The Relationship Between the Binomial and Poisson Distributions
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- เผยแพร่เมื่อ 17 ก.ย. 2024
- A look at the relationship between the binomial and Poisson distributions (roughly, that the Poisson distribution approximates the binomial for large n and small p). I work through some calculations in an example, showing that the approximate probability from the Poisson can be quite close to the exact probability from the binomial distribution.
(The example used involves albinism. Albinism affects all races, but the rates of albinism vary a little around the world. In Europe and North America, roughly 1 in 20,000 people have some form of albinism).
For those using R, here is the R code to find the probabilities for the example in this video:
Finding the probability of getting exactly 2 with albinism in a random sample of 1000 Europeans.
Binomial (exact):
dbinom(2,1000,1/20000)
[1] 0.001187965
Poisson (approximate):
dpois(2,1000*1/20000)
[1] 0.001189037
This is great. I just passed the chapter on Poisson in my textbook, basically wondering "Well isn't that pretty much binomial? Why complicate things?". This video answers all the questions I had.
me too!!
i spent 3 hours in a lecture hall listening to my lecturer and nothing made sense to me..but this 5 minutes has changed everything...thanks so much Prof.
What is the meaning of lambda in the Albinism example? How to put that in words? In other cases of Poisson distribution, Lambda is the average rate at which a particular event occurs over a unit time /space interval. What does lambda = 0.05 "mean" in the albinism example?
@@chetanagrawal1519 i know its too late but it might help someone else..here lamda means 0.05 people is expected no.(average) of people having albinism among 1000..i know it sounds weird..but if sample was 40000 then lamda would be 2.
This video is better than my textbook (and that seems to be case of everybody else in the comments). Thank you so much for it!
Very very nice videos! You just saved me before my exam day. Such great emphasis and energy in the way you speak, love it! It's hard to leave your videos without understanding the topic :D
+Anirban Mookherjee I'm very glad I could be of help. Thanks for the compliments!
You are very welcome Ershad, and thanks very much for the compliment!
So helpful. I did not have a clear idea of when to use which and I am learning to teach Statistics! Thank you JB
exam for this is in 11 days, starting today to study, yay no more last minute, your stuff is awesome, subbed
I am from Argentina and the video is quiet understandable. Congrats and thanks!
Thank you this is brilliant, so hard to find good explanations is statistics!
dude...thank you.....2 weeks of confusion erased in 60 seconds. incredible. it actually makes sense. i went from scared and nervous for my exam to excited.
I'm glad to be of help! Best of luck!
great video and straightforward explanation! thanks very much!
Excellent video tutorials. I understand them very well. Excellent teacher & teaching methodology . I am an O /A level teacher of Mathematics , Dhaka , Bangladesh. Many thanks Sir.
This video answers fundamental question like why do we use ...etc. This video is better of all other videos i saw on the same topic. Thanks professor.
Very good summary. Short and to the point. Thank you.
You are very welcome. Thanks for the compliment.
An excellent, intuitive explanation. This guy has done a terrific job of explaining a complex mathematical concept.
Thanks!
You're welcome Adel.
Math never gets old. I am watching this in 2021 and is the best than any other
Great explanation! Probably the video can be complemented by giving the value of the constant e=2.71828. Thank you so much!
Thank you for providing such an easily understandable and conceptual video!
You are very welcome!
Thank you! This was incredibly helpful. I have a final in one hour and was still struggling with comparing Binomial x Poisson.
You're welcome! I'm glad to be of help.
hi. Enjoying your lectures so much. Could you please add a tutorial for the exponential distribution
excellent...i dont knows difference between the two but finally got it...
1) n is large and p is small for poisson
2) when only mean is given not n and p
great video and great voice
I think
There is no difference, mean is also equal to np
Great video, thanks! Very clear and not all over the place like math videos on youtube.
What is the meaning of lambda in the Albinism example? How to put that in words? In other cases of Poisson distribution, Lambda is the average rate at which a particular event occurs over a unit time /space interval. What does lambda = 0.05 "mean" in the albinism example?
Helped me greatly with my assignment! Liked and subscribed!
Great videos, thanks!
Fantastic video. Thanks
Great explanation indeed. Many thanks for all effort put into the video. One question: at the end of the video, we are reasoning why and when we might need to use Poisson. We state, we may only have a mean value, while we don't know n and p. Even if it is true, how can we use it in the Poisson formula, since the formula itself requires n for calculation?
I'm saying we might have an approximate value of np, even if we don't know n and we don't know p.
@@jbstatistics Thanks a lot for the reply. Appreciate.
Wow, that was brilliant. Really opened my eyes thank you.
I have an exam coming on this 5th july, thank you boss
You could’ve also shown how the graphs relate in the two different distributions
Thank you very much. Good helping.
What is the meaning of lambda in the Albinism example? How to put that in words? In other cases of Poisson distribution, Lambda is the average rate at which a particular event occurs over a unit time /space interval. What does lambda = 0.05 "mean" in the albinism example?
The average number of people with albinism in the sample of 1000. You are correct in that some sources have lambda as the average rate per unit time, and then have the mean be lambda*t, but I'm using lambda as the average number of occurrences in the time frame under discussion. That can be a little unsatisfying sometimes, but it's simple and fairly common notation.
So in a way, the Poisson distribution assumes continuous time, i.e. an infinite number of Bernoulli trials. The expected value of successful outcomes is lambda. The binomial distribution is a way of discretizing this problem by dividing this an interval of continuous time in a bunch of discrete intervals, where each of these intervals is a Bernoulli trial. By setting the expected value of successful outcomes of the binomial distribution random variable to that of the Poisson distribution random variable, we get an approximation of the Poisson distribution. The finer this approximation, i.e. the more discrete Bernoulli trials we divide the continuous interval into, the closer we get to the Poisson distribution. The number of trials tends toward infinity and the success probability of each of these trials tends toward zero.
Phenomenal video. Thank you so much for your help.
You are very welcome!
this was really helpful thank you
In the last section, Why use this approximation, I think there is some error like binomial does not have any factorials and exponentials right ? please correct me if im wrong. BTW a very good lecture. Thank you.
The binomial coefficient appears in the binomial probability mass function, i.e. we need to calculate (n choose x ) = n!/(x!(n-x)!) in order to find a binomial probability. While conceptually this is always pretty straightforward, this number can sometimes be huge. It's typically more of a problem if someone is trying to break it down term by term on a calculator, as most calculators can't display above 70! (that's the crossover point where there starts to be 3 digits in the exponent rather than 2, when expressed in scientific notation).
Excellent video!
Thanks!
Thanks for this video :)
You are very welcome!
awesome vid
Great :D
well explained!
Thanks Doctor. It was helpful :)
thanks a lot! it really helps !!
Thanks a lot!
really helpful and understandable... thank you ^.^
+Fira Ramlee You are very welcome!
Awesome! Thanks!
You are very welcome!
Awesome thanks!
Thank u sir
What if you have a large n, a large p and a very small q relatively close to 0. With an indication that q*n
I'm not sure what you're getting at. If a random variable has a binomial distribution, then one can find probabilities using the binomial probability mass function. It doesn't matter what n or p are. If it's binomial, it's binomial.
@@jbstatistics : I see poisson can be used to approximate binomial, so is the normal used to approximate binomial when conditions like p*n and q*n are greater than 5 hold.So what approximation do you use when n is so large but your q is relatively small to meet the conditions for normal approximation. How do you find probabilities like P(X
@@yaweli2968 If it's a binomial r.v., then it's a binomial r.v. and probabilities should be found with the binomial probability mass function. Don't go looking for approximations when the exact method works. Yes, there are approximations that are sometimes reasonable and sometimes helpful. So sure, we can sometimes use the Poisson to approximate the binomial, and we can sometimes use the normal to approximate the binomial. But when we're looking for a probability involving a binomial random variable, the first thing we should be thinking of is using the binomial distribution to find that probability.
Suppose n = 1,000,000 and p = 0.995. We can find P(X pbinom(994900,1000000,.995)
[1] 0.07950808
Though the normal approximation would work reasonably well here:
> pnorm((994900.5-995000)/sqrt(1000000*.995*.005))
[1] 0.07917044
@@jbstatistics Thanks
how should i choose between the poisson or binomial distribution to describe a data set ?
my understanding so far is, poisson represents random events better.
anyone got a simple method
thanks my G
thanks!
You are welcome!
thanks
You are very welcome.
thanks a lot :))
You are very welcome!
references pls ??
👏👏👏👏
You'll never ever see this in the real world. what a waste. Great video though.
Excellent video!
Thanks Ronald!
hi. Enjoying your lectures so much. Could you please add a tutorial for the exponential distribution
+Ama opokua Asomani-Adem I'm glad to hear you're enjoying my videos! I hope to find some time in the not-too-distant future to make videos. When I do, the exponential distribution would be near the top of the list. Cheers.
Ohkk cool
What is the meaning of lambda in the Albinism example? How to put that in words? In other cases of Poisson distribution, Lambda is the average rate at which a particular event occurs over a unit time /space interval. What does lambda = 0.05 "mean" in the albinism example?