Math 6010 • Linear Models

Let \(X_i\sim\operatorname{EXP}(\theta)\) be a random sample of size
\(n\).

\textbf{7a)} (5 points) Find the MGF of \(\frac{2n\bar X}{\theta}\) and identify
its distribution.

\textbf{7b)} (10 points) Derive the generalized likelihood ratio test of
\(H_0:\theta=\theta_0\) against \(H_a:\theta\ne\theta_0\).

\textbf{7c)} (5 points) The following data are times (in hours) between
failures of air conditioning equipment in a particular airplane:
74, 57, 48, 29, 502, 12, 70, 21, 29, 386, 59, 27, 153, 26, 326. Test
\(H_0:\theta=125\) against \(H_a:\theta\ne125\).