Calculating expected values: $\mathbb{E}\{ X \vert Y > y \}$ Assuming I know the conditional expected value of the form $\mathbb{E}\{ X \vert Y=y \}$, how do I calculate $\mathbb{E}\{ X \vert Y >y \}$?

$$E(X|Y>y)=\dfrac{\sum_{k=y+1}^{+\infty} E(X|Y=k)\cdot Pr[Y=k]}{1-\sum_{k=-\infty}^y Pr[Y=k]}=\dfrac{\sum_{k=y+1}^{+\infty} E(X|Y=k)\cdot Pr[Y=k]}{\sum_{k=y+1}^{+\infty} Pr[Y=k]}$$
If $Y\ge 0$ then the first expression maybe more useful.
Notice that $Pr[Y>y]=\sum_{k=y+1}^{+\infty} Pr[Y=k]=1-\sum_{k=-\infty}^{y} Pr[Y=k]$ and
$$Pr[Y=k|Y>y]=\dfrac{Pr[Y=k]}{Pr[Y>y]}\text{ if } k>y \text{ and zero otherwise.}$$
