# Ewens's sampling formula

In population genetics, Ewens's sampling formula, describes the probabilities associated with counts of how many different alleles are observed a given number of times in the sample.

## Definition

Ewens's sampling formula, introduced by Warren Ewens, states that under certain conditions (specified below), if a random sample of n gametes is taken from a population and classified according to the gene at a particular locus then the probability that there are a1 alleles represented once in the sample, and a2 alleles represented twice, and so on, is

$\operatorname {Pr} (a_{1},\dots ,a_{n};\theta )={n! \over \theta (\theta +1)\cdots (\theta +n-1)}\prod _{j=1}^{n}{\theta ^{a_{j}} \over j^{a_{j}}a_{j}!},$

for some positive number θ representing the population mutation rate, whenever a1, ..., ak is a sequence of nonnegative integers such that

$a_{1}+2a_{2}+3a_{3}+\cdots +ka_{k}=\sum _{i=1}^{k}ia_{i}=n.\,$

The phrase "under certain conditions" used above is made precise by the following assumptions:

• The sample size n is small by comparison to the size of the whole population; and
• The population is in statistical equilibrium under mutation and genetic drift and the role of selection at the locus in question is negligible; and
• Every mutant allele is novel. (See also infinite-alleles model.)

This is a probability distribution on the set of all partitions of the integer n. Among probabilists and statisticians it is often called the multivariate Ewens distribution.

## Mathematical properties

When θ = 0, the probability is 1 that all n genes are the same. When θ = 1, then the distribution is precisely that of the integer partition induced by a uniformly distributed random permutation. As θ → ∞, the probability that no two of the n genes are the same approaches 1.

This family of probability distributions enjoys the property that if after the sample of n is taken, m of the n gametes are chosen without replacement, then the resulting probability distribution on the set of all partitions of the smaller integer m is just what the formula above would give if m were put in place of n.

The Ewens distribution arises naturally from the Chinese restaurant process.