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

2023-04-02

[public] 285K views, 49.2K likes, dislikes audio only

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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

Correction: 13:46 - The denominator should read 2πσ^2

Timestamps:

0:00 - The statistician's friend

3:44 - The classic proof

12:47 - The Herschel-Maxwell derivation

21:55 - Reflecting back on the proof

23:50 - A bonus problem

Thanks to these viewers for their contributions to translations

French: Jeremy Senn, Laurent Joubert

German: dreivmeister, lprecord, mathiashaebich

Hebrew: David Bar-On, Omer Tuchfeld

Hindi: prateekbansal97

Spanish: Joel Tovar

Turkish: kilobravo3

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