Probability Distributions
Probability Distributions tests the move from a random variable to its distribution, and the mean, variance and moment generating function of the binomial and Poisson models.
Discrete random variables
A probability mass function satisfies p(x) ≥ 0 and Σp(x) = 1. Then E(X) = Σx p(x), E(X²) = Σx² p(x), and V(X) = E(X²) − [E(X)]². The distribution function is F(x) = P(X ≤ x).
Binomial distribution
X ~ B(n, p) counts successes in n independent trials with constant p: P(X = x) = nCx p^x q^(n−x), q = 1 − p. Mean = np, variance = npq, mgf = (q + pe^t)ⁿ.
Poisson distribution
P(X = x) = e^(−λ)λ^x/x!, x = 0, 1, 2, …, with mean = variance = λ and mgf exp(λ(e^t − 1)). It is the limiting form of the binomial as n → ∞, p → 0 with np = λ fixed.
Worked example
If X ~ B(5, 0.4), find P(X ≥ 1). Use the complement: P(X ≥ 1) = 1 − P(X = 0) = 1 − (0.6)⁵ = 1 − 0.07776 = 0.92224.
Traps
Writing the variance as np instead of npq; forgetting that 'at least one' is usually a complement; using the binomial when p changes between trials; assuming mean equals variance for a binomial, which holds only for Poisson.