I am working on an exercise which requires me to compute $\mathbb{E}\log|X|$, where $X$ is uniformly distributed on the unit ball $\{u\in\mathbb{R}^{2}:|u|\leq 1\}$.
I know that this uniform distribution implies that $$\mathbb{P}(|X|\leq r)=r^{2}\ \text{for}\ 0\leq r\leq 1\ \ (*),$$ then I tried to use the formula $$\mathbb{E}|X|^{p}=\int_{0}^{\infty}px^{p-1}\mathbb{P}(|X|>x)dx,$$ to compute $$\mathbb{E}\log|X|=\int_{0}^{\infty}\mathbb{P}(\log|X|>x)dx,$$ what should I do next to use $(*)$?
Thank you!
If you want to use that formula, you need to notice that $\log|X| \leq 0$ so instead you would write
$$\begin{align*}\mathbb{E}\log|X| &= -\mathbb{E}[-\log|X|] \\&= -\int_0^\infty \mathbb{P}(-\log|X| > x)\,dx \\&= -\int_0^\infty \mathbb{P}(|X| < e^{-x})\,dx\end{align*}$$
Now note that $0 < x < \infty \implies 0 < e^{-x} < 1$ so you can use $(*)$ directly.