- My Summer of Math Exposition Explainer - Buffon's needle problem. Estimate $\pi$ by throwing items onto a grid. I made this video for SoME 2021 [https://youtu.be/e-RUyCs9B08]( https://youtu.be/e-RUyCs9B08) - The ABRACADABRA theorem: In an IID sequence of random variables (e.g. random letters a typerwriter), how long until a particular sequence (e.g. "ABRACADABRA") is observed? I made this video for SoME 2022 [https://youtu.be/t8xqMxlZz9Y](https://youtu.be/t8xqMxlZz9Y) - Moment Method Proof of the Central Limit Theorem. I did this video for SoME 2023 [https://youtu.be/oPQ4mNcqY7k](https://youtu.be/oPQ4mNcqY7k) - Future Ideas - Probability - [[How many dice rolls problem?]] Roll a $d$ sided dice until the sum of the dice so far exceed $d$. It turns out you need on average $\approx e=2.718...$ dice rolls to do this! - [[Why do Beta Random variables work?]] Explain how beta random variables work with no nasty integrals - The Devil's Staircase | Cantors random variable. A probability distribution that is neither discrete (i.e. $\mathbb{P}(X=a) = 0$ for every $a$) and yet it has no probabilituy density (i.e. there is no function $\rho$ so that $\mathbb{P}(X \in A) = \intop_A \rho(x) dx$ ) - Semi-circle law proof (similar to CLT moment method) - Why moments determine the distribution (Lagrange polynomial for discrete distributions and Bernstein Polynomial for continuous distributions) - Proof of Bernstein Polynomial convergence using sums of Bernoullis - Visual proof of Jensen's inequality - Future Ideas - Misc - Patterns in [[Pascal's Triangle]] . Explain why it repeats itself recursively when you look mod p. (An unlisted sneak peek of the animation for this is https://youtu.be/vlfwWxBwM6o) - How [[RSA public-key-private-key Encryption]] works by going from very simple Cesar Shift Cypher to "Baby RSA" to true RSA, all explained using [[Fermat's Little Theorem]]. - Vector Programming in NumPy - Example using the [[pairwise distance problem]] for a 250x speedup - Simple Fibonacci Ideas: Miles to Km using Fibonacci and how you can see this by looking at fixed points of the equation for the ratios $r_n = \frac{F_n}{F_{n-1}}$ that $r_{n} = 1 + \frac{1}{r_{n-1}}$ You can also get Binet's formula from this by doing a linear combination of the two fixed points. - Manim Explainer Video: Show integration by parts but "livestream" how to make it in manim