Poisson or Not? (When does a random variable have a Poisson distribution?)

A not-too-technical look at the conditions required for a random variable to have a Poisson distribution. It can be difficult to determine whether a random variable actually has a Poisson distribution, so here I look at a few examples and some visual illustrations that may help. There are no probability calculations carried out in this video. I assume that the viewer has already been introduced to the Poisson distribution, but I do a brief review at the start.

[@cagri5886]

At 9:32, I smiled because I am studying for my math final and students are spacing themselves apart as shown because of the "Social Distancing" due to coronavirus

[@daloshy]

I'm taking probability at McGill, my prof is a world leading expert in his field, and an incredible teacher. But honestly, your ability to visualize these concepts is incredible, classic notes on a blackboard does little to compare, not to mention zoom school. The fact that this is free hopefully pushes the envelope of what is expected from an undergrad university mathematics education. I have found myself throughout the entire semester, after each probability lecture, praying you had a video on the topic we just learned. Thank you for sharing your incredible ability for teaching with all of us.

[@jbstatistics]

Thanks so much for the kind words Dalyan. I'm very glad to be of help. You weren't to know this of course, but tonight was an especially good night for you to post such a nice comment.

[@mike7920]

Absolutely. Universities need to up their game with content like this out here for free

[@dd-kj6tb]

After watching all of your distribution videos, I find myself very confident in identifying and solving these types of problems. From top to bottom, everything is perfect! I'm glad that I found this channel. Thank you very much sir!

[@jbstatistics]

I'm glad to be of help!

[@Antonio-lt1sp]

I need to say it again - best Poisson distribution video ever!

[@jbstatistics]

Thanks for the wonderful compliment!

[@kartikkamboj295]

Thanks for doing this - I love the power of internet; people who actually want to learn can learn about such concepts while staying at home. And its because of people like you, that internet is so powerful. p.s:The count of dots in the 'unit square' was a very good example.

[@deoxyde]

You have an amazing archive of statistics videos. I watched this video in highschool (around 6 years ago) and since then I've kept coming back to this and several other of your videos to review. Today was another one of those days. Thanks so much for your work.

[@jbstatistics]

You are very welcome. I just stumbled upon these very kind words tonight as I fired up this video for the first time in a long time. I'm very glad I can be of help, and glad you've found these useful over the years.

[@vinodb5050]

Your videos and examples are easy to understand, thanks for your efforts.

[@jbstatistics]

You are very welcome. I'm glad you find my videos helpful!

[@5sf4bw86]

Best video for poisson distribution

[@carlofino4666]

this is the best way to understand Poisson distribution. Loved the bit of the three rectangles and how one violated the "randomness" and the other violated the independence condition

[@jbstatistics]

Thanks!

[@DreamWorker-jm5xn]

Ok, can i say this one last time: best Poisson distribution video I've ever seen.

[@ddd12343]

I was struggling to get many of statistical concepts for a long time, thankfully I found this channel. You are doing great job

[@jbstatistics]

I'm glad to be of help! Thanks for the kind words!

[@hamidgaikani9683]

The plane example was fantastic and really opened my eyes! Thanks 🙏🏻

[@prasadliyanage1855]

A great presentation. It is very rare to find a good account like the one you created. All the best with your good work!!

[@jbstatistics]

Thank you for the very kind words! I'm glad to be of help!

[@oracleofottawa]

Awesome quality videos. You should be a FULL PROF my friend. And that voice! If you don't get full tenure please consider the CBC in radio....

[@jbstatistics]

+oracleofottawa Thanks!!!

[@juarjuna4130]

What a good voice to give good explanations. Thank you!

[@jbstatistics]

You are very welcome, and thanks for the compliment!

[@aq11qap]

Such a great, clear explanation - thank you!

[@prosimulate]

I subscribed after I saw this video. The quality of the examples is just brilliant. It is explained in such a way that one can understand it on a deep level. Thank you very much Sir.

[@jbstatistics]

You are very welcome, and thanks so much for the kind words!

[@jankaas9256]

A very intuitive explanation. Thank you!

[@swethabj9819]

Your probability of teaching in very good manner is absolutely one

[@jesswong5173]

Thank you!!!! I LOVE UR VIDEOS😍pls do more concept videos!! It’s very interesting and your explanations are really 👍👍👍

[@jayjung89]

Truly wonderful series of videos in your channel professor!

[@jbstatistics]

+jayjung89 Thanks!

[@marcoventura9451]

Very good. Clever, reasoning and important example.

[@prathameshmahankal4180]

This is absolutely the best Poisson Distribution video on the internet. Stomped that like button! That makes me think, would 'the number of likes on this video' follow a Poisson distribution? Even though there is randomness, I assume there is dependence since more people would like it when it becomes viral at different times?

[@musclesmalone]

Your videos are phenomenal! Thank you

[@KeepingMyDayJob]

Excellent examples--nicely done!

[@jbstatistics]

Thanks Scott!

[@rate578]

Thank you so much! This is so helpful. You made the world a better place today.

[@jbstatistics]

You are very welcome. Thanks for the kind words!

[@ayushdwivedi3519]

It's a wonderful video for clearing out the concept on Poisson distribution..please upload these type of concept videos for other distributions as well........

[@jbstatistics]

Thanks for the compliment. I'll try to get more videos up in the near future.

[@yamansanghavi]

It is Superb. Best Lecture I ever had on Probability & statistics.

[@jbstatistics]

I can't ask for a nicer compliment than that! Thanks so much!

[@deepfakevasmoy3477]

Great video, thank you for clear explanation!

[@ThePeacefullOasis]

So far best video I found to grasp the concept behind the poison distribution. I like the cow example :)

[@jbstatistics]

Thanks! I like that one too :)

[@wawirebrian526]

this is one of the best videos for poisson distribution

[@jbstatistics]

Thank you. This is one of my faves.

[@dillonramoutar1133]

these videos has helped me to better understand stats

[@hongyesong6901]

AWESOME video!! If it's convenient,would you add the subtitles? it will be easier for understanding for Non-native English speakers.thank you so much~~

[@Songvbm]

Great Lecture. Could you please elaborate compound Poisson Distribution with example? And what is the process of fitting compound Poisson Distribution based on data?

[@vivekthirumala44]

Extremely helpful. Thanks a lot.

[@sambhavjain3655]

Thanks for a good explanation and approach to the problems (Real time problems)

[@jbstatistics]

You are very welcome!

[@brazilfootball]

Hi, my stats professor created a “master key’ for us to determine which test we should use, given a particular circumstance. One such circumstance is that if we have count data of fish, we either use a poisson or negative binomial distribution for modeling our data. The 3rd example in this video seems to match this circumstance - is that because fish count data can have an over-dispersion problem (too many zeros), making it a negative binomial distribution?

[@jimmydim2212]

Subbed because you are the man.

[@lashlarue7924]

Great explanation! Subscribed!

[@user-qq1sl8rz7r]

In the last example(Crashes & Fatalities) Since the figures are similar, can I say the variance of second example is too large to be a Poisson distribution? Then could I conclude that Y(Fatalities) can't be too far from zero? If I standardize the data, is it possible to turn a "not Poisson distribution data" into a "Poisson distribution data"? Thanks for all these awesome videos!

[@LearningToCodeAndDesign]

I’m learning a lot about the importance of the independence of occurrences of events

[@databloke2755]

Very well explained!

[@Fan-fb4tz]

Very great helpful video! Thank you! I am wondering how do you create that simulation process about students arriving at school? that is a really straightforward illustration!

[@jbstatistics]

Thanks for the compliment! I spent quite a bit of time on this video. I know very little about animation at the moment (though I'm hoping to learn). For the simulation with students arriving at the University Centre, I simulated the values and did the plotting in R. I'm sure there are better ways of getting a slicker video, but doing it in R allowed me to do the precise simulation that I said I was doing (rather than simply faking it and making it look about right).

[@24brophy]

Great example. Would love to see how you coded this, if that's asking too much. Thanks for the great videos!

[@alexiadrey938]

This is gold!

[@NorthernChinar]

Thank you so much for explaining

[@jbstatistics]

You are very welcome!

[@abdurrahmankhaled9212]

i really appreciate that , very helpful !

[@aimalrehman3657]

Thank you. It was very helpful!

[@jbstatistics]

You are very welcome!

[@Jrcoaca]

how about could you code something that executes a number of events in a random tendency, but with a tendency to group together, like at 10:15?

[@TheCoolcat0]

Excellent non-examples, thanks!

[@jbstatistics]

You're welcome!

[@fanyizhao9163]

Best explanation ever!

[@jbstatistics]

Thanks!

[@victorrice4549]

Very helpful, thank you!

[@jbstatistics]

You are very welcome!

[@chriscollins5320]

Great break down of whether a distribution is Poisson or not. Suggest you juxtapose with Binomial and other Discrete Distributions.

[@jbstatistics]

+Chris Collins Thanks Chris. I do have another video in which I give an overview of a number of discrete distributions and discuss the links between them. It's available here: kzbin.info/www/bejne/i6OyiYWsfJ6ij6c. It's one of my personal favourites.

[@alicenorman5236]

Great video!

[@sarthakagarwal9198]

please upload a video related to gamma distribution and beta distribution and their derivation

[@ajiththomas009]

will Poisson work for history of data (12 months spare part item demand) with lots of zero-values in between the demand? many thanks

[@ahidk7945]

thank you this is helpful video

[@rahulpriolkar8851]

This was Brilliant!!

[@jbstatistics]

Thanks!

[@heshanpalliyaguruge6333]

You are a statistics god. Thank you so much

[@user-bz8nm6eb6g]

Thank you. helped alot !

[@AnEasyWayToStatisticsbyDrTariq]

nice concept building lecture

[@shoulderescape]

Great video, one question: a problem that I have needs me to check with a 5% significanslevel, if a poisson distribution is good for the data, how can I do this?

[@diencai1812]

Awesome! thank you!

[@tnuts92]

Perfect thanks!

[@imnotdigginthis]

at 6:00 its five groups. doesn't really matter tho.

[@jbstatistics]

Robert Kelly Yes, there are 5 groups (and I say there are 4). I hadn't noticed that before -- I'm lucky that it's in a spot where misspeaking like that isn't really a problem. Cheers.

[@imnotdigginthis]

haha just nitpicking ;-)

[@GarrettPetersen]

At 5:10 it's the disappearing dots optical illusion.

[@MasoNao]

9:32 social distancing

[@dillonramoutar1133]

thanks so much

[@christiansinger2497]

Thanks alot!

[@theodoresweger4948]

What about a chi square to check if the observed fits the expected reasonably

[@GastroenterologyPINNs]

Great!👍

[@craigmunday3707]

Excellent

[@jbstatistics]

Thanks!

[@roseb2105]

sorry for this may be nnitpicking but if the cookie dough is mixed then the # of chocolate chips are not independent throught the cookie and therefore I thought cannot follow a poisson distribution if the dough is not mixed the probalite are independent and would follow poisson ( like if not mixed number of choclate chips from to spot to spot may differ)

[@adolfoketchum8157]

Good stuff

[@abcdxx1059]

really helpful

[@mostafaatallah7001]

Is the case of 'the number of deaths" are also NOT time-homogeneous? as the probability of deaths to occur varies dramatically for the same time periods lengths?

[@Javlini]

Thank you sir

[@jbstatistics]

You are very welcome.

[@fatriantobong2097]

what about lambda is a normal distribution?

[@sapphireseee1003]

Can anyone please tell me hpw the avg weight of customers arriving at a store kn a 10 minute is not poisson...? I thought it is poisson as the variables r random....can anyone plz tell clear it

[@jbstatistics]

Not everything that is random has a Poisson distribution. A random variable must result from a specific set of circumstances for it to have a Poisson distribution. One of the simpler conditions is that it represents a count (a Poisson random variable takes on the possible values 0, 1, 2, ...). Is the average weight of customers a count? Are the possible values of this random variable 0, 1, 2, ...?

[@bluesunflower2007]

Wondering about the number of fatal crashes. In the vicinity of one crash incident, airlines may be more cautious regarding the securities, and therefore lowering the accident rates. And on the other side of the spectrum, if the accident is due to some vulnerabilities in the software/hardware as in the case of 737 Max, the events may appear in groups instead.

[@jbstatistics]

There is often some form of subtle dependency, like the ones you mention. Like with many models in statistics, the practical situation might not perfectly match the assumptions, but overall the model can provide a reasonable approximation. It's almost always the case that the reality is a touch more complicated.

[@bluesunflower2007]

@@jbstatistics That's very true. Another issue here is about statistical inference. If the sample is not large enough, one cannot completely distinguish a real Poisson distribution from a somewhat correlated non-Poisson distribution.

[@kommineniyuvashankarbabu8910]

Good explanation!! What if there is an assumption the fatalities of a given crash are independent. Then number of fatalities in a year can be Poisson distribution because there will be survivors and also fatalities on given crash. There can be 1 , 2 , 3 fatalities from one crash doesn’t that make them independent.?

[@jbstatistics]

We can assume whatever we'd like, but sometimes it's impossible for those assumptions to be true. I can live under the assumption that I'll be a billionaire in two years, but that will likely lead to problems for me. In any practical sense, airplane crash fatalities cannot be occurring independently. Let's think again about an example I use in the video. Consider the next time you fly. For fatalities to be occurring independently, the information that the person next to you on that plane will die in a plane crash that day would say nothing about the likelihood that you also die in a plane crash that day. But that's simply not the reality. If you know the person next to you on that plane will die in a plane crash that day, that is very, very, very bad news for you, and dramatically increases the chance that you will also die.

[@dchpavan7296]

Usefulll one on poisson

[@mrcake4525]

i see the light!

[@roseb2105]

i dont understand the last scenario

[@abcdxx1059]

11:22 boieg : let us introduce ourselves

[@rodelcabauatanmanuel2249]

Perfect explanation sir

[@bireswarsaha4881]

Great

[@mahimkhan7769]

Driving hours per day ?

[@keijukamatthew3852]

Love how you explain statistical concepts so easily compared to my professor

[@jbstatistics]

Thanks! Sometimes it's just a "different strokes for different folks" sort of thing.

[@harendrasingh_22]

After 2000 are the crashes because of cost cutting after dotcom bubble? Impossible, but what else then?.

[@vineetverma6645]

Its a limiting case of binomial distribution.

[@jbstatistics]

Yes, as I discuss in this video and elsewhere. But in many practical cases that notion is not all that helpful.

[@ishmaeladdae1625]

Best

[@antoinemack6241]

I couldnt hit "Subscribe" fast enough

[@sindisosihlehle1354]

nice intro

[@jbstatistics]

Thanks!

[@aglaiawong8058]

why uniform distribution cannot be modeled by Poisson? Is it because the probability is the same across entire area and thus is not considered 'random'?

[@aglaiawong8058]

I'm speaking of the case in video where there are green boxes with 2 everywhere

[@jbstatistics]

A Poisson distribution is a very specific thing. There are infinite types of random variables that don't have a Poisson distribution. In that example, the count would be 2 very frequently. The points weren't scattered randomly and independently (if you knew where one point was, that gave you a *very* good idea where the other points were). It's the opposite of clumping together -- they're spread as far apart as possible. But even if there was some variability, and the count sometimes came up 1 and sometimes came up 3, that wouldn't mean its Poisson. The number of heads that comes up when a fair coin is tossed 20 times is a random variable that has a distribution, has variance, etc., but it doesn't have a Poisson distribution (it has a binomial distribution with n = 20 and p = 0.5).

[@aglaiawong8058]

@@jbstatistics I see I see! The explanation's very clear and helpful. Thanks jbstatistics!

[@jbstatistics]

@@aglaiawong8058 You are very welcome!

[@smoothbraindetainer]

That spike in 2001 is lookin a little uh... _peculiar_

[@prashantkhot10]

How to simulate 100000 occurrences of a random variable Y which is equal to sum of two different random variables X and Z, where X follows a normal distribution with mean = 50 and variance = 20 and Z follows a Poisson distribution with mean = 40. Draw a histogram of the simulation

[@jbstatistics]

This is easily done, but depends on what software you're using. This is clearly some question from a course or other homework type stuff, and the precise method of carrying it out depends on what software your instructor wants you to use.

[@prashantkhot10]

@@jbstatistics I have solved it in python. But wanted to understand it in excel as I am not sure whether it is correct or not? So wanted to confirm it with experts. Below is program ### Progrem for Poisson ## Importing main header files import seaborn as sb from scipy.stats import poisson X,Y,Z =[],[],[] print (type(X)) X = np.random.normal(20,20**0.5,100000) Y = poisson.rvs(mu=40, size=100000) Z = X+Y print (max(Z)) print (min(Z)) df_Z = pd.DataFrame(data = Z,columns =["Value-Z"]) df_Z.head() bins = np.array(range(0,101,5)) print (bins) df_Z["Bucket"] = pd.cut(Z, bins) df_Z.head() Histogram = np.histogram(Z,bins) print(Histogram) plt.hist(df_Z["Value-Z"], bins =Histogram[1] ) plt.show()

[@taotaotan5671]

selecting a reasonable model is more essential than fitting a model

[@Kerrosene]

You missed saying that the events have to be rare. It is after all an approximation of a binomial process.

[@jbstatistics]

I think saying that the events have to be rare is very misleading. For example: "Don't forget folks, we're talking about *rare* events here. Now, suppose we have a Poisson random variable that has a mean of 1 trillion occurrences per second..." Sure, the occurrences are rare in the sense of being equivalent to the binomial in the limit, but not rare in the everyday sense.

[@Kerrosene]

@@jbstatistics Yes, but the approximation here is that if we considered the unit of time to be a femtosecond, or something of that order, the probability of the event would go very low in one time unit and thus the number of trials would have to be so high as to make np = lambda true and equal to 1 trillion. I guess rarity is relative, taken in this sense. But my question is whether we can model a process with probability as high as 0.5 using Poisson distribution?

[@Kerrosene]

@@jbstatistics Scratch that question. I think the answer is along the same lines as my comment. Just consider the time interval to be so small that the probability of success in that interval becomes small and the approximation holds true.

[@jbstatistics]

@@Kerrosene I have other videos in which I explain the relationship between the Poisson and binomial distributions. Take as a given that I understand that aspect, otherwise you are completely wasting your time. Even in this video I very briefly discuss the relationship between the Poisson and binomial, as it does of course help to visualize the situation *at times*, and there I include the fact that we are talking about small p. I bring it up again in the airline fatalities example, as it is beneficial in that context. But I feel that it is misleading to always make it about the rare event aspect,, as we are not always thinking of the Poisson in terms of its relationship to the binomial. For example, consider this problem: "Suppose in a certain rainstorm, the number of raindrops hitting your car per second has approximately a Poisson distribution with a mean of 1000..." Do you not see how making a big deal of the "rare event" aspect (outside of the direct relationship with the binomial) could be misleading for students?

[@Kerrosene]

@@jbstatistics It is rare in a relative sense, isn't it? Of the billion raindrops that fall in a second only 1000 end up in your car. This makes the probability of the event so less that we can make the approximation (1 + 1/x)^x -->e as x -->infinity. We here are just saying that of the billion drops that fall on earth in one second, on average the 1000 fall on the car. I don't think that the rate of an event happening can be equated to the notion of rarity.

[@petergorelov6852]

Да, почему-то в учебниках по статистике такие вещи не объясняют...

[@KristofferEngstrom]

How does one take into account that e.g. some countries has way worse safety regulations when it comes to commercial ailiners ? What I mean, where does one draw the line of what is concidered random and influential ? Lets say one year a poor country with bad airplane safety sees a big increase in people flying ? That would drastically increase the number of fatal crashes, and it is influenced by the safety regulations in said country.

[@jbstatistics]

The example I used at the end was for U.S. commercial airliners. The situation would be essentially the same for any country with inferior safety standards, but the the rate at which these crashes occur would be higher. The Poisson discussion I had in the video would still hold. One big question in any Poisson situation is whether the occurrences are independent, and this can feel a little abstract and tough to think about. For commercial airline crashes, it seems like a reasonable assumption, but the crashes could conceivably can be related. If the loose safety standards of a country resulted in the hiring of a terrible mechanic to carry out essential maintenance, then it's possible a series of poorly done jobs would result in a clumping together of fatal accidents (before the cause was determined). Or, 9/11 style attacks, say. In situations like this, knowing there was a crash makes it more likely to see another crash in the near future. But, at least to a reasonable approximation, loose safety standards would increase the mean number of crashes (lambda), but not have much of an impact on independence.