Expectation of ratio of two distributions is the ration of the expectations

Question

Given three random variables $X$ ,$Z$ and $Y$, does this hold in general: $E(\frac{X}{Z+X+Y})= \frac{E(X)}{E(X+Y+Z)}$? If no, then in what circumstances will this hold?

Answer 1

Let $S=X+Y+Z$ and $$ T=\frac{X}{S} $$ If $S$ is independent of $T$, then $$ EX=ES\times ET $$ and $$ ES=\frac{EX}{ET}=\frac{EX}{EX+EY+EZ} $$ An example of where this might arise in practice occurs in the formulation of the dirichlet distribution