The concept

When you want to compute $E[X + Y]$, you can just do $E[X] + E[Y]$. But what if you want $X + Y$ by itself? It turns out, this addition operation needs a sort of pairwise operation. For every value, $k$, you have multiple ways that $X + Y = k$.

You create the distribution $X+ Y$ by convolving the distributions with each other, which is equivalent to

where $Z = X + Y$

Common Convolutions

• Convolutions of bernoulli is binomial

• Convolutions of gaussian is gaussian (this one is very cool and useful)