Actuarial science is in some ways older than probability and statistics itself and was in many ways instrumental in the development of probability and statistics. One of the earliest applications is to life distributions or mortality laws, in order to be able to determine life insurance fees. But there are many more uses, including claim frequency and claim size distributions.

Lifetime Distributions

GompertzMakehamDistribution — exponential mortality rate

WeibullDistribution — power mortality rate

UniformDistribution — de Moivre's mortality rate

BenktanderGibratDistribution  ▪  BenktanderWeibullDistribution

Insurance Claim Frequency Distributions

PoissonDistribution — number of claims when independent and constant rate

BetaBinomialDistribution  ▪  BetaNegativeBinomialDistribution  ▪  GeometricDistribution  ▪  LogSeriesDistribution  ▪  NegativeBinomialDistribution

Insurance Claim Severity Distributions

ExponentialDistribution — size of claims when independent and constant rate

ParetoDistribution — heavy tail claim sizes, for outsized claims

SplicedDistribution — splicing together body and tail claim distribution

WeibullDistribution  ▪  GammaDistribution  ▪  HalfNormalDistribution