Binomial Distribution

A binomial random variable X has parameters m (the number of independent trials) and q (the probability of success on each trial). Each trial is a Bernoulli random variable with mean q, and the binomial is the sum of m independent Bernoullis. The Bernoulli is the special case m = 1.

The SOA Loss Models tables write the parameters as m and q. Earlier probability courses often use n and p. The mathematics is the same; only the letters change.

Binomial is the right model when three conditions hold: a fixed number of trials, the same success probability on every trial, and independence across trials. The classic actuarial uses are number of policies with a claim in a portfolio of fixed size, number of survivors in a fixed cohort over a fixed period, and number of defaults among a fixed loan pool.

Pick Poisson instead when the count of trials is itself random or very large with small per-trial probability. Pick hypergeometric instead when sampling is without replacement from a small finite population so trials are not independent. Pick negative binomial when the observed variance exceeds the observed mean, which is incompatible with a binomial fit.

With independent observations X_1, ..., X_n drawn from Binomial(m, q) and m known, the maximum likelihood estimator for q is the sample mean divided by m. With m = 1 the MLE is just the sample proportion of successes.

If m is unknown, both m and q must be estimated jointly. The MLE for m has no clean closed form; in exam settings m is virtually always given.

A book of 20 independent annual policies has a 0.08 probability that each one produces at least one claim during the year. Let X be the number of policies that produce a claim. Then X is Binomial(20, 0.08).

Probability that exactly two policies produce a claim is C(20, 2)(0.08)^2(0.92)^{18} = 190 × 0.0064 × 0.2229 ≈ 0.271. The mean is 1.6 and the variance is 1.472, so the coefficient of variation is √1.472 / 1.6 ≈ 0.758.

A cohort of 100 lives at age 65 each survive one year with probability 0.97, independently. The number of survivors S at age 66 is Binomial(100, 0.97). The expected number of survivors is 97; the variance is 100 × 0.97 × 0.03 = 2.91.

The probability that fewer than 95 survive can be approximated by the normal distribution because m × q × (1 − q) is not too small. With continuity correction, P(S ≤ 94) ≈ Φ((94.5 − 97) / √2.91) = Φ(−1.466) ≈ 0.071.

If X is Binomial(m_1, q) and Y is Binomial(m_2, q) and X and Y are independent with the same q, then X + Y is Binomial(m_1 + m_2, q). The convolution is closed only when q is shared.

If m_1 = 8, m_2 = 12, q = 0.25, then X + Y is Binomial(20, 0.25) with mean 5 and variance 3.75. If the two q values differed, no binomial would fit the sum; the variance formula would still hold but the distribution would no longer be binomial.

Bernoulli is the m = 1 case. The Poisson distribution is the limit when m grows large and q shrinks with mq held at a constant λ; this is why Poisson is sometimes called the law of rare events.

Hypergeometric replaces independent trials with sampling without replacement; the binomial is the with-replacement analogue. As the population grows relative to the sample size, the two distributions converge.

Inside Loss Models, the binomial is a member of the (a, b, 0) class with a = -q/(1-q) and b = (m+1)q/(1-q). That parameterization is what enables the Panjer recursion for compound binomial aggregates.