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
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!
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! 🎉
@Ramendra(UA-cam 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!
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.
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.
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.
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!
@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)
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
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.
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
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.
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].
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.
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
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
I'm glad I could be of help! And "You're making a difference" is a very nice thing to hear. Thanks.
@@jbstatistics Still making it
Videos like this make me realise how much money I'm wasting on tuition. You've outdone my university, thank you.
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!
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! 🎉
You have no idea how I am crying inside with this class. Thank you!!!
@Ramendra(UA-cam 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!
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.
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.
+Mohammad Pourheydarian Thanks so much for the wonderful compliment!
Dear Video Creator:
Very brilliantly expounded...my highest thumbs up!!!
I feel blessed to have found your channel - once again a heartfelt thanks!!
+queenforever You're welcome, and thanks for the kind words!
Amazingly explained in simple words. Just what I was searching for.
I'm glad to be of help!
life saver!
you've simplified this distribution. Thank you!
Thank you so much for your videos! It helps to see the extra steps worked out that I couldn't see in my textbook.
It actually blows my mind that I can get a level of education as good as this for free on youtube.
Thanks for the very kind words! I'm glad to be a part of the movement.
Great vid, ive watched nearly all of your distribution videos. helped me out a lot on the understanding the proofs, much appreciated!
You are very welcome Jimmy. I'm glad I could help!
A very clear and easy to understand derivation. Good work.
You explained this with such aplomb that it was simple to follow your reasoning and logic. Great video!
Thank you for the very kind words Harry!
so breaking e/\a into a summation is the key concept, Thank You !
Thank you jbstatistics! Your video is amazingly clear and concise. Certainly one of the best explanations I have seen.
Thanks for sharing such method in such a simple way
At 5:28 , shouldn't the X(x-1) be put in the power of lambda and the factorial too?
So simple, that even I can understand this! Thanks Man! Gr8 job done!
Thanks a lot. Short but clear explanation.
Thanks for posting this. Your explanation was very clear and easy to follow.
You are very welcome!
Thank you so much!! So much simpler than anything else I have seen.
Thank you so much for simplifying it by explaining every step this was very helpful.
You're just amazing in your way of teaching
great explanation.You have helped me a lot.thank you ☺
+Maeda Beegun You are very welcome!
This is so so helpful. Thank you so much! Best video I've seen over a topic I am trying to learn!!
You are very welcome. Thanks for the compliment!
Hi just a question at about 8:33, isnt E(x^2 - x) just E(x^2) - E(x)? I'm a little confused
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.
That's very simple! easier than what my teacher explained ! thank. you so much ! we really appreciate that
Please make more videos
So simple with great explanation! Thank you.
You are very welcome!
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!
Awesome video sir.....you make things so clear and easy
Respect from India☺
Thanks for the compliment!
YOu r amazing this video is the proof of quality explanation.......keep posting
Thanks for the kind words.
Your videos are so helpful! Thanks for making them.
You are very welcome. Thanks for the compliment!
Sometimes you need a quick review of something you learned 40 years ago. Thank you!
I can't understand why a=lamda at 3:37..
That was very beautiful! I love elegant proofs......
outstanding as always !
@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)
We're simply taking a constant outside of the summation. sum lambda^x = sum lambda*lambda^(x-1) = lambda sum lambda^(x-1).
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
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.
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
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.
Really well made video, thank you a lot!
thanks so much, it is so easy to understand
Why E [ X ( X- 1 ) ] ??? is it a property or just a algebraic manipulation?
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].
Thank you very much for sharing your work, very coherent :)
Excellent video
Great tutorial. Thanks!
very clear and straightforward,thanks :-)
Thanks for the kind words.
sir could you tell me summation of (2/
)/r!, or for any variable a, a/
/r! ?
Would say this again, "You're making a difference".
thank you ... this is very clear and helpful
That is amazing, cool method
Happy to Subscribe!
thank you. I couldn't find it elsewhere
I'm glad to be of help!
Thanks for posting this video
This video is amazing. Thanks!
The derivation of variance could have been more straightforward. Just substitute the term E(x ^ 2); you'll eventually reach m ^ 2 - m.
can any one tell me where is the [e^a=sigma a^y/y! ] link? thanks
ua-cam.com/video/alEjOQN0lYA/v-deo.html
Where were you bro 😭
That's beautiful...👍👍
thank you for making these videos....
You're welcome!
Sir please give me answer
If mean=2 then find E(X+1)! .this is Poisson distribution
Nice explanation
Nice video!
wow. thanks very much. it was really a help to me.
Helped a lot, thank you bro
thank you, this is very helpful and clear.
You are very welcome!
Thanks a lot for this!
Thanks for help
thanks for making dis so simple
You're welcome. I just try to give you the real deal, in an understandable way.
Brilliant!
THANK YOU😭
You're welcome!
Thank you so much!
Just amazing thanks alot
Thank you! This video helps me a lot :D
You are very welcome!
Thanks for the help !
Thnk u soo much
very very useful!
I'm glad to hear it!
Great Explanation
you're amazing
Thanks!
Amazing!
Thanks!
nice one...
Thank you 😀
+Samah Salah You are welcome!
Thankyou sir
Food explanation... But can u speak little bit louder from next..
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.
Thank u sir
Thank you so much
You are very welcome!
I love you bro
stats is hard :(
it is very nice
Thanks!
oof I feel kinda stupid that I didn't get this
supper
my exam is in one hour
You sound like Mark Zuckerberg
Thank you so much!!