If $U_1, U_2$ are iid $Unif(0,1)$, how to compute $E[U_1\mid|U_1-U_2|<a]$ for $a < \frac{1}{2}$?

Question

If $U_1, U_2$ are iid $Unif(0,1)$, I am wondering how to compute $E\left[U_1\mid|U_1-U_2|<a\right]$ for $a < \frac{1}{2}$?

I saw that the answer is:

$$ \mathbb{E}\left[U_1\mid|U_1-U_2|<a\right] = \int_0^a \int_0^{a+u_i} u_iu_j du_jdu_i + \int_a^{1-a} \int_{-a+u_i}^{a+u_i} u_iu_j du_jdu_i + \int_{1-a}^1 \int_{-a+u_i}^{1} u_iu_j du_jdu_i \\ $$

but am not sure why.

Answer 1

Realize that here $(U_1,U_2)$ and $(1-U_1,1-U_2)$ have equal distributions so that:$$\mathbb E\left[U_1\mid|U_1-U_2|<a\right]=\mathbb E\left[1-U_1\mid|(1-U_1)-(1-U_2)|<a\right]=1-\mathbb E\left[U_1\mid|U_1-U_2|<a\right]$$ Final conclusion:$$\mathbb E\left[U_1\mid|U_1-U_2|<a\right]=\frac12$$