Adding random variable, and continuous convolutions (Early view for patrons)

Hey everyone,

Continuing with the theme of probability videos, which I plan to eventually coalesce into a proper series sometime this year, this next video covers adding random variables. That is to say, convolutions.

Initially, I intended for this to be "part 2" to the convolution video from a few months ago, this time focused more exclusively on the continuous case. It builds up two visualizations, and for each one, it's very helpful to have freshly seen what it looks like for a simple discrete example (rolling a pair of dice is irresistibly relatable). The ultimate plan is for this to become a chapter in a probability series, and it would be a bit awkward to make that other video a prerequisite or to insert it into the series, given that it was mostly not about probability. So at the risk of a bit of redundancy, I decided to include the discrete case in this video, which is part of why this video took a few more weeks than expected.

I had also planned on ending the video by walking through how the diagonal slice point of view lends itself to a clean visual intuition for why convolving two Gaussian distributions gives another Gaussian, and how this relates to the central limit theorem. After incorporating the warmup examples, it was all running a bit too long, so that will be its own video. The aim is to get that out to you pretty soon.

Separating out that example may actually be the best way to do it anyway, though, because hopefully many people will be able to predict what that argument looks like, and rediscovery is always much more fun than being told something directly. A gap between videos does more to encourage someone to try it out themselves than a simple "pause and ponder".

As always, let me know if you spot any errors!

Grant