I have a question about conditional expectations. In my problem, $X$ and $Y$ and independent standard normal random variables, and $Z = X^3 + Y^2$. I am asked to find $E(Z|X)$.
I was thinking of using the tower property of conditional expectations (or the law of total expectation) to find that:
$$ E(Z) = E[E(Z|X)] $$
where
$$ E(Z) = E(X^3+Y^2) = E(X^3) + E(Y^2) = 1 $$
using the fact that for standard normal variables, $E(X^3) = 0$ and $E(Y^2) = 1$. This then implies that:
$$ E(Z) = E[E(Z|X)] = 1 $$
which then implies that $E(Z|X) = 1$.
I'm not entirely sure if my approach is "too easy" or if there's an appropriate way of calculating this conditional expectation. Any hints or suggestions would be greatly appreciated.
just write it out: $$ E[Z\mid X]=E[X^3+Y^2\mid X]=E[X^3\mid X]+E[Y^2\mid X] $$ the first term is simply $X^3$ for $E[h(X)\mid X]=h(X)$, the second one is $E(Y^2)=1$ by independence.