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Nov 22nd, 2008 01:33
John Martin, Knud van Eeden,
  Knud van Eeden  22 December 2004  00:09 am  Math: Probability: Distribution: Poisson: Can you derive the Poisson distribution L^k . e^(L) / k!?   The idea is to reduce the amount of information (by letting it cancel away), when N grows very large.   The Poisson distribution is directly derived from the Bernouilli chance formula. The Poisson distribution can be viewed as the limiting case of a binomial distribution as N approaches infinity and p approaches zero, but N . p remaining constant.   from Bernouilli: chance on k out of N, given a chance p is: N k N  k ( ) . p . (1p) (1) k  given: L = N . p {=average rate} (2) (L gives also the expected number of successes, but is also equal to the average rate (e.g. 16 incoming telephones per day on average)) The purpose of the work out below is to replace any occurrence of p (and thus 1p) (as it shows to be inconvenient to work with these values) by this average rate L. This L is the average vertical value of the Poisson curve. As for almost every curve a mean value can be calculated, this is a quite general value. So instead of p you replace it to work with L.  Another way of looking at L is: Imagine an interval of 1 unit of time divided into N tiny subintervals, each so small that it is not physically possible for more than one event to occur in a subinterval. Poisson: ... 0 1 2 3 4 N1 N > number of intervals If p is the probability that an event does occur in a subinterval, then the average number of events in 1 unit of time will be proportionally larger, and because there N subintervals here, the number of events will be N times as large. So you scale with a factor N. p + p + p + p + p + p + p + p + p + p + p + p + p + p + ... + p   + as much p's as there are subintervals, thus N totally + or thus N . p And L = N . p  Now, (1) can be written out, using the definition of N over k combinations, as: N.(N1).(N2).(N3)...(Nk+1) k Nk  . p . (1p) k.(k1).(k2).(k3)...1 or further as, using the law of the exponents (you can split the difference of powers in a quotient) N.(N1).(N2).(N3)...(Nk+1) k N  . p . (1p) (3) k.(k1).(k2).(k3)...1  k (1p) Working towards the goal is to get average the e expression in. You know maybe that x (another) definition of e is: x 1 1 (1 + ) = e (4) x lim x > +infinity An analog expression is: x 1 1 (1  ) = e (5) x lim x > +infinity and similarly: p x +p (1 + ) = e (6) x lim x > +infinity and similarly: p x p (1  ) = e (7) x lim x > +infinity Now, you can write (3) also as (as the law of the products says that you can exchange the factors) k N N.(N1).(N2).(N3)...(Nk+1) p . (1p)  .  (8) (1p).(1p).(1p).(1p)...(1p) k.(k1).(k2).(k3)...1   + k factors + Now, from (2) follows equivalently: p = L / N, (9) so you replace p by L and N instead L k L N N.(N1).(N2).(N3)...(Nk+1) (  ) . (1  ) N N  .  (10) (1L).(1L).(1L).(1L)...(1L) k.(k1).(k2).(k3)...1         N N N N N    + k factors + This can be written as:  A. You have, by using the law of the exponents (as you can distribute the power of a quotient over the nominator and the denominator) k L k L (  ) =  N k N 1 1 By writing  as  (11) k N . N . N . N . N . N ... N   + k factors + and writing this underneath every corresponding factor in the N . (N1)... product. This denominator will cancel against the numerator, when N grows very large, as every term (Ni) becomes more and more equal,  N thus growing towards 1 for every factor. And the product of all 1's is again 1, which is of no influence in a product, so can be cancelled.  B. We write the form N (1  p) in the equivalent form L N (1  ) N as this is approaches (when N grows larger and larger), according the definition and canonic form (7) to L e (12)  C. As every factor (1L) converges to 1  0 (in other words 1), when  N N grows larger and larger (as L grows then to 0),  N this product (1L) . (1L) . (1L) ...    N N N grows to (10) . (10) . (10) ... or thus to 1 . 1 . 1 ... = 1 (13)  D. According to the definition of faculty, k.(k1).(k2)...1 is of course k! (I did expand this, but it was not really necessary here) (14) Now putting (11), (12), (13) and (14) in (15) leaves the much reduced form (given that N grows larger and larger): + k factors +   k L N N .(N1).(N2).(N3)...(Nk+1) L . (1  ) N  .  (15) N . N . N . N ... N k.(k1).(k2).(k3)...1  (1L).(1L).(1L).(1L)...(1L)         N N N N N    + k factors + Or thus: k L 1 L . e  .  1 k! So this leaves k L L . e =  k! (given: that L is constant, and N grows very large) further: k maybe any value between 0 and N  Note that the sample size N as well as the probability p have completely dropped out of the expression, which has the same functional form for all values of L.  Or thus in words: k (average over your interval) = (average over your interval) . e  k!   Book: see also:  [book: Spiegel, Murray R.  Probability and statistics  McGrawHill (Schaum series)  p. 129 'The Poisson distribution']   Internet: see also:  Poisson Distribution http://home.ched.coventry.ac.uk/Volume/Vol4/derive.htm  Poisson Distribution http://mathworld.wolfram.com/PoissonDistribution.html  Math: Probability: Distribution: Link: Overview: Can you give an overview of links? http://www.faqts.com/knowledge_base/view.phtml/aid/32917/fid/815http://www.webs4soft.com/Links4.htm http://hotelsinindia.webs4soft.com/hotelsindelhi.htm http://indiatravel.webs4soft.com/Resources.htm http://indianmovies.webs4soft.com/KuchKuchHotaHai.htm http://realestate.webs4soft.com/propertytips.htm http://foodhealthcaretips.webs4soft.com/Resources.htm http://www.websitecompanyindia.com/seoLinks.htm http://www.websitecompanyindia.com/seoLinks.htm http://indianjewelry.websitecompanyindia.com/dimondring.htm http://www.bestindiaeducation.com/LinkExchange.htm http://bestjobconsultant.bestindiaeducation.com/PARTNERS.htm http://eloctronicandmobilestore.bestindiaeducation.com/mobilelinks.htm http://easyloanservice.bestindiaeducation.com/HomeInsuranceLinks.htm http://www.rajhealthcenter.com/CosmeticSurgery.htm http://creativebusinessgroup.rajhealthcenter.com/BusinessLinks.htm http://onlinefreeinternetgames.rajhealthcenter.com/GamesLinks.htm http://topbeautytips.rajhealthcenter.com/BeautyLinks.htm http://www.indiatourpoint.com/TravelLinks.htm http://fourwheelerbuytips.indiatourpoint.com/AutoLinks.htm http://watchonlinefreecricketmatch.indiatourpoint.com/SportsLinks.htm http://directory.indiatourpoint.com/ http://www.bestlifeindia.com/Resources1.htm http://directory.bestlifeindia.com/ http://freeorkutscrapandsms.bestlifeindia.com/SMSLinks.htm http://onlinegiftshop.bestlifeindia.com/GiftLinks.htm http://www.freemusicpoint.com/MusicLinks.htm http://lovedating.freemusicpoint.com http://onlineartpresentation.freemusicpoint.com/ http://onlinefurnitureshop.freemusicpoint.com/