The Binomial and Poisson Models
This domain covers the two workhorse models for counting events - deaths, claims, lapses, arrivals - and the assumptions that make each appropriate.
The binomial model
- X ~ Bin(n, p): n independent trials, constant p, two outcomes. P(X = x) = C(n,x) p^x (1-p)^(n-x), E[X] = np, Var(X) = np(1-p).
- Maximum likelihood gives p-hat = x/n with Var(p-hat) = p(1-p)/n.
- Life-table link: if l_x independent lives each have the same q_x, then d_x ~ Bin(l_x, q_x). Note Var(d_x) = l_x q_x (1-q_x), smaller than the Poisson variance when q_x is not negligible.
The Poisson model
- X ~ Poisson(lambda): P(X = x) = e^(-lambda) lambda^x / x!, with mean equal to variance equal to lambda.
- Counts over disjoint intervals add; lambda-hat = observed events / total exposure.
- Poisson is the limit of binomial as n grows, p shrinks and np tends to lambda - which is why it suits large populations with rare events.
- Poisson process: constant rate, independent increments, exponential inter-arrival times with mean 1/lambda, gamma waiting times.
- Overdispersion: observed variance above the mean indicates extra heterogeneity (frailty, epidemics, clustering). Use quasi-Poisson with a scale factor, or a negative binomial.
Worked example
1,000 lives are observed for one year and 12 die. q-hat = 12/1,000 = 0.012. The binomial standard error is sqrt(0.012 x 0.988 / 1000) = 0.00344, so an approximate 95% interval is 0.012 +/- 1.96 x 0.00344, i.e. (0.0053, 0.0188). Treating deaths as Poisson with exposure 1,000 gives sqrt(12)/1000 = 0.00346 - almost identical here because q is small, but they diverge when q is large.
Traps that cost marks
- Forgetting exposure as an offset when rates rather than counts are of interest.
- Using binomial when lives are not independent, or Poisson when q is not small.
- Applying the normal approximation without a continuity correction.