Expected value of maximum of discrete and continuous random variable

Question

Just looking for confirmation of what I believe to be true. Let $\tilde{x}\in\{x^L,x^H\}$ with $Pr(x^L)=p$ and $Pr(x^H)=1-p$. Also let $\tilde{y}$ be a continuous random variable with unspecified distribution function. Then, is the following correct?

$E[max\{\tilde{x},\tilde{y}\}] = p\cdot E[max\{x^L,\tilde{y}\}] + (1-p)\cdot E[max\{x^H,\tilde{y}\}]$.

Thanks.

Answer 1

It's true if $\bar x$ and $\bar y$ are independent, because then $E[\max\{x^L,\bar y\}]=E[\max\{\bar x,\bar y\}\mid\bar x=x^L]$, and the same for $x^H$. It's not normally true otherwise.