E(X)/E(Y) compared to E(X/Y)

Question

Is there any sort of inequality stating the relationship between the two?

1) if X and Y are independent they are equal, I think. Since 1/Y will be independent to X too as well right?

2) but what if two are dependent?

Answer 1

If $X, Y$ are independent then $$ E\left[\frac{X}{Y}\right] = E[X]E\left[\frac{1}{Y}\right]$$ In general $E[\frac{1}{Y}]$ is not the same as $\frac{1}{E[Y]}$. Let's assume $E[X]$, $E[\frac{1}{Y}]$ are finite.

The function $1/y$ is strictly convex over the domain $y>0$. So if $Y>0$ with prob 1, then by Jensen’s inequality we have: $$ E\left[\frac{1}{Y}\right] \geq \frac{1}{E[Y]} $$ with equality if and only if $Var(Y)=0$. So if $X,Y$ independent and if $Y>0$ with prob 1 then

• $E[X]=0 \implies E\left[\frac{X}{Y}\right] = 0= \frac{E[X]}{E[Y]}$.

• $E[X]>0 \implies E\left[\frac{X}{Y}\right] \geq \frac{E[X]}{E[Y]}$ with equality if and only if $Var(Y)=0$.

• $E[X]<0 \implies E\left[\frac{X}{Y}\right] \leq \frac{E[X]}{E[Y]}$ with equality if and only if $Var(Y)=0$.