|Re: "How to" for WP-34s Poisson distribution|
Message #14 Posted by Paulo MO on 7 Oct 2011, 4:37 p.m.,
in response to message #6 by Dominic Richens
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)