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Grouped exponential. Suppose a type of unit has an exponential life distribution with meanθ,. Suppose that each unit has the same inspection times For a sample of I units, suppose that rJ fail in period

(a) Write the sample log likelihood in terms of theY.

(b) Derive af/aθ, and solve the likelihood equation for 8. (c) Use the group 10 data in Table 3.1 of Chapter 10; calculate the

(d) Derive a* f/a8*, and evaluate it for the data.

(e) Give the formula for the local estimate of Van(θ).

(f) Evalute the local estimate of Van(θ) for the data in (c). (θ) Calculate two-sided positive approximate 95% confidence limits

(h) Derive the formula for the true asymptotic variance of 8.

(i) Evalute the ML estimate of Van(θ) for the data, using (c) and (h).

(j) Calculate positive approximate 95% confidence limits for O,, using (i).

(k) On Weibull paper, plot the data, the fitted exponential distribution, and approximate 95% confidence limits for percentiles.

Table

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