Standard normal; normal approx of Poisson

Standard normal distribution

If X is normal with mean μ and standard deviation σ, then Z = (X − μ)/σ has the standard normal distribution with mean 0 and variance 1, written Z ~ N(0, 1). Its pdf is

f(z) = (1/√(2π)) e^(−z²/2), −∞ < z < ∞.

The curve is bell shaped, symmetric about 0, with total area 1. Areas under it are tabulated, so probabilities for any normal variable are found by standardising.

Normal approximation to the Poisson distribution

If X ~ Poisson(λ), then mean = variance = λ. When λ is large, the Poisson distribution becomes nearly symmetric and bell shaped, and

Z = (X − λ)/√λ ≈ N(0, 1), i.e. X ≈ N(μ = λ, σ² = λ).

Condition: λ must be large, usually λ ≥ 10 (some books use λ > 15 or 20). The larger the λ, the better the approximation.

Because Poisson is discrete and the normal is continuous, a continuity correction is applied, e.g. P(X ≤ a) ≈ P(Z ≤ (a + 0.5 − λ)/√λ).