Let $X_1,X_2$ and $X_3$ be independent and identically distributed random variables with $E(X_1)=\frac{15}{4}$.
If $\psi:(0,\infty)\to(0,\infty)$ is defined through the conditional expectation $\psi(t)=E(X_1^2\mid X_1^2+X_2^2+X_3^2=t),t>0$.
Then what is the value of $E(\psi((X_1+X_2)^2))$ ?
I don't know how to approach this problem. Please help.
Hint: Since the r.v. are independent and identically distributed, then by symmetry:
$$\mathsf E(X_1^2\mid X_1^2+X_2^2+X_3^2=t){~=~\mathsf E(X_2^2\mid X_1^2+X_2^2+X_3^2=t)\\~=~\mathsf E(X_3^2\mid X_1^2+X_2^2+X_3^2=t)}$$