Let $ X \sim \operatorname{Exp}(\lambda =1), Y\sim U(1,2) $ be independent continuous variables. What is $E(\frac{x}{y})$?

Question

I'm struggling to understand how to start the following:

Let $ X \sim \operatorname{Exp}(\lambda =1), Y\sim U(1,2) $ be independent continuous variables. What is $E(\frac{x}{y})$?

Thanks :)

Answer 1

$E(\frac X Y |Y)=\frac 1 Y EX=\frac 1 Y$. Taking expectation we get $E\frac X Y=E\frac 1 Y=\int_1^{2} \frac 1 y \, dy=\ln\, 2$.

Answer 2

Guide:

If $X$ and $Y$ are independent then so are $f(X)$ and $g(Y)$ where $f$ and $g$ are measurable functions.

Consequence for expectation (if it exists):$$\mathbb Ef(X)g(Y)=\mathbb Ef(X)\mathbb Eg(Y)$$

This can be applied to find: $$\mathbb E[XY^{-1}]=\mathbb EX\mathbb EY^{-1}$$ It only remains now to find $\mathbb EX$ and $\mathbb EY^{-1}$.