Why π is in the normal distribution (beyond integral tricks)
1,541,864
Published 2023-04-02
Help fund future projects: www.patreon.com/3blue1brown
Special thanks to these supporters: www.3blue1brown.com/lessons/gaussian-integral#than…
An equally valuable form of support is to simply share the videos.
The artwork in this video is by Kurt Bruns, aided by Midjourney
Here are several other good posts about the classic Poisson proof
vcubingx: • Why does pi show up here? | The Gauss...
BriTheMathGuy: • This isn't a Circle - Why is Pi here?
Dr. Alter's math library: idan-alter.github.io/2023/02/20/Gaussian-Integral.…
And if you'd like to see many other variations on approaching this integral, take a look at this expository paper from Keith Conrad: kconrad.math.uconn.edu/blurbs/analysis/gaussianint…
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
------------------
These animations are largely made using a custom python library, manim. See the FAQ comments here:
www.3blue1brown.com/faq#manim
github.com/3b1b/manim
github.com/ManimCommunity/manim/
You can find code for specific videos and projects here:
github.com/3b1b/videos/
Music by Vincent Rubinetti.
www.vincentrubinetti.com/
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/album/the-music-of-3bl…
Stream the music on Spotify:
open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe
Various social media stuffs:
Website: www.3blue1brown.com/
Twitter: twitter.com/3blue1brown
Reddit: www.reddit.com/r/3blue1brown
Instagram: www.instagram.com/3blue1brown
Patreon: patreon.com/3blue1brown
Facebook: www.facebook.com/3blue1brown
All Comments (21)
-
The follow-up to explain how this fits into the central limit theorem: https://youtu.be/d_qvLDhkg00 That video, in turn, benefits from a little prerequisite knowledge about convolutions, which I cover here: https://youtu.be/IaSGqQa5O-M
-
"Who ordered another dimension" 😂 classic mathematician path to solving a problem
-
Who else came here just to listen Grant speak Korean?
-
Grant, as a lowly college lecturer with insufficient funds to donate to your cause, I must nonetheless congratulate you on another masterpiece. Your visualizations are second to none and your teaching is beyond fantastic. Thank you for your contributions to mathematics.
-
I think pi gets really sad whenever e is not around. It's not just love. It's a rather complex relationship. No wonder, both of them seem to be a bit irrational. Especially e gets fouriously impotent when pi is not around, despite pi's negativity.
-
1:45 put such a wide smile on my face and reminded me why I love this channel so much. So often the explanation for why a thing is is entirely proof based. I love proofs, and coming up with proofs is a wonderful experience of problem solving, but on their own they cannot satisfy my disbelief. Stuff like this, using the concepts and reasoning /within/ the proofs to make a point, is exactly what I love about math. Thank you, grant!
-
Him saying "feel less out of the blue" at 22:10 after deriving the proof visually and in BLUE is like the 10th bonus point for this channel. I love it so much
-
That we live in an age where educators have the opportunity to unpack the meaning and history behind some of the greatest mathematical discoveries for a substantially large audience is a privilege that we should all be infinitely grateful for.
-
Just watched the video again with your Korean AI voice. As a Korean, I'm genuinely blown away—it sounds incredibly natural! Imagining how this will broaden accessibility to your fantastic math content is truly exciting 👏
-
Grant, I’m a mathematician and math educator. I’ve of course seen the integration proof of the computation of the area under e^{-x^2}, but never in my life have I either seen or come up with such an elegant demonstration for why we MUST expect pi to show up in the Gaussian. Thank you, sir. Actually this gives me an idea for an in-class activity for my future analysis students…
-
Finally, the much awaited 3b1b statistics series is on a roll!
-
The Korean version sounds very natural. The pipeline works incredibly well!
-
More than perhaps any video in 3b1b, this one shows how learning math history makes one a better mathematician. What a great lesson!
-
This makes me think: a series on statistics would be excellent. I am sure there is a lot of visualisation behind all the sum of squares and F statistics, of interactions and everything, that are never taught. Even "serious" books barely talk about the intuition of the sum of squares beyond how they are derived from LRT.
-
I'm guessing Grant is building up to a 23 minute video explaining a function that describes everything everywhere
-
I love everything about your videos. Your amazing animations, masterful scripts, pleasant and well recorded voice, tight editing. It all comes together to create some of the best educational content the world has ever seen. Thank you for sharing this for free and enriching the intellectual lives of so many people. ❤
-
I love the pieces of art that aren’t normally part of the “aesthetic” of 3b1b but somehow still fit right in.
-
Just wanna Say, Thanks for doing whatever you are doing. Never stop 3B1B
-
I doubt anyone can possibly make a better visualization for explaining this proof. The quality of your videos is truly on another level
-
Im here for the Korean AI dub
