# 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.

- 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?
- For a random variable , find .
- Let the conditional pdf of given be given by . Find .
- Let be uniformly distributed over and . Check if the random variables and are correlated?
- Check for the stationarity of the random process if and are constants and q is a uniformly distributed in .
- 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. - 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.
- Compute the mean time to failure of the component having a failure rate , is a constant.
- Compare and contrast the Latin Square Design with the Randomised Block Design.
- What is meant by process control in industrial statistics?

PART B — (5 ´ 16 = 80 marks)

- (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)

- (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)

- (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.

- (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)

- (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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