10.7 Exercises

1. You are given a set of 10 processors that are believed to follow a Poisson failure process, with failure rate ? per hour per processor. You run the processors for a week, and obtain the following numbers of failures for each processor: 2, 4, 2, 1,1, 2, 3, 2, 0, 2.

   1. What is your estimate for the value of ??

   2. Construct a 95% confidence interval for ? using Equation 10.1.

   3. Construct a 95% confidence interval for ? using the fact that for the Poisson distribution E(x) = Var(x) = ?.

   4. Explain the difference between the results of parts b and c.

2. You are given a set of 10 processors that are believed to follow a Poisson failure process, with failure rate ? per hour per processor. The prior density of ? is a uniform distribution over the range [0.001,0.002].

   1. You run these processors for 100 hours without any of the processors failing. What is the best estimate for the value of ? (the mean of the posterior density of ?)

   2. You continue the experiment for a total of 10,000 hours without observing any failures. What is your best estimate for ??

   3. Suppose you were to run this experiment for a very long time without any processor failing. What do you think the posterior density function for ? would be?

3. This question follows up on our comments on the difficulty of validating the reliability of a life critical...