Chapter 8.5, Problem 47E

Refer to Exercise 8.46. Assume that Y₁, Y₂, …, Y_n is a sample of size n from an exponential distribution with mean θ.

a Use the method of moment-generating functions to show that $2{\displaystyle {\sum }_{i=1}^n{Y}_i}\text{/}\theta$ is a pivotal quantity and has a χ² distribution with 2n df.

b Use the pivotal quantity $2{\displaystyle {\sum }_{i=1}^n{Y}_i}\text{/}\theta$ to derive a 95% confidence interval for θ.

c If a sample of size n = 7 yields $\bar{y}=4.77$, use the result from part (b) to give a 95% confidence interval for θ.

8.46 Refer to Example 8.4 and suppose that Y is a single observation from an exponential distribution with mean θ.

a Use the method of moment-generating functions to show that 2Y/θ is a pivotal quantity and has a χ² distribution with 2 df.

b Use the pivotal quantity 2Y/θ to derive a 90% confidence interval for θ.

c Compare the interval you obtained in part (b) with the interval obtained in Example 8.4.