Theoretical Distributions (Binomial, Poisson, Normal)
What the domain covers
The exam tests conditions, parameters and calculations for binomial, Poisson and normal distributions, including standardisation and normal-area interpretation. You should recognise which model fits a given random experiment.
Core rules
- Binomial: fixed n independent trials, two outcomes, constant p. P(X = x) = C(n,x)p^x q^(n−x); mean np, variance npq.
- Poisson: counts of rare events in a fixed interval; P(X = x) = e^(−λ)λ^x/x!; mean = variance = λ. It approximates binomial when n is large and p small.
- Normal: continuous, symmetric, bell-shaped; mean μ, SD σ. Standardise z = (x − μ)/σ. Total area = 1; about 68% lies within μ ± σ and about 95% within μ ± 2σ.
- For 'at least' use 1 − P(X < k); for 'at most' use cumulative probability.
Worked example
For a binomial variable with n = 4 and p = 0.5, P(X = 2) = C(4,2)(0.5)²(0.5)² = 6 × 0.0625 = 0.375. If X is Poisson with λ = 3, P(X = 0) = e^(−3) ≈ 0.0498. For a normal variable with μ = 50 and σ = 10, x = 65 gives z = 1.5; the area to the right is about 0.0668 from the standard normal table.
Traps
Do not use Poisson when n is small and p is not small. In binomial, 'exactly' and 'at least' differ. For normal approximations to discrete data, apply a continuity correction. Poisson λ must match the interval: double the interval, double λ. Do not confuse variance npq with SD.