How to find the moment generating function of Y|X=x

Question

I am used to work with simple MGF and I know that

$$MGF(t)= \mathbb{E}[e^{xt}]$$

but now I need to work with the MGF of $Y|_{X=x}$, I know that:

$f(y|x)= \frac{1}{2-x}$

$f(x,y) = \frac{1}{2}$

$f(y)=\frac{y}{2}$

And that $0< x < y <2$

Answer 1

Just as $\mathsf M_X(t)=\mathsf E(e^{Xt})$, so too $\mathsf M_{Y\mid X=x}(s)=\mathsf E(e^{Ys}\mid X=x)$.

So apply the usual definition for expectation using $f_{Y\mid X}(y\mid x)$.