Normal definition; approximation conditions

Normal distribution: a continuous random variable X is normally distributed with mean μ and standard deviation σ if its pdf is

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

written X ~ N(μ, σ²). Its curve is a symmetric bell shape, centred at μ, with mean = median = mode = μ, total area 1, and points of inflection at μ ± σ.

Normal approximation to the Binomial

If X ~ B(n, p), then for large n it is approximated by N(μ = np, σ² = npq) with q = 1 − p.

Conditions:

• n is large (n ≥ 30 is commonly used), and
• p is not too close to 0 or 1, so that np ≥ 5 and nq ≥ 5 (some books use 10).

Use the continuity correction: P(X = a) ≈ P(a − 0.5 < X < a + 0.5), and z = (x ± 0.5 − np)/√(npq).

Normal approximation to the Poisson

If X ~ Poisson(λ), then for large λ it is approximated by N(μ = λ, σ² = λ).

Condition:

• λ is large, commonly λ ≥ 10 (some books accept λ > 5), and the continuity correction is applied: z = (x ± 0.5 − λ)/√λ.