Conditional variance of the convolution of two R.V.?

Question

Let $X,Z \sim U[0,1]$ independent. I want to compute the conditional variance of $Y=X+Z$. First, I notice that $$f(y) = \begin{cases} y & \text{for $0 < y < 1$} \\ 2-y & \text{for $1 \le y < 2$} \\ 0 & \text{otherwise.} \end{cases}$$

And that $E[Y|X]=1/2+X$. I think I may compute the conditional variance as follows: $$E[(Y-E(Y|X))^2|X)=E[Y^2+E(Y|X)^2-2YE(Y|X)|X]=...$$ using linearity of expectations etc. But is there an easier way?

Answer 1

Note that $$ \text{Var}(Y\mid X)=E(Y^2\mid X)-E(Y\mid X)^2. $$ See for example here. As you noted $$ E(Y\mid X)=X+E(Z\mid X)=X+EZ=X+1/2 $$ by linearity and independence of $X$ and Z. Moreover $$ \begin{align} E(Y^2\mid X)&=E(X^2+2XZ+Z^2\mid X)\\ &=X^2+2XE(Z\mid X)+E(Z^2\mid X)\\ &=X^2+2XEZ+EZ^2\\ &=X^2+X+\frac{1}{3} \end{align} $$ where we used independence of $X$ and $Z$ together with the pull-out property in the third line. So $$ \text{Var}(Y\mid X)=X^2+X+\frac{1}{3}-\left(X+\frac{1}{2}\right)^2 $$