Conditional Expectations, Variances: Stuck on a practice exercise

Question

Let $$ be a random variable with PDF (). Let $ = ^2$ be another random variable. The following should be given in terms of the PDF $()$:

• $[|]$.
• $[|]$.
• $(|)$.
• $(|)$.
• The conditional CDF of $Z$ given $Y$.

$E[Y|Z] = E[Z^2|Z] = Z^2$, but i'm not sure how to follow on the remaining questions. Any help is greatly appreciated.

Answer 1

$var(Y|Z)=0$, since $Y$ is defined by $Z$.

Given $Y$, $Z$ has two possible values. $E(Z|Y)=\sqrt{Y}\frac{f(\sqrt{Y})-f(-\sqrt{Y})}{f(\sqrt{Y})+f(-\sqrt{Y})})$

To get $var(Z|Y)$, get second moment $E(Z^2|Y)=Y$ therefore $var(Z|Y)=Y(1-(\frac{f(\sqrt{Y})-f(-\sqrt{Y})}{f(\sqrt{Y})+(f(-\sqrt{Y})})^2)$