How to compute $\mathbb{E}\left[Y_1Y_2 \mid |U_1-U_2| <a\right]$ for $Y_i\sim N\left(\beta U_i, \sigma^2\right)$ and $U_i \sim Unif(0,1)$?

Question

Suppose we have that $U_1, U_2$ are iid $Unif(0,1)$ random variables and that

$$ Y_1\sim N\left(\beta U_1, \sigma^2\right) $$

is a Normal random variable independent of another Normal random variable:

$$ Y_2\sim N\left(\beta U_2, \sigma^2\right) $$

for a fixed $\sigma^2$.

I am wondering how to compute the conditional expectation:

$$ \mathbb{E}\left[Y_1Y_2 \mid |U_1-U_2| <a\right] $$

given that $Y_1, Y_2, U_1, U_2$ are all independent and that $a < \frac{1}{2}$?

Answer 1

Suppose you know that $U_1$, $U_2$, obtain the values $u_1$, $u_2$, respectively. Then, because of the independence of the variables,

$$ E[Y_1 Y_2 | U_1 = u_1, U_2 = u_2] = \beta^2 u_1 u_2. $$

Therefore, we need to calculate

$$ \int_{u_1 = 0}^1 \int_{u_2 = u_1}^{u_1 + a} \beta^2 u_1 u_2 \text{d} u_2 \text{d} u_1 + \int_{u_2 = 0}^1 \int_{u_1 = u_2}^{u_2 + a} \beta^2 u_1 u_2 \text{d} u_1 \text{d} u_2 $$

This can be simplified to

$$ 2 \beta^2 \int_{u_1 = 0}^1 u_1 \int_{u_2 = u_1}^{u_1 + a} u_2 \text{d} u_2 \text{d} u_1, $$

which is very easy to calculate.