What is the average distance of two points in a disc? (PART 1)

This seemingly simple question actually encompasses very rich connections between different topics in mathematics, including statistics / probability theory, Jacobian / multivariable calculus, and even differential equations. This video is the first part in a video series tackling this question, which tries to make the problem more mathematically precise by breaking down the words “random“ and “average“, which can have precise mathematical definition, using probability density function, and the concept of mean of a function. The “joke“ in the beginning of the video is inspired by (read: “copied from“) CGPGrey’s “How many countries are there?“ Tom Scott’s “How many languages are there?“ and Tom Scott’s “The Never-Used Road Where The BBC Crash Cars“ Even though this problem “highlights the unity and utility of the undergraduate mathematics curriculum“ (from the paper below), I would assume you know nothing, so don’t worry if you are not in university / have a degree in mathematics! If you are in college / university, hopefully the first few videos can be a nice revision and application of the concepts, and possibly a new perspective on the concepts. The paper that I am following (very readable, for an undergraduate at least): I do notice that MindYourDecisions made a similar video ( a few years ago but for a unit *square* instead. I still make this video series because (1) the unit disc version is much harder to tackle in the sense that we are not even attempting to evaluate the integral, and (2) Presh’s video seemed to pull pdf’s and Jacobian out of nowhere, which might be confusing to people who have not gone to college to study mathematics, and genuinely quite a different level of difficulty from his other videos, so I am going to actually explain what those are. Thanks to all my subscribers, because this channel has grown a lot since the beginning of the year, and this cannot happen without your support! Merry Christmas / happy holidays! We shall see next year, which hopefully will be better :) Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels: If you want to know more interesting Mathematics, stay tuned for the next video! SUBSCRIBE and see you in the next video! If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don’t use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe! Social media: Facebook: Instagram: Twitter: For my contact email, check my About page on a PC. See you next year!