I'm having trouble with the following questions:
Suppose $X,Y$ ~ $U(0,1)$. Find $E[X^2|X+Y]$ and $E[XY|X+Y]$.
So far, I have: $E[X^2|X+Y] = \int x^2 f_{xy|x+y=z}$ Similarly: $E[XY|X+Y] = \int xy f_{xy|x+y=z}$
Furthermore: $f_{xy|x+y=z} = P((X=x,Y=y)\cap(X+Y =z))/P(X+Y =z)$
I'm lost on what: 1. The conditional density should be and 2. What the bounds of integration should be on both expectations.
Let $Z=X+Y$. Assuming that $X$ and $Y$ are independent, the joint density of $X$ and $Z$ is $$ f(x,z)=1_{0\leq x\leq 1, x\leq z\leq x+1}$$ and the density of $Z$ is $f_Z(z)=z$ if $0\leq z\leq 1$, $f_Z(z)=2-z$ if $1\leq z\leq 2$. These densities can be computed by finding the areas of appropriate regions in the unit square, then differentiating.
Next, by defintion $$ \mathbb{E}[X^2|Z=z]=\int x^2f_{X|Z}(x|z)\;dx=\frac{1}{f_Z(z)}\int x^2f(x,z)\;dx $$
If $0\leq z\leq 1$ this integral becomes $$ \frac{1}{z}\int_0^z x^2\;dx=\frac{z^2}{3} $$ while if $1\leq z\leq 2$ it becomes $$ \frac{1}{2-z}\int_{z-1}^1x^2\;dx=\frac{z^2-z+1}{3} $$ (after some simplifications).
By symmetry, $\mathbb{E}[Y^2|Z]=\mathbb{E}[X^2|Z]$. And since $\mathbb{E}[(X+Y)^2|Z]=Z^2$ and $(X+Y)^2=X^2+2XY+Y^2$, we can use the work we've already done to determine $\mathbb{E}[XY|Z]$.