I can't seem to figure out how to solve that. If X and Y are two independent random variables X and Y sampled uniformly from the [0, 1], then what is the variance of $(X+Y)^2$ ?
I know that $Var((X+Y)^2 = Var(X^2 + 2XY + Y^2)$ but after that we can't separate them as the terms are not independent, correct?
It seems like a pretty simple thing to solve, but I can't figure out how to proceed.
Let $Z=(X+Y)^2$. The variance of $Z$ is $E(Z^2)-E(Z)^2$. Now \begin{align} E(Z^2)=E((X+Y)^4)&=E(X^4+4X^3Y+6X^2Y^2+4XY^3+Y^4)\\ &=E(X^4)+4E(X^3)E(Y)+6E(X^2)E(Y^2)+4E(X)E(Y^3)+E(Y^4) \end{align} using linearity of expectation and independence. To proceed further use $E(X^k)=E(Y^k)=1/(k+1)$ (by a simple integration). The calculation of $E(Z)$ is similar, but shorter.