The Binomial Distribution: Mathematically Deriving the Mean and Variance

[@DSEC_DGDharshan]

Wow sir, you just saved me , before I stumbled upon your video, I searched many websites for the derivation but never was it as elegant and simple as yours, I hope youtubers who post such academic content get more recognition and paid properly instead of many of my lecturers who swallow money unjustly

[@mooreg87]

This was excellent! I'm in mathematical statistics 1 and this video helped clarify quite a bit. The relationships at work are made very clear, thank you for your help.

[@jbstatistics]

Thanks for the feedback and compliment Cuong! I'm glad you found this video helpful.

[@jbstatistics]

You're welcome Amogh! I'm glad you found it helpful. Cheers.

[@just4listening]

I have searched for this for days. Thank you so much. Learning and understanding feels so good.

[@abubardewa939]

Give that man a cookie :) ... That was almost the standard of Euler.Well done !

[@jbstatistics]

+Abu Bardewa Thanks!

[@kausalyaakannan7064]

Best of besttttt❤ I m in statistic field for 2 years and only understand about this in detail today. I dont know how to express my gratitude for providing this tutorial😭😭 Thank you so muchhhhh💓💓

[@katiehung3632]

Your explanation by far is the most easy to understand and helpful! Thanks a lot!

[@khcrafts1972]

You saved my life.. Excellent tutorial, crystal-clear explanations. Thank you very much sir!

[@jbstatistics]

You are very welcome. Thanks for the kind words!

[@jbstatistics]

Thanks Wilson! I'm very glad to be of help. Cheers.

[@fengyun1991]

Hi Prof, your descriptions are very detailed and clearly explained. Keep up the good job! It's benefiting students like us! :)

[@feelgood0108]

I think nobody else should have explained in a better way than you sir. Thank you

[@jbstatistics]

You are very welcome. Thanks for the compliment!

[@akibkhan9916]

This is amazing. The way the logic and the calculation are presented is very elegant. I loved the second method.

[@jbstatistics]

Thanks for the very kind words!

[@johnchan9598]

good presentation. I get confused from my professor but now everything are clear.

[@jbstatistics]

Great! I'm glad to be of help.

[@gidaygebrehiwot3308]

This video enables me to confidently continue loving and doing advanced mathematical statistics problems. Thank you for your brief video. Please send me related videos.

[@jbstatistics]

I'm glad I could help! I have many videos available on KZbin, and will be adding more this year.

[@jbstatistics]

You are very welcome. I'm glad to be of help.

[@cujo006]

Thanks for explaining why m replaces n on top of sigma! Very important and nowhere else (book or online) have I seen it.

[@jbstatistics]

You're welcome Philip. Thanks for the feedback!

[@all_sonagomes485]

Watching this a night before my exam, phew! saved!

[@jbstatistics]

I hope your exam went well!

[@VikasSingh-tw8wu]

Hassssss..... (relief).... finally after watching bunch of videos I got a good tutorial.

[@sahildalvi6892]

Thankyou for this. Was struggling to understand but you made it so easy.

[@firasb-ck1dj]

This is an awesome explanation , clear and comprehensive

[@jbstatistics]

You are very welcome Marcela!

[@danastantasaoufmridula222]

thank u so much ! you save me this time for passing my exam ...

[@jbstatistics]

Thanks Rohan!

[@ramamurthyambhikapathy3661]

Excellent and clear explanation sir. ..very easy to understand..

[@sumayyah4273]

You r toooo goood man.. Hats off.. (Y) I UNDERSTOOD WAT THE TEACHER EXPLAINED THROUGH YOUR VIDEO.. KEEP IT UP!!

[@kaushik540]

excellent explanation and method

[@miyabialenabarth9265]

Thanks for the concise logical, step by step explanation!

[@gagadaddy8713]

The derivation is brillant!

[@bestseller24987]

It's great and very useful. A lot of calculations but it really helps with the concept.

[@GVINEELA-pr7ux]

Your explanation is Very Good Sir Thank you so much for making this video Sir🤗

[@abel5545]

An amazing explanation to something I was told to memorize in school ! Thanks for making this :D

[@ShaunWong1997]

Wow you are amazing. The way you derive is genius :D Thanks a lot for helping.

[@sparrshnagdda1190]

Extremely helpful and detailed!

[@KartikyanDogra]

thanks you helped me in clarify my doubt perfectly.👍

[@jbstatistics]

You are very welcome!

[@allengrant5874]

A fantastic explanation of a concept by which I was confounded! Thank you.

[@mike-yj5mm]

For the variance proof, why the probability does not change accord to the expectation's change. (I mean E(x^2) = sigma(x^2 * p(X^2)), but in the video, you use the E(x^2) = sigma(x^2 * p(X))). Appreciate for any idea.

[@jbstatistics]

Good question. The law of the unconscious statistician tells us that E(g(X)) = sum(g(x)p(x)), where the summation is over all possible values of X and p(x) = P(X=x). As you bring up, we could also use E(g(X)) = sum(g(x)f(g(x))), where the summation is over all possible values of g(X) and f(g(x)) = P(g(X) = g(x)). It's typically easier to use the law of the unconscious statistician (and we do, almost unconsciously at times), rather than have to work out the distribution of g(X).

[@varungupta5841]

Beautifully done. Thank you very much.

[@mikemayo4938]

Thank you thank you thank you. Put all my doubts as ease!

[@jbstatistics]

You are very welcome!

[@jimmyleo4718]

Thanks man! Excellent tutorial with good explanations.

[@yosefronaldo]

Thank you so much for the explanation!

[@uvaismohammad4216]

Thanks... You are really doing great job

[@dharmendrachaurasia5469]

Gd explaination each and every steps Finally I got it thanks

[@jbstatistics]

You are very welcome!

[@premkrishna1821]

Excellent explanation I get clarified from this Thank you for video

[@emperorpingusmathchannel5365]

9:04 Why did you write p(x)*(x^2-x) instead of p(x^2-x)*(x^2-x) in the series? Since x(x-1) is just another value. Shouldn't it be multiplied with the probability of x(x-1) happening?

[@jbstatistics]

The Law of the Unconscious Statistician tells us that E[g(X)] = sum g(x)p(x). We don't need to work out the distribution of g(X). We *could* do it that way, but it's an unneeded extra step.

[@kautukraj]

Very helpful, thanks a lot.

[@christopherrayeroux843]

from what country are you? that awesome. thanks for imparting your talent

[@syedahmedali7417]

you are outstanding...my God great ...

[@hassanghani1878]

Thanks . God bless you. great methodology, easier to understand

[@nikitasingh6127]

Good explaination! Thanks a lot sir! keep up the good work.

[@poojagoyal869]

Best tutorial....Thanks a ton sir....

[@jbstatistics]

You are very welcome!

[@rochakmehrotra7121]

Brilliant! You explained it very smoothly

[@abhayraj3568]

A great thank for so clearly explaining this proof.

[@roshanroy4566]

awesome work sir really glad now, can you do such ones for the remaining distributions too

[@jbstatistics]

Thanks! I do have a similar video for the Poisson distribution. I'll find some time to make more videos one of these days, and fill in some of the gaps in the content. Cheers.

[@mandrinnd10]

This is great . What program are you using to write this?

[@jbstatistics]

The base is a Latex/Beamer presentation. I annotate with Skim, and record & edit with Screenflow.

[@anabananacapacabana2427]

Thank you so much for all your wonderful videos and energy, they've really helped me a lot!I just have a really quick question: how come I can convert the power of p from p^x to p^(x-1) and then for variance from p^x to p^(x-2)? I understand how this is done with factorials but very confused about how the same method can be applied to powers. I would really appreciate any explanation and thank you in advance!

[@davemcintosh3359]

he took out p to the left of the sum sign... using power laws, p^a times p^b = p^(a+b). If you know that, and you know that p = p^1, then it follows that p times p^(x-1) = p^1 times p^(x-1) = p^(1+x-1) = p^x. Just a trick to get p outside the summation sign. hope that helps

[@shannusharmila4013]

Helpful video, please explain derivation part of the binomial distribution

[@HabibKhan-jr9id]

Very Clear Explanation. Thanks

[@jbstatistics]

You are very welcome!

[@kajolyadav890]

I love you 💕 I was about to cry ❤ thanks for this😘

[@jbstatistics]

You are very welcome!

[@rakibulhashanrabbi]

Thanks for detailed information

[@johnwalu9564]

i just appreciate you have made bit exact and precise!!

[@jbstatistics]

I'm glad I could be of help!

[@manishkumaryadav4303]

Thanks a lot sir for the wonderful explanation...It helped me a lot

[@k-6779]

thank you !!! :) This video helped me a lot with my assignment. haha

[@fouadamr301]

i want ask you on the variance of lgistic model ?

[@61raindrops]

Thanks. This is so helpful!

[@anuragdixit6237]

Great job man! Keep it up!!!!

[@jbstatistics]

Thanks!

[@PreetanjaliRay]

Great explanation! Thanks!

[@TheRandomgigi]

in findeing E(X^2) I tried to do the same trick as in E(x) i took the first part out of the summation to start at k=2 and then fron N! i took out N(N-1) in the end igot NP(1-P)^(1-N)+(NP)^2-N*P^2 which is probably wrong yet i dont understand why ?

[@mazvta]

Hello. May you please do a video on deriving the moment generating function of a Binomial random variable?

[@ngonidzashekimbini7177]

Wakapasa here mazvita

[@aniani6660]

Hello! Does anyone know how to prove the relationship between the central moment and the usual moments? E[(X-E[X])^n]=? using the binomial formula.

[@tsangpakho]

very very impressive, thank you.

[@jinanks2330]

Nice explanation.........yure really good at what you do......chow

[@jbstatistics]

Thanks!

[@lemyul]

Why did you change E[x^2] to E[x(x-1)]? Are they equal?

[@jbstatistics]

No, they're not equal. I discuss why I do that around 8:50.

[@amoghbarakol4723]

Found it very helpful! Thank you :)

[@VarunKaushik18]

Amazing video! Many thanks...

[@jbstatistics]

You are very welcome!

[@TURBOKNUL666]

Very very nice! Thank you!

[@Uppurak]

at 6:25 shouldn't the n on top of the summation sign be n-1 then being replaced by m?

[@jbstatistics]

+Kabilan T When we are summing over the values of x, x goes from 1 to n, and this is what the limits of summation are given as. When I change the variable to y = x -1, the lower limit of summation becomes 0, and the upper limit becomes n - 1, which is just m. I state in the video, "when x is equal to n, y is equal to n - 1, or m." I could have crossed out the n, replacing it with n-1, then crossed out the n-1, replacing it with m, but I skipped the first step.

[@Uppurak]

+jbstatistics thank you! the video helped me a lot!

[@jbstatistics]

+Kabilan T I'm glad to hear it!

[@toufeeqali2182]

Please explain that how is that(the formula mentioned to be Recalled) equal to 1?

[@jbstatistics]

I'm not sure what part you're asking about. (p + (1-p))^m = 1^m = 1. If you want more information about the binomial theorem, a google search will yield a lot of detailed information.

[@toufeeqali2182]

I am asking about the binomial expansion of [p+(1-p)]^m. It is a request to show it's proof of being equal to 1 in some other video Thanks.

[@pasindumalshan5457]

Thank you so much. ❤

[@ebeharris4942]

Vera level thala 💥❤️

[@marthatompkins6279]

I don't quite grasp how m can replace n as the upper boundary for the summation.....help!

[@jbstatistics]

The original summation is from x = 1 to n. We are setting y = x-1. When x = 1, y = x-1 = 1-1 = 0, so we need our summation to start at y = 0. When x = n, y = n-1, and I've simply defined m = n-1 (to make a few things a little cleaner and easier to see). So our summation ends at y = n-1 = m. Cheers.

[@marthatompkins6279]

Got it! Thank you!!!!jbstatistics

[@tvvt005]

6:28 shouldn’t the limit n change to m+1?

[@mysteriousperson0845]

We are summing from x=1 till x=n If y = x-1 and m = n-1 then we can rewrite x=1 as y+1=1 and we can rewrite x=n as y+1 = m+1 so technically we are summing from y+1=1 till y+1=m+1 in other word, from y=0 till y=m

[@jasonduffin7897]

Is this deriving them from first principles?

[@jbstatistics]

No, I use a few known relationships along the way. For the method starting at 2:58, I use the binomial theorem (as stated in the video), use the relationship E[(X-mu)^2] = E(X^2) - [E(X)]^2, and implicitly assume the law of the unconscious statistician in calculating E(X^2).

[@JustMoseyinAround]

Mind Blasting!

[@Jbroglydecap]

BRILLIANT

[@ailema7773]

wow. great. really helpful. tanks man.

[@karthiklogan9384]

Omg, I love it. My professor sucks

[@jbstatistics]

I'm glad to be of help. But sometimes it's a "different strokes for different folks" sort of thing. Some of my students think I suck :)

[@ishwarchandachom1804]

M bit confused Since (a+b)^n= sumation nCr a^n-r b^r As i found in text books

[@selinashen2993]

really helpful!! thanks alot!

[@frostwobs]

Impressive! I mean, I never expected videos of this quality to be on KZbin, that's precisely what I need. One question though, I've noticed from the video bar that you haven't made videos in the last 2 years. Will you be making new videos?

[@jbstatistics]

Thanks for the compliment Jonathan! I've been busy with other things for the past couple of years, but will get back to video production in the not-too-distant future. Cheers.

[@soumyatiwari1679]

You are awesome hats off to you thanks

[@jbstatistics]

+soms tiw Thanks! I'm glad I could help.

[@rohanghige]

Amazing...Nice explanation..:)

[@vadimkokielov2173]

Hi, you’ve incidentally proved something else...that the binomial PMF is a real PMF. Thats because the probabilities all sum to 1😎in my proof i took that for granted (it can be a separate proof or a lemma as you had it)

[@balaram1257]

thank u very pls upload more videos for other topics

[@GTmxsp]

Thanks a lot for Soontorn HW

[@pulkitgupta3477]

Good flow man

[@jbstatistics]

Thanks!

[@josephwheelerton]

I would really appreciate it if someone explained how a binomial rv can be thought of as the sum of Bernoulli random variables. That does not seem so obvious to me.

[@jbstatistics]

A Bernoulli random variable represents the number of successes in a single Bernoulli trial. A binomial random variable is the number of successes in n independent Bernoulli trials. # of successes in n trials = # of successes on the first trial + # of successes on the second trial + ... + # of successes on the nth trial (which will be a sum of 1's and 0's). For example, suppose we toss a fair coin once. The number of heads is a Bernoulli random variable (that takes on the values 0 and 1, with probabilities 1/2 and 1/2). The number of heads in 20 tosses is a binomial random variable (that takes on the values 0, 1, 2, ..., 20). Number of heads in 20 tosses = # of heads on the first toss + # of heads on the second toss +...+ # of heads on the 20th toss.

[@josephwheelerton]

Thanks so much, that was very helpful.

[@PomegranateAmazing79]

very good explanation

[@jbstatistics]

Thanks!

[@erfanmuhammed7915]

Good work

[@93000042]

Well done. Thanks!

[@jbstatistics]

You are very welcome!