How do you DERIVE the BELL CURVE?

In this video, I'll derive the formula for the normal/Gaussian distribution. This argument is adapted from the work of the astronomer John Herschel in 1850 and the physicist James Clerk Maxwell in 1860.
A similar argument is presented in this video: • Normal distribution's ...
References / Further Reading:
1. The Evolution of the Normal Distribution: www.ww.ingeniousmathstat.org/s...
2. The Central Gaussian, or Normal, Distribution: www-biba.inrialpes.fr/Jaynes/c...
3. J.C. Maxwell "Illustrations of the Dynamical Theory of Gases" (p. 152-154): hermes.ffn.ub.es/luisnavarro/n...
4. en.wikipedia.org/wiki/Gaussia...
5. en.wikipedia.org/wiki/Functio...
Erratum:
32:35 - the left-hand side should be positive, not negative.

[@MomStillah]

Thank god someone cares to explain this equation that just floats around in the math realm with no explanation from teachers other than, "here!"

[@univuniveral9713]

Come on, it is not an equation floating around. There are many derivations in books. Actually this derivation is rather too long.

[@wurttmapper2200]

@@univuniveral9713 The best derivations are the easiest to understand imo

[@univuniveral9713]

@@wurttmapper2200 True

[@donegal79]

Just because you were too lazy to seek out a proof. Hey, but too easy to blame your teachers. Dufus.

[@jameslapp554]

@@donegal79 the job of a good teacher is to point a student in the right direction and not just hand wave complex topics. Plus, he did go and look for the proof as is evident in him watching this video.

[@malignusvonbottershnike563]

This video made me so happy, best 35 minutes of my day for certain, maybe my whole week. Cheers for this; if you see this comment 3 years later, you're a legend for taking the time to explain this so clearly.

[@inversemetric]

My physics professor from Greece pronounced it "φ"

[@yerr234]

wrong, i'm pretty sure it is pronounced "φ"

[@wolfpackdot]

@@yerr234 What's funny is that my professors here in Germany, even though one is from Russia and the other is a German, both pronounce it "φ".

[@bukunmiodukoya5039]

You're all funny😂

[@naman4067]

It's φ not φ

[@HisMajesty99]

This video is absolutely brilliant! I've always wanted to know how the normal distribution curve was derived, and your explanation was perfect! Thanks so much!!! Math is beautiful

[@stevenson720]

"Maths is beautiful" your so right. 😁

[@stevenson720]

@Ask why? ? differential equations. It's the maths for working out when more than one thing is changing at the same time. As opposed to normal calculus which only changes one thing. Like how far a ball goes if you throw it, if you change both the angle and how hard you throw.

[@tzywg3399]

J. I can’t really link your comments to your profile photo... it’s so illogical

[@knivesoutcatchdamouse2137]

@@stevenson720 I know my question is about 11 months too late, but regarding your reply, I.e. "differential equations ... for working out when more than one thing is changing...", etc. , isn't that actually what multivariable calculus is actually for? I suppose it really depends on whether we're talking about ordinary diff equations or partial diff equations, since ODE's deal with one independent variable, while PDE's are for multiple independent variables. I'm no math major, just an independent learner and lover of maths, so my response might not be 100% accurate, but if it's not, anyone may feel free to correct me.

[@Adam-Friended]

"We are more likely to find a dot near the bulls eye." You've oblivious never seen my wife play darts.

[@Math_oma]

+FriendEd More likely to strike someone else in the eye...amirite?

[@jorgejimenez4325]

wait what are you doing here

[@HiQuantumSKY]

It was a good joke though... But the last equation, the general one covers that too. Bad at throwing darts?? The value of σ is more. Or if your good at throwing darts but your shots clusters somewhere other than bull's eye. Your μ will be different. So never loose faith in maths, specially in general maths.

[@youtubeyoutubian7099]

It’s beautiful to observe that the number of KZbin “likes” decrease as the video is of educational nature and not about useless make-up tutorials etc. This itself IS a proof that the number of curious people actually wanting to understand and therefore watching this helpful video completely are from the “other side” of the Gaussian Distribution. ;-) Thanks for the fantastic job!!

[@Jocularious]

Dude this was a great video, keep up the great work!! I love how at 3 am in the morning I am binging on your videos, goes to show that you have skill

[@kerensahardesty9851]

This is fantastic! Everything is explained and paced so well; no other video online has derived the normal distribution so clearly as you have.

[@Barba_007]

I'm so grateful that I found this absolute gem of KZbin, keep posting videos. You probably inspired thousands of people to be more interested in math/science

[@nurulc]

What I love about this presentation, unlike some others I have seen, is that it does not skip over any steps. Each step is very clear and easy to follow. Wonderful job sir., thank you.

[@marvinbcn2]

This video is a brilliant illustration of Einstein's famous sentence: "Everything should be made as simple as possible, but not simpler". The derivation is beautiful, elegant and crystal clear. Thanks so much for sharing your knowledge!

[@HareSrinivasa]

I have always wondered about this formula. Your explanation is the most concise and understandable one even to a novice like me. Thanks a million.

[@fedexos11]

Amazing, finally someone that explains completely and holistically how to derive the Gaussian density function. Thank you!

[@andreweinhorn]

One of the best videos on the internet, love it!

[@Dawg89Kirkland]

Great vid. Cleared up much of the mystery of where normal distribution comes from. Have forgotten most of my calculus but was still able to follow along!

[@DeadlyCuddlez]

This was wonderful, thank you, I'm excited to see the rest of your channel now!

[@Math_oma]

I just started a Patreon if you appreciate the work done on this channel: www.patreon.com/Mathoma Thanks for viewing the channel!

[@giantneuralnetwork]

Really well explained. I'm relatively new to this type of thinking and it was illuminating! On the fly definitions of new functions that lead you in any direction you'd like, seems really powerful, but also like a puzzle. Thanks for making this video!

[@sreejanichatterjee9399]

I don't know what to say but you just saved my life. I have been looking for a proper derivation and not the 5 mins ones for months. Thank you thank you thank you so so so much.

[@jacoboribilik3253]

I love this video, this derivation is spectacular, I love it when mathematics links various seemingly unrelated concepts with each other and yields this beauty. That being said, I also think Gauss' proof in the article you provide is far easier and more accesible to students like myself.

[@maxwall2924]

Wow, this was awesome. I'm reading E. T. Jayne's Probability Theory. In Chapter 7, he performs this derivation, but as is often the case, he assumes the reader is as fluent as he is with functional analysis. This video really helped me fill in the gaps. Can't wait to watch the rest of your videos. Thanks a bunch!

[@vemana007]

Awesome explanation...I have been trying to understand the function behind normal curve all this while and this is so beautifully explained...thanks a ton

[@carlossp1]

This is the most clear and complete derivation of the Normal distribution I've seen. Thanks for sharing

[@wyw4466]

Great works in this video! After watching this video, I just can't appreciate enough the original inventor of this function Carl Friedrich Gauss!

[@spanishlanguageeducational3737]

Brilliant!!!! This is a MUST for any student of statistics. The way statistics are normally taught... many unanswered assumptions... which are answered by this video.

[@standidderen9270]

Wow, amazing explanation that cleared up everything about the normal distribution for me! You’re calm way of teaching is very clear and enjoyable, thank you!

[@lucasmoratoaraujo8433]

Thank you! Most simple and well demonstrated video about the subject that I've seen. Congratulations!

[@viniciuscaina8400]

I'm from Brazil and i just found this class right here. I ask to my professor how to derivate this formula and she did not know it. This was one of the most impressive class that i ever saw, Thank you so much!!!!

[@rileyjeffries1282]

I am incredibly new to statistics and have never actually taken a course, but I have taken physics and engineering courses that apply physics; it's pretty neat to see that the dirac delta function is a limiting case of the normal distribution in which lambda approaches infinity, which I realized as soon as you showed how lambda transforms the shape of the graph. Very cool video!

[@kg6395]

We need great explainers like you... Awesomely explained.

[@soffer]

Seriously Mathoma... thank you so much sir. Thank you. You are doing God's work. You will ride shiny and chrome in Valmatha. Excellent video.

[@ahmedidris305]

Crystal clear explanation, thank you Sir for the great work!

[@SK-ck3qb]

This is an outstanding video. You have explained all the details with clarity. Thank you!

[@rajendramisir3530]

Absolutely! I enjoy this fresh look of the derivation of this class of Gaussian functions. I like the way you explained

[@davidkombe3718]

This was truly beautiful. Thank you so much for the great content!

[@charliebrannigan1275]

Thanks so much for the clarity of the video.

[@willdavis2053]

Man, what an incredible video! Loved your derivation.

[@Kriegsherr2]

Yes. This video Is perfect. I've been looking for this for years. You are wonderful. Thank you so much. Excellent explanation!!!

[@AndreanoCelentano]

Beautiful! Thank you so much. I wish this was around when I went through my MS.

[@prajwalchauhan6440]

Amazing... This video really helps to understand the Gaussian distribution a lot better. Thank you.

[@libertarianspirit]

I'm software developer. I'm pretty comfortable with discrete math and terrible with calculus. But your explanation is so clear that I was able to understand most of it. Thank you for such high quality content. This video deserves more views! I'm subscribed (which happens rarely!).

[@paula.agbodza5135]

Absolutely brilliant! You make Mathematics look like what it's meant to be, simple. Thank you for this great video.

[@zairoxs]

It took me 2 days to understand the concept behind this topic for my statistics class. This video cleared everything up for me. Thx!!

[@akenny4308]

I have been looking for a video like this for so long, a clear derivation from scratch of the normal distribution

[@atakan716]

Thank you for taking your time and explaining it beautifully!

[@jackhuangly89]

Fantastic explanation, step by step and does not assume the viewer knows any particular mathematical derivation. best video i've seen on the derivation of the normal distribution!

[@sureshapte7674]

this is a fantastic lecture and literally reveals mathemagic of normal distribution curve, i am going to see this again and again and...

[@stevenson720]

Clear clean well described well paced, excellent. Thank you.

[@invisi.]

this is so well presented, thank you so much for this

[@Willzp360]

Amazingly satisfying video; I've enjoyed your videos for a long time and this was particularly good. Everything is explained at just the right level and the derivation was so logical it all felt obvious by the end. Thank you for putting so much effort into your videos!

[@Math_oma]

+Will Price That's very kind of you to say. My pleasure.

[@imrematajz1624]

Beautiful derivation of a ubiquitous formula: the hat trick comes at 22:48! To complete the discovery of this treasure, I am now deep diving to revise my Euler integral. Thanks so much!

[@skeletonrowdie1768]

you nailed it with this video tho! it's so cool to see that this derivation is actually an insight derived from a multivariable case.

[@justforknowledge6367]

Thank you for the brilliant derivation from nearly the First Principles. Thank you indeed. I deeply expect that there should have been a "History of Mathematics" YT channel, along with providing the reasons for the clever decisions taken at crucial steps to derive historically important equations. These steps have nothing to do with computation, but just a "Leap of Intelligence", because of which mathematics has prospered for so long. Began with Pythagoras's proof of root2 being an irrational number.

[@aztecterp]

Thank you for this video! Also, you have a very steady hand when drawing the normal distribution. I can't do it that well.

[@gyeonghunkang7215]

Great video!! Helped me immensely in understanding where the normal dist. pdf came from. Thnx a lot😆😆

[@davidmwakima3027]

I like how you're writing the integral signs! Anyway this is a super clear video, thank you so much! I'm reading Maxwell's 1859 paper and I wasn't sure where the Ce^Ax^2 came from.

[@diegosantoyo3322]

Esto es bellisimo, me ayudo de una manera impresionante en mi clase de diseño de experimentos. Simplemente gracias!, no lo hubiera logrado sin ti This is beautiful, it helps me in an impressive way in my experiment design class. Simply thank you!, I wouldnt achieve it without you

[@adityakhedekar9669]

best video for derivation of gaussian distribution ever.

[@CBenky]

Beautiful explanation

[@jerekabi8480]

Thanks. Very clearly explained.

[@pianoman47]

This was a great explanation! Thank you!

[@rodrigoserra2112]

Thanks for this! Really, really helpful! Will you make videos about other distributions?

[@sureshapte7674]

thanks for wonderful presentation

[@rineeshparai1780]

I actually enjoyed watching this video. I expected to learn, but i never expected to enjoy the derivation of PDF. This was fun!

[@Ryan-sj5qp]

Amazingly done

[@lilagraham6481]

For the transition from y = 0 to the general case, the 1D equation can be generalized to 2D due to radial symmetry, which makes the x axis equivalent to any other line going through (0,0). Regarding the number of dimensions, a minimum of two is necessary to specify coordinate independence and radial symmetry, which together give the form of an exponential. Lovely, unique video. Thanks!

[@labalimbu829]

wow i am watching your video at 12/25/2020 and i suppose your video is my Christmas gift. so beautiful explanation.

[@momozhang8190]

Thanks a lot! the explanation​ is fantastic!

[@zhengyuanwei4544]

most beautiful prove I saw!!!

[@libertyhopeful18]

hey great idea for a video. i found in undergrad that my stats prof didnt really care to elaborate on this, and so stats ended up being my least favorite math class. turns out you need to understand this stuff to do my actual job, so this is much appreciated

[@Caradaoutradimensao]

no words to thank you enough

[@silvera1109]

Thanks! Very nicely explained indeed.

[@dongpengcao3938]

很好的推导，非常感谢

[@foggs2275]

fantastic video , really enjoyed that.

[@azimuth4850]

Loved it. And to think that's the just beginning....

[@tjdahlke]

Very well done!

[@bca-ig1nb]

Awesome derivation . This really helped. Thanks.

[@kashiramk.c3046]

Absolutely brilliant👍

[@duckymomo7935]

oh, thats how you get e^-(x^2)

[@Math_oma]

+Mi Les Indeed.

[@madhukareppalapelly4239]

You are awesome bro.. Finally someone cared about proof..

Huge thanks, that was a great video

[@soumyajain6982]

Just blown!!! Thank you so so much!!😁😁😁

[@guvencagil5043]

It's a very good video. Thank you. I just wish the frequent commercials weren't as loud.

[@ThankYouAYODHYA]

😮😮😮👏👏👏👌 awesome... for years this has been troubling me how to derive normal distribution equation... trust me I had night mares... 😅😅😅 I am too bad with just mugging up... thanks a ton... I used to wonder how we are able to connect two independent sample space through mysterious Z... now its quite clear. 1. The sample space has to behave like a normal distribution phenomena 2. Anyway the probability density curve will have area as 1... and that makes me to understand t distribution even better... thank you...

[@hsyang5252]

really FANTASTIC!!

[@akaskaul]

Thanks a ton. I loved the explanation. Will you also sometime try to do a derivation for multivariate Gaussian distribution?

[@stratpap637]

perfect video...congratulations!!

[@sedgieroobets]

Excellent! Thank you!

[@Lukas-zl5zs]

Really nice video, thanks

[@alpha_kappa_357]

i was so confused how the "2" comes in the formula now i finally understand thanks :)

[@rhke6789]

excellent pesentation of the derivation. congrats

[@lutfilutfi3310]

excellent derivation very intuitive,i needed it for understanding gaussian regression

[@vasishtapolisetty639]

Wonderful video! Keep up the good work :)

[@nindergardjason9271]

great explanation

[@BoZhaoengineering]

Class teacher often omit the derivation of Normal Distribution. I always wonder how the bell curve formula is derived. Here the answer I am. Thanks a lot.

[@hdmat101]

This video is quite interesting, I got bored one afternoon and I searched the derivation for Normal distribution because I was learning probability questions involving Normal distributions in school where we use a table to find the values for standardized normal distribution where n=1 s=0 using a table . This is really good except that I lack the attention span to understand most of this

[@HarshitaSolanky-bc1yf]

We love you from the bottom of our hearts. Everything is soooooooooo clear unlike other videos on youtube. I am trying to learn data analytics, on my own from youtube for free Cause I want a neww skill your standard deviation

[@panospanagiotou3233]

Thank you. Really helpful

[@BarkanUgurlu]

thanks! this has been what I am looking for. finally some mathematical rigor!

[@aydinahmadli7005]

I feel enlightened, thank you 😄✋