Conditional expectation $E(\xi^2\mid\xi\eta)$ where $\xi, \eta$ iid $N(0,1)$

Question

Suppose $\xi, \eta \sim N(0,1)$ are two independent random variables. Then how can we calculate the composite conditional expectation $E(\xi^2\mid\xi\eta)$? I thought about calculating the distribution/density function of random vector $(\xi^2, \xi\eta)$ but the mapping is not bijective.

Answer 1

Following Ragib's hint, the joint distribution of $U,V$ is $p(u,v)=\frac{1}{|u|}e^{\frac{u^2+\frac{v^2}{u^2}}{2}}$, $u\not =0$. Therefore by symmetry $E(U^2\mid V=v)=\frac{\int_0^{\infty}u^2p(u,v)du}{\int_0^{\infty}p(u,v)du}=\frac{|v| K_1(|v|)}{K_0(|v|)}$, $K_{\nu}$ Macdonald.