Binomial Distribution: Properties, Formulas, and Exam P Applications

Definition and Parameters

The binomial distribution models the number of successes X in n independent Bernoulli trials, each with success probability p. The PMF is P(X = k) = C(n, k) * p^k * (1-p)^(n-k) for k = 0, 1, ..., n. The mean is E[X] = np and the variance is Var(X) = np(1-p).

In actuarial contexts, "success" might mean a policyholder files a claim. If 20 policies each have a 0.1 probability of claim independently, then the number of claims follows Binomial(20, 0.1) with mean 2 and variance 1.8.

Key Properties

The moment generating function (MGF) of X ~ Binomial(n, p) is M(t) = (1 - p + p*e^t)^n. This is useful for proving that the sum of independent binomial random variables with the same p is also binomial. If X ~ Bin(m, p) and Y ~ Bin(n, p) independently, then X + Y ~ Bin(m+n, p).

The mode of a binomial is the most likely value, approximately floor((n+1)p). The distribution is symmetric when p = 0.5 and becomes increasingly skewed as p moves away from 0.5.

Exam P Applications

Typical Exam P problems ask you to compute P(X >= k) or P(a <= X <= b). For small n, compute directly. For large n, the normal approximation X is approximately Normal(np, np(1-p)) can be used when np >= 5 and n(1-p) >= 5. With the continuity correction, P(X >= k) is approximately P(Z >= (k - 0.5 - np) / sqrt(np(1-p))). Practice both exact and approximate calculations, as Exam P tests both.