(a) State and prove the Neyman-Pearson lemma.

(b) Let $X$ be a real random variable with density $f(x)=(2 \theta x+1-\theta) 1_{[0,1]}(x)$ with $-1 \leqslant \theta \leqslant 1 .$

Find a most powerful test of size $\alpha$ of $H_{0}: \theta=0$ versus $H_{1}: \theta=1$.

Find a uniformly most powerful test of size $\alpha$ of $H_{0}: \theta=0$ versus $H_{1}: \theta>0$.