The Binomial Distribution: Mathematically Deriving the Mean and Variance
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- Опубликовано: 22 авг 2024
- I derive the mean and variance of the binomial distribution. I do this in two ways. First, I assume that we know the mean and variance of the Bernoulli distribution, and that a binomial random variable is the sum of n independent Bernoulli random variables. I then take the more difficult approach, where we do not lie on this relationship and derive the mean and variance from scratch.
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
Give that man a cookie :) ... That was almost the standard of Euler.Well done !
+Abu Bardewa Thanks!
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.
I have searched for this for days. Thank you so much. Learning and understanding feels so good.
Thanks for the feedback and compliment Cuong! I'm glad you found this video helpful.
You're welcome Amogh! I'm glad you found it helpful. Cheers.
You saved my life.. Excellent tutorial, crystal-clear explanations. Thank you very much sir!
You are very welcome. Thanks for the kind words!
Your explanation by far is the most easy to understand and helpful! Thanks a lot!
Watching this a night before my exam, phew! saved!
I hope your exam went well!
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💓💓
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.
I'm glad I could help! I have many videos available on RUclips, and will be adding more this year.
Thanks for explaining why m replaces n on top of sigma! Very important and nowhere else (book or online) have I seen it.
You're welcome Philip. Thanks for the feedback!
Hi Prof, your descriptions are very detailed and clearly explained. Keep up the good job! It's benefiting students like us! :)
This is amazing. The way the logic and the calculation are presented is very elegant. I loved the second method.
Thanks for the very kind words!
A fantastic explanation of a concept by which I was confounded! Thank you.
I think nobody else should have explained in a better way than you sir. Thank you
You are very welcome. Thanks for the compliment!
Thanks Wilson! I'm very glad to be of help. Cheers.
Thankyou for this. Was struggling to understand but you made it so easy.
good presentation. I get confused from my professor but now everything are clear.
Great! I'm glad to be of help.
excellent explanation and method
This is an awesome explanation , clear and comprehensive
Hassssss..... (relief).... finally after watching bunch of videos I got a good tutorial.
You are very welcome. I'm glad to be of help.
You r toooo goood man..
Hats off..
(Y)
I UNDERSTOOD WAT THE TEACHER EXPLAINED THROUGH YOUR VIDEO..
KEEP IT UP!!
thanks you helped me in clarify my doubt perfectly.👍
You are very welcome!
Wow you are amazing. The way you derive is genius :D Thanks a lot for helping.
Excellent and clear explanation sir. ..very easy to understand..
An amazing explanation to something I was told to memorize in school !
Thanks for making this :D
thank u so much ! you save me this time for passing my exam ...
Thanks for the concise logical, step by step explanation!
Your explanation is Very Good Sir
Thank you so much for making this video Sir🤗
Thank you thank you thank you. Put all my doubts as ease!
You are very welcome!
Thanks for detailed information
Thank you so much for the explanation!
Thanks Rohan!
awesome work sir really glad now,
can you do such ones for the remaining distributions too
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.
The derivation is brillant!
You are very welcome Marcela!
Very helpful, thanks a lot.
It's great and very useful. A lot of calculations but it really helps with the concept.
Best tutorial....Thanks a ton sir....
You are very welcome!
Extremely helpful and detailed!
Thanks man! Excellent tutorial with good explanations.
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!
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
Beautifully done. Thank you very much.
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?
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.
Gd explaination each and every steps
Finally I got it thanks
You are very welcome!
Good explaination! Thanks a lot sir! keep up the good work.
i just appreciate you have made bit exact and precise!!
I'm glad I could be of help!
A great thank for so clearly explaining this proof.
Thanks... You are really doing great job
Excellent explanation
I get clarified from this
Thank you for video
Thanks . God bless you. great methodology, easier to understand
Very Clear Explanation. Thanks
You are very welcome!
Thanks. This is so helpful!
Thank you so much. ❤
you are outstanding...my God great ...
Found it very helpful! Thank you :)
thank you !!! :) This video helped me a lot with my assignment. haha
Very nice ! Thanks a lot !
You are very welcome.
Helpful video, please explain derivation part of the binomial distribution
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.
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).
Thanks a lot sir for the wonderful explanation...It helped me a lot
I love you 💕 I was about to cry ❤ thanks for this😘
You are very welcome!
You are awesome
hats off to you
thanks
+soms tiw Thanks! I'm glad I could help.
Impressive! I mean, I never expected videos of this quality to be on RUclips, 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?
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.
Great job man!
Keep it up!!!!
Thanks!
Brilliant! You explained it very smoothly
Thanks a lot for Soontorn HW
Great explanation! Thanks!
Sick ass videos dude!
Thanks! I'm glad you like them.
Very very nice! Thank you!
Nice explanation.........yure really good at what you do......chow
Thanks!
very very impressive, thank you.
Thank you.
Thanks a lot👍
Mind Blasting!
Amazing video! Many thanks...
You are very welcome!
Perfect!
Good flow man
Thanks!
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)
really helpful!! thanks alot!
BRILLIANT
from what country are you?
that awesome. thanks for imparting your talent
Vera level thala 💥❤️
This is great . What program are you using to write this?
The base is a Latex/Beamer presentation. I annotate with Skim, and record & edit with Screenflow.
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 ?
wow. great. really helpful. tanks man.
Amazing...Nice explanation..:)
amazing!
thanks a lot
thank u very pls upload more videos for other topics
excellent
Well done. Thanks!
You are very welcome!
Good work
Beautiful
Thanks!
very good explanation
Thanks!
Hello. May you please do a video on deriving the moment generating function of a Binomial random variable?
Wakapasa here mazvita
M bit confused
Since (a+b)^n= sumation nCr a^n-r b^r
As i found in text books
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.
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.
Thanks so much, that was very helpful.