The Poisson Distribution: Mathematically Deriving the Mean and Variance

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  • Опубліковано 23 гру 2024

КОМЕНТАРІ • 135

  • @kyliegrimes5504
    @kyliegrimes5504 Рік тому +9

    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
    @shankararjune4074 8 років тому +82

    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
      @jbstatistics  8 років тому +21

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

    • @emiliocanton
      @emiliocanton 4 роки тому +4

      @@jbstatistics Still making it

  • @TomDulson
    @TomDulson 10 років тому +7

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

  • @user-nw4jm8np4g
    @user-nw4jm8np4g Рік тому +1

    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!

  • @weslielegaspi196
    @weslielegaspi196 2 місяці тому

    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! 🎉

  • @rosalindayanez6865
    @rosalindayanez6865 3 роки тому

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

  • @jbstatistics
    @jbstatistics  11 років тому +3

    @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!

  • @MrCobozco
    @MrCobozco 2 роки тому

    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.

  • @mohammadpourheydarian5877
    @mohammadpourheydarian5877 8 років тому

    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
      @jbstatistics  8 років тому

      +Mohammad Pourheydarian Thanks so much for the wonderful compliment!

  • @charlescharles1270
    @charlescharles1270 4 роки тому

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

  • @Maha_s1999
    @Maha_s1999 8 років тому +2

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

    • @jbstatistics
      @jbstatistics  8 років тому +1

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

  • @sumreenaziz8693
    @sumreenaziz8693 5 років тому +2

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

  • @njabulonzimande2893
    @njabulonzimande2893 4 роки тому +1

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

  • @lisagraifer
    @lisagraifer 3 роки тому +1

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

  • @noahrubin375
    @noahrubin375 3 роки тому

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

    • @jbstatistics
      @jbstatistics  3 роки тому +1

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

  • @jimmy000
    @jimmy000 11 років тому +1

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

    • @jbstatistics
      @jbstatistics  11 років тому

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

  • @GaMeON159753456
    @GaMeON159753456 5 років тому

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

  • @harryhughes7761
    @harryhughes7761 7 років тому +1

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

    • @jbstatistics
      @jbstatistics  7 років тому +2

      Thank you for the very kind words Harry!

  • @ramendrathakur6861
    @ramendrathakur6861 11 років тому

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

  • @DmitriArkhipov
    @DmitriArkhipov 11 років тому +3

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

  • @GoodLuckForever-wi9kb
    @GoodLuckForever-wi9kb Рік тому

    Thanks for sharing such method in such a simple way

  • @ReddiSathvik
    @ReddiSathvik 4 роки тому

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

  • @gopinatht7677
    @gopinatht7677 4 роки тому

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

  • @ipshitabanerjee1323
    @ipshitabanerjee1323 4 роки тому

    Thanks a lot. Short but clear explanation.

  • @mattjohnlewis
    @mattjohnlewis 8 років тому +1

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

  • @davidbishop8870
    @davidbishop8870 7 років тому

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

  • @poonamdasilva7540
    @poonamdasilva7540 4 роки тому +1

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

  • @yourdipak7994
    @yourdipak7994 6 років тому

    You're just amazing in your way of teaching

  • @thetravellerfuture
    @thetravellerfuture 8 років тому +2

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

    • @jbstatistics
      @jbstatistics  8 років тому

      +Maeda Beegun You are very welcome!

  • @learnlovelaugh4life
    @learnlovelaugh4life 7 років тому

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

    • @jbstatistics
      @jbstatistics  7 років тому

      You are very welcome. Thanks for the compliment!

  • @ugestacoolie5998
    @ugestacoolie5998 7 місяців тому

    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
      @jbstatistics  7 місяців тому

      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.

  • @Adam-gp3ij
    @Adam-gp3ij 5 років тому

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

  • @johnnyd.j.6068
    @johnnyd.j.6068 7 років тому +1

    So simple with great explanation! Thank you.

  • @jacoboribilik3253
    @jacoboribilik3253 6 років тому +1

    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!

  • @AmanSingh-gs4ue
    @AmanSingh-gs4ue 6 років тому

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

  • @asmashafiullah8720
    @asmashafiullah8720 7 років тому

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

  • @jacobp9470
    @jacobp9470 7 років тому +1

    Your videos are so helpful! Thanks for making them.

    • @jbstatistics
      @jbstatistics  7 років тому +1

      You are very welcome. Thanks for the compliment!

  • @jamesfkey
    @jamesfkey 9 років тому

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

  • @橡皮擦-e1j
    @橡皮擦-e1j Місяць тому

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

  • @jtflypegasus
    @jtflypegasus 10 років тому

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

  • @jeanconstantinople2328
    @jeanconstantinople2328 8 років тому

    outstanding as always !

  • @codework-vb6er
    @codework-vb6er 2 роки тому

    @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
      @jbstatistics  2 роки тому

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

  • @anandkulkarni2111
    @anandkulkarni2111 7 років тому +2

    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
      @jbstatistics  7 років тому +5

      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
      @stewiegriffinfan18 6 років тому

      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
      @blakejhonshen2710 6 років тому

      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.

  • @cronos6890
    @cronos6890 3 роки тому

    Really well made video, thank you a lot!

  • @Dr.RokiaAbdein
    @Dr.RokiaAbdein 5 років тому

    thanks so much, it is so easy to understand

  • @pra601
    @pra601 6 років тому

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

    • @jbstatistics
      @jbstatistics  6 років тому

      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].

  • @allanabela2248
    @allanabela2248 10 років тому

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

  • @emadharazi5044
    @emadharazi5044 4 роки тому

    Excellent video

  • @blazcukelj7273
    @blazcukelj7273 Рік тому

    Great tutorial. Thanks!

  • @wishmaherath404
    @wishmaherath404 6 років тому

    very clear and straightforward,thanks :-)

  • @ramendrathakur6861
    @ramendrathakur6861 11 років тому

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

  • @kautukraj
    @kautukraj 3 роки тому +1

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

  • @ahmedros3713
    @ahmedros3713 8 років тому

    thank you ... this is very clear and helpful

  • @samstreet6595
    @samstreet6595 2 роки тому

    That is amazing, cool method

  • @gopinatht7677
    @gopinatht7677 4 роки тому

    Happy to Subscribe!

  • @pavelooo2289
    @pavelooo2289 5 років тому

    thank you. I couldn't find it elsewhere

  • @88NA
    @88NA Рік тому

    Thanks for posting this video

  • @colecofer8474
    @colecofer8474 8 років тому

    This video is amazing. Thanks!

  • @aronquemarr7434
    @aronquemarr7434 4 роки тому +1

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

  • @weicarlos1719
    @weicarlos1719 7 років тому

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

    • @johnnyd.j.6068
      @johnnyd.j.6068 7 років тому

      ua-cam.com/video/alEjOQN0lYA/v-deo.html

  • @ahsansacademy6796
    @ahsansacademy6796 Місяць тому

    Where were you bro 😭

  • @jtflypegasus
    @jtflypegasus 2 роки тому

    That's beautiful...👍👍

  • @irtizahasan3537
    @irtizahasan3537 8 років тому

    thank you for making these videos....

  • @dml39prudhvi65
    @dml39prudhvi65 5 років тому

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

  • @i_m_hk
    @i_m_hk 6 років тому

    Nice explanation

  • @zuckirosenfuchs5441
    @zuckirosenfuchs5441 3 роки тому

    Nice video!

  • @ailema7773
    @ailema7773 10 років тому

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

  • @Mark-li3my
    @Mark-li3my 7 років тому

    Helped a lot, thank you bro

  • @challakishore6975
    @challakishore6975 8 років тому

    thank you, this is very helpful and clear.

  • @tarmopungas
    @tarmopungas 2 роки тому

    Thanks a lot for this!

  • @protidinjob7193
    @protidinjob7193 5 років тому +1

    Thanks for help

  • @surnermahiteja9942
    @surnermahiteja9942 8 років тому

    thanks for making dis so simple

    • @jbstatistics
      @jbstatistics  8 років тому

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

  • @ednakazuya12
    @ednakazuya12 5 років тому

    Brilliant!

  • @nerdymathematician
    @nerdymathematician Місяць тому +1

    THANK YOU😭

  • @electrikkblue
    @electrikkblue 3 роки тому

    Thank you so much!

  • @maqboolurrahimkhan
    @maqboolurrahimkhan 3 роки тому

    Just amazing thanks alot

  • @annalam8624
    @annalam8624 7 років тому

    Thank you! This video helps me a lot :D

  • @kaushal32755
    @kaushal32755 10 років тому

    Thanks for the help !

  • @Quizpop12
    @Quizpop12 5 років тому

    Thnk u soo much

  • @liangchihuang7775
    @liangchihuang7775 7 років тому

    very very useful!

  • @JismalJamal
    @JismalJamal 10 років тому

    Great Explanation

  • @ahmedshalapy7923
    @ahmedshalapy7923 7 років тому

    you're amazing

  • @MsSprulez
    @MsSprulez 6 років тому

    Amazing!

  • @alanbchen
    @alanbchen Рік тому

  • @amanmwangwa8987
    @amanmwangwa8987 8 років тому

    nice one...

  • @samah241
    @samah241 9 років тому

    Thank you 😀

  • @conceptofcivilengg.withabh1728
    @conceptofcivilengg.withabh1728 3 роки тому

    Thankyou sir

  • @mdjunaid1664
    @mdjunaid1664 4 роки тому

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

    • @jbstatistics
      @jbstatistics  4 роки тому

      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_m_hk
    @i_m_hk 6 років тому

    Thank u sir

  • @mohdyunus1512
    @mohdyunus1512 7 років тому

    Thank you so much

  • @BadEnoughDudeRescues
    @BadEnoughDudeRescues 6 років тому

    I love you bro

  • @hp-qx7tf
    @hp-qx7tf 7 місяців тому

    stats is hard :(

  • @dr.gavisiddappaanandhalli6563
    @dr.gavisiddappaanandhalli6563 7 років тому

    it is very nice

  • @DistortedV12
    @DistortedV12 3 роки тому

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

  • @erandaweerasingha404
    @erandaweerasingha404 6 років тому +1

    supper

  • @uncommon2848
    @uncommon2848 4 роки тому

    my exam is in one hour

  • @Diego-ss7sg
    @Diego-ss7sg 4 роки тому

    You sound like Mark Zuckerberg

  • @sophiechavigny9725
    @sophiechavigny9725 Рік тому

    Thank you so much!!