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Poisson definition; limiting case of binomial

Short answer

Poisson distribution has P(X=x)=e^-lambda lambda^x/x!, mean=variance=lambda. It is the limit of B(n,p) as n->infinity, p->0 with np=lambda, giving the same Poisson probability mass function.

Define Poisson distribution. Write the limiting case of Poisson distribution as a Binomial distribution.

Poisson distribution

A discrete random variable X, the number of occurrences of a rare event in a given interval (of time, length, area), follows a Poisson distribution with mean λ if

P(X = x) = e⁻ˡ λˣ / x!, x = 0, 1, 2, …, λ > 0

Mean = variance = λ.

Poisson as the limiting case of the binomial

Let X ~ B(n, p) and let n → ∞, p → 0 so that np = λ stays constant. Then

P(X = x) = C(n, x) pˣ (1 − p)ⁿ⁻ˣ

Put p = λ/n:

P(x) = [n(n-1)...(n-x+1)/x!] (λ/n)ˣ (1 - λ/n)ⁿ⁻ˣ
     = (λˣ/x!) · [1(1-1/n)...(1-(x-1)/n)] · (1-λ/n)ⁿ · (1-λ/n)⁻ˣ

As n → ∞: each factor (1 − k/n) → 1, (1 − λ/n)ⁿ → e⁻ˡ and (1 − λ/n)⁻ˣ → 1. Hence

lim P(X = x) = e⁻ˡ λˣ / x!

So for large n and small p, B(n, p) ≈ Poisson(λ = np).

✓ Answer checked by Claude Sonnet 5.5

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