The Poisson Distribution: Mathematically Deriving the Mean and Variance

[@kyliegrimes5504]

I cannot thank you enough, I spent 10 days by myself trying to prove this on my own and this was exactly what I needed

[@shankararjune4074]

you made this proof so simple and straightforward. Thanks a lot. Please keep posting more stats and probability videos like this. You're making a difference

[@jbstatistics]

I'm glad I could be of help! And "You're making a difference" is a very nice thing to hear. Thanks.

[@emiliocanton]

@@jbstatistics Still making it

[@TomDulson]

Videos like this make me realise how much money I'm wasting on tuition. You've outdone my university, thank you.

[@weslielegaspi196]

I was working on my exercises when I came across this. Our Professor left us to try and find the variance on our own. When I got to the part where the lower limit of the summation had to be change, I started to get lost. Thanks to your vid! It still helps even after a decade! 🎉

[@user-nw4jm8np4g]

I'm 10 years late but thank you! Many explanations glossed over the fact that e^x can expand and instead assumed you already knew the definition.. good for people reviewing, but its difficult for those trying to learn. Thanks for being explicit about what you're using!

[@mohammadpourheydarian5877]

Explains well, goes one step at a time, is a natural teacher. At once I learned everything he said and now I can teach the same because I learned it. He is organized. He does not assume anything. Job well done.

[@jbstatistics]

+Mohammad Pourheydarian Thanks so much for the wonderful compliment!

[@rosalindayanez6865]

You have no idea how I am crying inside with this class. Thank you!!!

[@sumreenaziz8693]

Amazingly explained in simple words. Just what I was searching for.

[@jbstatistics]

I'm glad to be of help!

[@charlescharles1270]

Dear Video Creator: Very brilliantly expounded...my highest thumbs up!!!

[@jbstatistics]

@Ramendra(KZbin is not letting me reply to your post). The sum, from r=0 to infinity, of a^r/r! is e^a (this is given at the start of the video). And you are welcome!

[@MrCobozco]

I don't know why I'm told to buy textbook that doesn't go over these derivations and then tested on it. This explanation was so succinct. Thank you and subbed.

[@Maha_s1999]

I feel blessed to have found your channel - once again a heartfelt thanks!!

[@jbstatistics]

+queenforever You're welcome, and thanks for the kind words!

[@DmitriArkhipov]

Thank you jbstatistics! Your video is amazingly clear and concise. Certainly one of the best explanations I have seen.

[@lisagraifer]

Thank you so much for your videos! It helps to see the extra steps worked out that I couldn't see in my textbook.

[@njabulonzimande2893]

life saver! you've simplified this distribution. Thank you!

[@jimmy000]

Great vid, ive watched nearly all of your distribution videos. helped me out a lot on the understanding the proofs, much appreciated!

[@jbstatistics]

You are very welcome Jimmy. I'm glad I could help!

[@harryhughes7761]

You explained this with such aplomb that it was simple to follow your reasoning and logic. Great video!

[@jbstatistics]

Thank you for the very kind words Harry!

[@ramendrathakur6861]

so breaking e/\a into a summation is the key concept, Thank You !

[@davidbishop8870]

Thank you so much!! So much simpler than anything else I have seen.

[@poonamdasilva7540]

Thank you so much for simplifying it by explaining every step this was very helpful.

[@gopinatht7677]

So simple, that even I can understand this! Thanks Man! Gr8 job done!

[@mattjohnlewis]

Thanks for posting this. Your explanation was very clear and easy to follow.

[@jbstatistics]

You are very welcome!

[@GaMeON159753456]

A very clear and easy to understand derivation. Good work.

[@learnlovelaugh4life]

This is so so helpful. Thank you so much! Best video I've seen over a topic I am trying to learn!!

[@jbstatistics]

You are very welcome. Thanks for the compliment!

[@Adam-gp3ij]

That's very simple! easier than what my teacher explained ! thank. you so much ! we really appreciate that Please make more videos

[@yourdipak7994]

You're just amazing in your way of teaching

[@noahrubin375]

It actually blows my mind that I can get a level of education as good as this for free on youtube.

[@jbstatistics]

Thanks for the very kind words! I'm glad to be a part of the movement.

[@GoodLuckForever-wi9kb]

Thanks for sharing such method in such a simple way

[@jacobp9470]

Your videos are so helpful! Thanks for making them.

[@jbstatistics]

You are very welcome. Thanks for the compliment!

[@asmashafiullah8720]

YOu r amazing this video is the proof of quality explanation.......keep posting

[@jbstatistics]

Thanks for the kind words.

[@AmanSingh-gs4ue]

Awesome video sir.....you make things so clear and easy Respect from India☺

[@jbstatistics]

Thanks for the compliment!

[@ipshitabanerjee1323]

Thanks a lot. Short but clear explanation.

[@johnnyd.j.6068]

So simple with great explanation! Thank you.

[@jbstatistics]

You are very welcome!

[@thetravellerfuture]

great explanation.You have helped me a lot.thank you ☺

[@jbstatistics]

+Maeda Beegun You are very welcome!

[@jacoboribilik3253]

excellent video, I'm becoming obsessed with probability and statistics and this is gold. Could you please make a video on the derivation of the Normal Distribution please? I've not found a video on that so far, it would be very much appreaciated!

[@ReddiSathvik]

At 5:28 , shouldn't the X(x-1) be put in the power of lambda and the factorial too?

[@jamesfkey]

Sometimes you need a quick review of something you learned 40 years ago. Thank you!

[@jtflypegasus]

That was very beautiful! I love elegant proofs......

[@Dr.RokiaAbdein]

thanks so much, it is so easy to understand

[@blazcukelj7273]

Great tutorial. Thanks!

[@jeanconstantinople2328]

outstanding as always !

[@88NA]

Thanks for posting this video

[@cronos6890]

Really well made video, thank you a lot!

[@allanabela2248]

Thank you very much for sharing your work, very coherent :)

[@kautukraj]

Would say this again, "You're making a difference".

[@wishmaherath404]

very clear and straightforward,thanks :-)

[@jbstatistics]

Thanks for the kind words.

[@ahmedros3713]

thank you ... this is very clear and helpful

[@anandkulkarni2111]

always this question raises : how did poisson come up/end up with that equation ? to me that underscores the real science i.e art of thinking and arriving towards an approach. without that its just memorizing shit

[@jbstatistics]

This video, as the title states, is about deriving the mean and variance of the Poisson distribution. It has nothing to do with the history of Poisson the man or the history of the Poisson distribution. I simply work through the derivation of the mean and variance, explaining the steps and logic along the way. If you think I'm encouraging students to simply memorize, I strongly disagree.

[@stewiegriffinfan18]

If I had to make a comment it would be, why is the mean and the variance of a distribution be the same? You usually end up with quite a bit of overdispersion when you do that

[@blakejhonshen2710]

Anand, though you posted this like a year ago, I figured on the off chance you're still curious, I'd post a quick few notes. The Poisson distribution is derived as the direct result of taking a binomial distribution for a continuous space as opposed to a discrete space. You start by defining lambda as a sort of density of when successes will occur in a certain process. For example, say I wanted to mark down all the cars that pass by on a specific stretch of road on Mondays. Then I could sit there with my pencil in hand and mark down every car that passes by, and come up with a result that, say, on average, three cars pass by per hour on my road. Say I wanted to find the probability that two cars will pass by in a given half-hour segment. In a regular binomial distribution, this would be impossible to define clearly, as the intervals upon which a success or failure can occur is discrete; you can't have 'half a trial'. So instead, you take the binomial distribution and let lambda=np, and thus p=lambda/n, and substitute in the binomial formula. You then take the limit as n approaches infinity - i.e. as the time intervals become smaller and smaller - and you eventually find P(X=k) for a continuous distribution. When you take the limit, due to the nature of the binomial distribution, out pops the Poisson distribution. There are plenty of places online where you can find a much more in-depth proof of the actual limit simplification process itself, but hopefully this should provide a reasonably good explanation as to the nature and derivation of the poisson process - it's taking the binomial process and applying it to a continuous time interval.

[@emadharazi5044]

Excellent video

[@ramendrathakur6861]

sir could you tell me summation of (2/ )/r!, or for any variable a, a/ /r! ?

[@colecofer8474]

This video is amazing. Thanks!

[@gopinatht7677]

Happy to Subscribe!

[@irtizahasan3537]

thank you for making these videos....

[@jbstatistics]

You're welcome!

[@aronquemarr7434]

The derivation of variance could have been more straightforward. Just substitute the term E(x ^ 2); you'll eventually reach m ^ 2 - m.

[@samstreet6595]

That is amazing, cool method

[@pavelooo2289]

thank you. I couldn't find it elsewhere

[@jbstatistics]

I'm glad to be of help!

[@橡皮擦-e1j]

I can't understand why a=lamda at 3:37..

[@ugestacoolie5998]

Hi just a question at about 8:33, isnt E(x^2 - x) just E(x^2) - E(x)? I'm a little confused

[@jbstatistics]

Yes, it is, and I say that and use that there. At that point I had found E(X(X-1)) = lambda^2, and then use that in finding Var(X). Finding E(X(X-1)) is a helpful little trick, as it allows us to cancel the x(x-1) terms from the x! in the denominator of the Poisson pmf. If you try to find E(X^2) directly you will run into problems.

[@ailema7773]

wow. thanks very much. it was really a help to me.

[@challakishore6975]

thank you, this is very helpful and clear.

[@jbstatistics]

You are very welcome!

[@Mark-li3my]

Helped a lot, thank you bro

[@pra601]

Why E [ X ( X- 1 ) ] ??? is it a property or just a algebraic manipulation?

[@jbstatistics]

I discuss why I use E[X(X-1)] in detail from 5:00 through 6:00. It is a trick that allows us to cancel terms in x!, thus making it easier to find E[X^2].

[@protidinjob7193]

Thanks for help

[@tarmopungas]

Thanks a lot for this!

[@jtflypegasus]

That's beautiful...👍👍

[@electrikkblue]

Thank you so much!

[@annalam8624]

Thank you! This video helps me a lot :D

[@jbstatistics]

You are very welcome!

[@dml39prudhvi65]

Sir please give me answer If mean=2 then find E(X+1)! .this is Poisson distribution

[@surnermahiteja9942]

thanks for making dis so simple

[@jbstatistics]

You're welcome. I just try to give you the real deal, in an understandable way.

[@codework-vb6er]

@2:30 you "take one single lambda out front" can you derive the proof to that Σ( lambda^(x) / (x-1)! , x=1, x=inf) => lambda * Σ( lambda^(x-1) / (x-1)! , x=1, x=inf)

[@jbstatistics]

We're simply taking a constant outside of the summation. sum lambda^x = sum lambda*lambda^(x-1) = lambda sum lambda^(x-1).

[@ahsansacademy6796]

Where were you bro 😭

[@zuckirosenfuchs5441]

Nice video!

[@i_m_hk]

Nice explanation

[@ednakazuya12]

Brilliant!

[@nerdymathematician]

THANK YOU😭

[@jbstatistics]

You're welcome!

[@maqboolurrahimkhan]

Just amazing thanks alot

[@kaushal32755]

Thanks for the help !

[@weicarlos1719]

can any one tell me where is the [e^a=sigma a^y/y! ] link? thanks

[@johnnyd.j.6068]

kzbin.info/www/bejne/l52om4KHg5Wfj6M

[@liangchihuang7775]

very very useful!

[@jbstatistics]

I'm glad to hear it!

[@ahmedshalapy7923]

you're amazing

[@jbstatistics]

Thanks!

[@MsSprulez]

Amazing!

[@jbstatistics]

Thanks!

[@conceptofcivilengg.withabh1728]

Thankyou sir

[@alanbchen]

[@mohdyunus1512]

Thank you so much

[@jbstatistics]

You are very welcome!

[@samah241]

Thank you 😀

[@jbstatistics]

+Samah Salah You are welcome!

[@Quizpop12]

Thnk u soo much

[@i_m_hk]

Thank u sir

[@BadEnoughDudeRescues]

I love you bro

[@JismalJamal]

Great Explanation

[@amanmwangwa8987]

nice one...

[@dr.gavisiddappaanandhalli6563]

it is very nice

[@jbstatistics]

Thanks!

[@erandaweerasingha404]

supper

[@mdjunaid1664]

Food explanation... But can u speak little bit louder from next..

[@jbstatistics]

I honestly have no idea whether you're being serious or sarcastic. It seems normal on my end, and volume controls moves it from very quiet to very loud.

[@hp-qx7tf]

stats is hard :(

[@DistortedV12]

oof I feel kinda stupid that I didn't get this

[@uncommon2848]

my exam is in one hour

[@Diego-ss7sg]

You sound like Mark Zuckerberg

[@sophiechavigny9725]

Thank you so much!!