Let $X, Y$ be a two discrete random variables, $X$ takes values from set $\{0,1,2,...,n\}$, and $Y$ takes values from set $\{0,1,2,...,n,n+1\}$. $X, Y$ are independent.
Let $Z = \max\{X, Y\}$
Is there a general formula for $E[Z]$?
I know that $P(Z\le t)=P(X\le t)P(Y\le t)$, since they are independent. I know also that: $$E[X]=\sum_{m=0}^{n}(1-P(X\le m))$$ $$E[Y]=\sum_{m=0}^{n+1}(1-P(Y\le m))$$ and $$E[Z]=\sum_{m=0}^{n+1}(1-P(X\le m)P(Y\le m))$$ But i don't know how to relate these equations.
However i belive that there is an answer, because if $X, Y$ were a continuous variables, then we would do the following.
$F_{Z}(t)=F_{X}(t)F_{Y}(t)$
Then we would find a derivative of this product of distirbutions, to get $f_{Z}(t)$ and finally calculate expected value.
Is it possible to convert expected value of discrete random variables into the expected value of continuous variables?