What is a simple hypothesis? Define the terms size and power for a test of one simple hypothesis against another.

State, without proof, the Neyman-Pearson lemma.

Let $X$ be a single random variable, with distribution $F$. Consider testing the null hypothesis $H_{0}: F$ is standard normal, $N(0,1)$, against the alternative hypothesis $H_{1}: F$ is double exponential, with density $\frac{1}{4} e^{-|x| / 2}, x \in \mathbb{R}$.

Find the test of size $\alpha, \alpha<\frac{1}{4}$, which maximises power, and show that the power is $e^{-t / 2}$, where $\Phi(t)=1-\alpha / 2$ and $\Phi$ is the distribution function of $N(0,1)$.

[Hint: if $X \sim N(0,1), P(|X|>1)=0.3174 .]$