Uniform distribution with unknown interval boundary Suppose that a random variable $Y$ is uniformly distributed on the interval $[-a,a]$, with $a > 0$. Suppose that the random variable $X$ is uniformly distributed on the (stochastic) interval $[-|Y|,|Y|]$, where $Y$ is introduced as above. How can we find $\mathbb{E}[X\mid Y]$ and $\mathbb{E}[|X|]$? Thanks a lot.

Hint:  $$\operatorname{E}[|X|] = \operatorname{E}[\operatorname{E}[|X|\mid Y]] = 2 \int_{y=0}^a \operatorname{E}[|X| \mid Y = y] f_Y(y) \, dy.$$
