X+Y, in probability, is a beautiful mess

Visualizing continuous convolutions

Continuing with the theme of probability videos, where the ultimate plan is to fold these into a proper “Essence of Probability” series, the newest video is all about adding random variables.

More specifically, the main focus here is on how to visualize a convolution between two functions, building up two ways to think about it. The first is what you’ll see in more standard explanations out there, but the second seems to be much less commonly discussed. In the end, I heavily hint at how to use this second view to do a rather pleasing (and important!) computation.