Conditional expectation of a linear combination of a discrete random variable and a parameter

Question

I have seen this calculation in a paper, and I didn't understand the development: $\mathbb{E}[y+X|X\leq M-y]\Pr[X\leq M-y]=\sum_{l=0}^{M-y}(y+l)\Pr(X=l)$. $M$ and $y$ are two parameters and $X$ is a discrete random variable. Please can you explain to me how this result is obtained?

Answer 1

For an event $A$ of non-zero probability, the conditional expectation of a nonnegative random variable $Z$ given the event $A$ is such that $$ \mathbb E[Z\vert A]=\frac{\mathbb E[Z1_A]}{\mathbb P(A)}\cdot $$

Applying the formula above with $Z=y+X$ and $A=\{X\le M-y\}$ yields (assuming that $X$ is valued in $\mathbb N$) $$ \mathbb E[y+X\vert X\le M-y]\mathbb P(X\le M-y)=\mathbb E[(y+X)1_{\{X\le M-y\}}]=\sum_{l=0}^{M-y}(y+l)\mathbb P(X=l). $$