1. a. Let X and Y be independent random variables having the exponential distribution with parameters \(\lambda\) and \(\mu\) respectively. Find the density function of X + Y.
b. Suppose that we have two independent random samples: \(X_1, \dots, X_n\) are exponential (\(\lambda\)), and \(Y_1, \dots, Y_m\) are exponential (\(\mu\)).
Find the likelihood ratio test of \(H_0: \lambda = \mu\) versus \(H_1: \lambda \neq \mu\).