Inequalities involving $E\left(\frac DC\right)$ and $E\left(\frac CD\right)$

Question

Consider 4 positive random variables, let's call them A,B,C,D and suppose their expectations exist. If the following is true $$ \mathbb{E}\Big[ \frac{D}{C} \Big] > \frac{\mathbb{E}[B]}{\mathbb{E}[A]} $$

Can I state the following , $$ \mathbb{E}\Big[ \frac{C}{D} \Big] < \frac{\mathbb{E}[A]}{\mathbb{E}[B]} $$

to be true?

Thanks a lot to anyone who can shed light on this point.

Answer 1

Your question can be refrased as:

If the expectations exist, then do we always have $\mathsf EX\mathsf EX^{-1}<1$ for positive random variable $X$?

The answer is: "no".

E.g. let $X$ have distribution $\mathsf P(X=2)=\frac12=\mathsf P(X=\frac1{2})$ so that $X$ and $X^{-1}$ have equal distribution with $\mathsf EX=\frac54=\mathsf EX^{-1}.$

It fits in your question by taking $C=1=A=B$ and $D=X$.