Why π is in the normal distribution (beyond integral tricks)

From 3Blue1Brown.

Where’s the circle? And how does it relate to where e^(-x^2) comes from?
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The artwork in this video is by Kurt Bruns, aided by Midjourney

Here are several other good posts about the classic Poisson proof

vcubingx: https://www.youtube.com/watch?v=9CgOthUUdw4
BriTheMathGuy: https://www.youtube.com/watch?v=S79KPrIm_Gc
Dr. Alter’s math library: https://idan-alter.github.io/2023/02/20/Gaussian-Integral.html

And if you’d like to see many other variations on approaching this integral, take a look at this expository paper from Keith Conrad: https://kconrad.math.uconn.edu/blurbs/analysis/gaussianintegral.pdf

0:00 – The statistician’s friend
3:44 – The classic proof
12:47 – The Herschel-Maxwell derivation
21:55 – Reflecting back on the proof
24:39 – A bonus problem


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