Standard Distributions: Binomial and Poisson
Binomial and Poisson are the two discrete distributions prescribed for this paper. Questions ask you to check whether the model applies, compute probabilities, find mean and variance, or fit expected frequencies to observed data.
Binomial
Conditions: a fixed number n of trials, each with two outcomes, constant probability p of success, and independent trials. Then P(X = x) = C(n, x)p^x q^(n−x), x = 0, 1, …, n, with q = 1 − p. Mean = np, variance = npq, standard deviation = √(npq). A useful recursion is P(x + 1)/P(x) = [(n − x)/(x + 1)] × (p/q).
Poisson
Conditions: n large, p small, trials independent, with λ = np moderate. Then P(X = x) = e^(−λ)λ^x/x!, x = 0, 1, 2, …; mean = variance = λ. Poisson is the limiting form of the binomial, and independent Poisson variables add, their parameters adding. Expected frequency of a value = N × P(X = x).
Worked example
Binomial with n = 5, p = 1/5, q = 4/5: P(X = 2) = C(5,2)(0.2)²(0.8)³ = 10 × 0.04 × 0.512 = 0.2048. Poisson with λ = 2: P(X ≥ 1) = 1 − P(X = 0) = 1 − e^(−2) = 0.8647.
Traps that cost marks
Using npq as the Poisson variance; applying the Poisson when p is not small or n is not large; forgetting to rescale λ when the interval changes; computing “at least one” directly instead of by complement; omitting the binomial coefficient; treating the number of trials as variable in a binomial setting.