"How to" for WP-34s Poisson distribution



#6

Have a look at this:

Users' Guide: Poisson distribution

It took me a long time to figure out the Poisson solver, mainly because the format the parameters needed to be in was not what I was used to. I assume I've got it right, since the answer is the same as this Poisson calculator.

Comments?


#7

I did not loose any time over this (and maybe I should ;-) ) but when the User's Guide your link is pointing to says that "Assuming call inter-arrival times follow the Poisson distribution,...", probably what is meant is "Assuming that the number of call arrivals follows a Poisson distribution,...". A Poisson inter-arrival time would be non-sensical. This will, BTW, imply that the inter-arrival time is exponentially distributed.

Who cares, right? :-)

-Paulo


#8

I care! Bring it on!

yeah, it is nonesense - thanks for the correct wording.

Two other things possibly wrong with the example:

  • I think the more accurate model for blocking is Erlang-B, which is Poisson pdf(x,u)/cdf(x,u)
  • I'm not sure 8000 is the right number for sample size. I think the answer is correct because the number is so large. But if I enter "1" as the sample size and 200 as the mean I get "Parameter Error".

  • #9

    Please read the footnote on page 16 of THE MANUAL. May help ;-)

    Walter


    #10

    In the manual I just downloaded, there is no footnote on page 16.

    #11

    Yes, read that, know that much already :-)

    I'm expecting to specify the mean and have it tell me the probability of X (as the footnote on page 48 says). I don't understand the lambda = n * p0 part. What does sample size n have to do with a Poisson distribution?

    In my example I used 8000 as the sample size but that's bogus - the 200 was a projection (the amount of traffic I think I'm going to have if I add more users). I could have used anything that kept the value of J =< 1 such than 200 = J * K and I get the same answer.

    It works like the footnote on pg 48 if I always put p0 = 1 in J and the mean in K. I think the Remark for Poisson should say "Alternatively, Poisson's lambda = n*p0, may be in K if J = 1.

    Edited: 7 Oct 2011, 6:56 a.m.


    #12

    Would someone be so kind and point me in the right direction: I really don't find a footnote on page 16 of the WP34s manual. Are you talking about some other manual?

    Edited: 7 Oct 2011, 7:22 a.m.


    #13

    Could be page 13 - whatever page "STATISTICAL DISTRIBUTIONS, PROBABILITIES ETC. " is on in your version. I think Marcus and I are both using the new 2.2 version, not the 2.1 version in the zip file:

    https://sourceforge.net/projects/wp34s/files/doc/


    #14

    Thanks, it is page 13. I was already using the recommended version.

    #15

    I thought this might cause a bit of confusion eventually.

    I wanted a single parameter lambda for the Poisson distribution.
    Walter wanted the two parameters.


    - Pauli


    #16

    I assume Walter had some very convincing reasons for this. But at the moment I simply cannot see which. ;-)

    I am a big fan of POLA and KISS. Using two parameters for the Poisson distribution sure is possible, but not what I would consider consistent with the mentioned policy of least astonishment. Please, let's do it the way most users would expect and as also stated on the Wolfram Mathworld website on this subject. Especially the part between equations (8) and (9): "Note that the sample size N has completely dropped out of the probability function, which has the same functional form for all values of nu" (resp. lambda, since nu or lambda = N*p).

    Finally: if really two values p0 and n are given, pressing [x] returns the usual single parameter lambda. On the other hand, splitting a given lambda into n and a 1 that has be be stored in a separate register is awkward and an additional effort for most users who expect the usual single-parameter Poisson distribution.

    Dieter


    #17

    Quote:
    I assume Walter had some very convincing reasons for this. But at the moment I simply cannot see which. ;-)

    I don't remember the reasons, long long past now. Set p = 1 and n becomes lambda which didn't seem such a hardship to provide both view points.


    - Pauli


    #18

    Okay - so that's how I'm documenting it in the Users' Guide. I think the comment in the Owner's Manual should be changed to say p0 = 1 the mean can be put in K.

    #19

    This lambda= n*p0 is probably a left-over of the binomial distribution. The thing is: when you have a bunch of n repeated experiments, each with p0 probability of success (aka Bernoulli trials), then the mean number of successes is of course, n*p0, and the probability of k successes is given by the binomial distribution.
    Now, for cases where n is unknown but large, instead of using the CORRECT binomial distribution, we may use a Poisson, which is a fine approximation to the real thing, since this Poisson is the mathematical limit of the binomial when the number of experiments goes to infinity, but MAINTAINING the same mean n*p0. The parameter of this Poisson distribution will then be precisely the very same value of n*p0 of the original binomial. But n (sample size) or p0 (prob of individual success) will have no meaning or expression in the Poisson world. Only their product n*p0 (the mean number of successes) remains, as THE Poisson parameter..
    The advantage of using the Poisson is huge. For example, in these traffic problems, we have no notion of n (the number of costumers who may or may not call), nor can we reliably estimate p0 (the probability that one of them decides to call). No way we can use a proper binomial, thus. BUT it is very easy to estimate the mean number of calls arriving per hour (just count them), which is all we need to go ahead and solve the problem via a Poisson approximation.

    Hurra for Poisson (who would have been a medical doctor, if he had been an obedient son)

    Best
    -Paulo


    #20

    I think you wrote a very nice and convincing explanation why the Poisson distribution on the 34s should be changed to the usual single-parameter definition. ;-) It's simply the way most users will expect this distribution to work, and you explained the mathematical background behind this.

    Dieter


    #21

    I totally agree with you, Dieter. Single parameter is the way to go. In a Poisson process, if someone forces us to state what N and p0 are, the only mathematically consistent answer is: N=infinity, p0=0. Having to, for example, put p0=1 in the 34s to make the thing work, would be a source of constant and unbearable mental agony :-)

    But hey, if that is the price to have a 34s, it's a small price;
    Long live the 34s team! ;-)

    Paulo


    #22

    They may live even a bit longer after this oddity has been corrected - it will give peace to their minds. ;-)

    This is one of the advantages of such a project with the user community involved: there is no need to stop at the second best solution: we can go for the best. :-)

    Dieter

    #23

    Exactly that's the way it is described on page 48 of the manual. The parameters used for the binomial distribution may remain unchanged in J and K for Poisson as well. Alternatively, you may use the canonical Poisson's parameter in J while you store 1 in K.

    HTH

    Walter


    #24

    But it's actually the other way around. The 1 goes into J and the mean goes into K because anything greater than 1.0 in J causes "Invalid Parameter".


    #25

    Bonsoir Dominic,

    Looks like there's a mismatch between documentation and firmware. We'll get that settled. Thanks for pointing that out.

    Walter

    Edit: Changed the manual. Not committed yet.

    Edited: 9 Oct 2011, 6:52 p.m.


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