Poisson definition; approximation to binomial

Poisson distribution

A discrete random variable X that counts the number of occurrences of a rare event in a fixed interval of time, length, area or volume, has a Poisson distribution with parameter λ if

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

Here λ > 0 is the average number of occurrences, and mean = variance = λ. Examples: calls received per minute, defects per metre of cable.

Conditions for Poisson approximation to the binomial

A binomial B(n, p) can be approximated by a Poisson distribution with λ = np when:

1. The number of trials n is very large (n → ∞, usually n ≥ 20 or more).
2. The probability of success p is very small (p → 0, usually p ≤ 0.05).
3. The product λ = np is finite and moderate (commonly less than 5 to 10).

Then C(n, x) pˣ qⁿ⁻ˣ ≈ e⁻ⁿᵖ (np)ˣ / x!.

Example: if 0.2% of bulbs are defective, the number of defectives in a lot of 1000 is approximately Poisson with λ = 1000 × 0.002 = 2.