From: horst.kraemer@snafu.de (Horst Kraemer)
Subject: Re: Poisson from Bernoulli?
Date: Wed, 10 Feb 1999 13:39:04 GMT
Newsgroups: sci.math

On Wed, 10 Feb 1999 00:29:56 -0800, tmcal@ucdavis.edu (Tyrrell
McAllister) wrote:

> I know that the poisson distribution is the limit of the bernoulli
> distribution with particular values for p and q, but does anyone here know
> how that limit is taken?  How does one deal with the binomial part?

              n!
B(n,k,p) =  --------  p^k * (1-p)^(n-k)   ; k = 0,1...,n
            k!(n-k)!



Now one takes the limit for n->oo and p such that  n*p = m = const and
fixed k, i.e. p will decrease when n increases. Now substitute p by
m/n. This yields


              n!         (m/n)^k
B(n,k,p) =  --------  * ---------  * (1-m/n)^n 
            k!(n-k)!    (1-m/n)^k


            m^k                  n*(n-1)*..*(n-k+1)
         = ----- * (1-m/n)^n  *  ------------------
             k!                      (n-m)^k

lim (1-m/n)^n = exp(-m)
n->oo

and 

          n*(n-1)*..*(n-k+1)
lim       ------------------  = 1
n->oo         (n-m)^k

                            m^k
Therefore lim B(n,k,p)  =   ---  * exp(-m)
              n->oo          k!
              n*p = m


Regards
Horst