Poisson Distribution – Advanced

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

image3292image3293 image3294

Now
image3295

where z = -p. But
image3297

and so
image3298

Also
image3299

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

image3301

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

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