Question: Question 5: Let \({{X}_{1}},...,{{X}_{n}}\) be i.i.d. observations from a Gamma(1, \(\mu \) ), and \({{Y}_{1}},...,{{Y}_{m}}\) be i.i.d. observations from a Gamma(1, \(\theta \) ). Also assume that X’s are independent of Y’s.

a) Formulate the Likelihood Ratio Test for testing Ho: \(\mu =\theta \) vs. H1: \(\mu \ne \theta \).

b) Show that the test in part (a) can be based on the following statistic

\[T=\frac{\sum\limits_{i=1}^{n}{{{X}_{i}}}}{\sum\limits_{i=1}^{n}{{{X}_{i}}}+\sum\limits_{j=1}^{m}{{{Y}_{j}}}}\]