let $X,Y$ be independent random variables that follow a continuous uniform distribution with bounds $[a_1,b_1], [a_2,b_2]$ respectively, ${1}_A$ be the indicator function, $A$ be the event $X<l, Y < m$ where $m,l$ are real numbers, and $g(X)$ a function of the rv $X$. I was trying to find the expected value $E[g(X)1_A]$ but i was having trouble with it since the event A depends on 2 different random variables. I have been stuck on this for a while and would appreciate any help
Thanks in advance
If $X$ and $Y$ are independent, then $$\mathbb E\left[g(X)\mathbf 1_A\right] = \mathbb E\left[g(X)\mathbf 1_{X\le m}\mathbf 1_{Y \le l}\right] = \mathbb E\left[g(X)\mathbf 1_{X\le m}\right]\mathbb E\left[\mathbf 1_{Y \le l}\right]$$
Can you finish?
Now If for example $X \sim \mathcal U\left([0, 3]\right)$, $Y \sim \mathcal U\left([1, 5]\right)$ and you want to compute $$\mathbb E\left[X^2 \mathbf 1_{X\le 2Y} \mathbf 1_{Y \le 2}\right] = \int_{1}^{2}\int_{0}^{\min \left\{2Y, 3\right\}} x^2\frac1{3-0}\mathrm dx\frac1{5-1}\mathrm d y = \int_{1}^{\frac32}\int_{0}^{2Y} x^2\frac13\mathrm dx\frac14\mathrm d y + \int_{\frac32}^{1}\int_{0}^{3} x^2\frac13\mathrm dx\frac14\mathrm d y + $$