How to calculate $E(X^Y)$ where X and Y are both random variables?

Question

My personal guess:

The general rule: $$E(X^Y)=E_Y(E_X(X^y)|Y=y)$$ or $$E(X^Y)=\iint x^y f(x,y) dx dy$$

When $X$ and $Y$ are independent:

$$E(X^Y)=E(X)^{E(Y)}$$

Is it correct?

Answer 1

Suppose $X$ and $Y$ are i.i.d. with $P(X = 1) = P(X = 2) = \frac{1}{2}$.

What is $E(X^Y)$ and what is $E(X)^{E(Y)}$?