Standard Probability Distributions
Domain 4: Standard Probability Distributions
This domain tests the binomial, Poisson and normal distributions: the conditions under which each applies, mean and variance, and routine probability computation including use of the standard normal table.
Core rules
- Binomial B(n, p): n independent Bernoulli trials with constant p. P(X = x) = C(n, x)p^x q^(n−x), q = 1 − p; mean np, variance npq. It is symmetric only when p = 0.5.
- Poisson P(λ): P(X = x) = e^(−λ)λ^x/x!, mean = variance = λ; it is the limit of B(n, p) with n large, p small and np = λ fixed; sums of independent Poisson variables are Poisson.
- Normal N(μ, σ²): symmetric and unimodal with mean = median = mode = μ. Standardise z = (x − μ)/σ and read areas from the N(0, 1) table.
- Areas: about 68.27% within μ ± σ, 95.45% within μ ± 2σ, 99.73% within μ ± 3σ; P(Z > 0) = 0.5 by symmetry.
Worked example
Ten items are inspected, each defective independently with probability 0.1. Then P(X = 2) = C(10, 2)(0.1)²(0.9)⁸ = 45 × 0.01 × 0.430467 = 0.1937. Reasoning: choose which 2 of the 10 are defective, attach (0.1)² to those and (0.9)⁸ to the other eight. Mean = 10(0.1) = 1 and variance = 10(0.1)(0.9) = 0.9. Poisson check with λ = 1: e^(−1)(1)²/2! = 0.1839, close to 0.1937, so the approximation is acceptable here because n is large and p small.
Traps
- Quoting np as the variance of a binomial.
- Applying the binomial to sampling without replacement from a small lot, where the hypergeometric model is required.
- Using λ instead of √λ when standardising a Poisson count.
- Substituting σ² for σ in z = (x − μ)/σ.
- Omitting the continuity correction when approximating a discrete distribution by the normal.