| course |
course_year |
question_number |
tags |
title |
year |
Statistics |
IB |
67 |
|
Paper 1, Section II, H |
2021 |
(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 $H_{1}: \theta>0$.