Conditional expectation of summation two random variables

Question

Suppose $X$~Unif[-1,1], $Y$~Unif[4,10], and $Z=X+Y$ where $X$ and $Y$ are independent. Find $E[X|Z=z]$.

I know the way that uses convolution, but I am seeking for a tricky solution.

Answer 1

The distribution of $(X,Y)$ is uniform on the rectangle $[-1,1] \times [4,10]$.

The constraint $x+y=z$ appears as a diagonal line cutting across this rectangle, and $x$ takes values from $\max\{z-10, -1\}$ to $\min\{z-4,1\}$ along this line. The conditional distribution of $X$ given $Z=z$ is intuitively uniform along this interval, so the conditional expectation is $$\frac{1}{2} (\max\{z-10,-1\} + \min\{z-4,1\}).$$ You can check that this intuition matches the rigorous calculation.