents or in the engines.

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