## Wednesday, November 4, 2009

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) |

15. (i) Six identical components with constant failure rates are connected in high level |

redundancy with 3 components in each subsystem. Find the component MTTF to |

provide a system reliability of 0.90 after 100 hours of operation. (6) |

(ii) Five elements and are connected as show |

Calculate the system reliability. |

(iii) State the relationship between various forms of maintenance. (4) |

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