## Wednesday, November 4, 2009

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) |

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) |

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 |

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 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 |

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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