Normal distribution; approximation of Poisson

Normal distribution: a continuous distribution with pdf

f(x) = (1 / (σ√(2π))) · e^(−(x − μ)² / (2σ²)), −∞ < x < ∞

where μ is the mean and σ the standard deviation. Its curve is bell-shaped and symmetric about μ, with mean = median = mode = μ, total area 1, and about 68.27%, 95.45% and 99.73% of values within μ ± 1σ, 2σ, 3σ.

Normal approximation of the Poisson distribution

If X ~ Poisson(λ), the mean and variance are both λ. When λ is large (generally λ ≥ 10; some books use λ > 5), the Poisson distribution is closely approximated by the normal distribution with

μ = λ and σ² = λ, i.e. X ≈ N(λ, λ)

Because the Poisson is discrete and the normal is continuous, a continuity correction of 0.5 is used:

• P(X = a) ≈ P(a − 0.5 < X < a + 0.5)
• P(X ≤ a) ≈ P(X < a + 0.5)
• P(X ≥ a) ≈ P(X > a − 0.5)

and the standard variable is z = (x ± 0.5 − λ)/√λ.

Example

Calls arrive at a rate λ = 25 per hour. Find P(X ≤ 30).

z = (30.5 − 25)/√25 = 5.5/5 = 1.10

P(X ≤ 30) ≈ P(Z < 1.10) = 0.8643

The exact Poisson value is 0.8633, so the approximation is good.