(a) Show that if $W_{1}, \ldots, W_{n}$ are independent random variables with common $\operatorname{Exp}(1)$ distribution, then $\sum_{i=1}^{n} W_{i} \sim \Gamma(n, 1)$. [Hint: If $W \sim \Gamma(\alpha, \lambda)$ then $\mathbb{E} e^{t W}=\{\lambda /(\lambda-t)\}^{\alpha}$ if $t<\lambda$ and $\infty$ otherwise.]

(b) Show that if $X \sim U(0,1)$ then $-\log X \sim \operatorname{Exp}(1)$.

(c) State the Neyman-Pearson lemma.

(d) Let $X_{1}, \ldots, X_{n}$ be independent random variables with common density proportional to $x^{\theta} \mathbf{1}_{(0,1)}(x)$ for $\theta \geqslant 0$. Find a most powerful test of size $\alpha$ of $H_{0}: \theta=0$ against $H_{1}: \theta=1$, giving the critical region in terms of a quantile of an appropriate gamma distribution. Find a uniformly most powerful test of size $\alpha$ of $H_{0}: \theta=0$ against $H_{1}: \theta>0$.