Theorem 1: If the probability p of success of a single trial approaches 0 while the number of trials n approaches infinity and the value μ = np stays fixed, then the binomial distribution B(n, p) approaches the Poisson distribution with mean μ.

Proof: Let  be the pdf for a binomial distribution as described in the statement of the theorem. Then

Now

where z = -p. But

and so

Also

Thus as p → 0 and n$\infty$, with μ = np fixed, we have

which completes the proof.

### 2 Responses to Poisson Distribution – Advanced

1. GEORGE ONYANGO says:

good work. i would like to get these lecture notes for statistical analysis through my email.
thank you.

• Charles says:

George,
Sorry, but I can’t make the lecture notes available via email. I plan to provide a book which contains these lecture notes which you can purchase for a modest fee, but the book is not yet available.
Charles