Let $X = X_1 + X_2 + X_3$ be a discrete random variable and $Y$ and $Z$ be continuous random variables distributed uniformly from over the ranges $[a,b]$ and $[c,d]$, respectively (with $Y$ and $Z$ independent of each other). There are no restrictions on the values of $a,b,c$ or $d$. How would I work out $\mathbb{E}[X | Y, Z]$?
I guess we could start with,
$\mathbb{E}[X | Y, Z] = \mathbb{E}[X_1 | Y, Z] + \mathbb{E}[X_2 | Y, Z] + \mathbb{E}[X_3 | Y, Z]$,
but I am not sure how to deal with the conditional being on both Y and Z.
Edit: $X_1, X_2 = f(Y,Z)$ and $X_3 = g(Z)$