Yule–Simon distribution

Yule–Simon distribution is a probability distribution named after the statisticians Udny Yule and Herbert A. Simon. It is a discrete probability distribution that is useful in various fields such as bibliometrics, linguistics, and ecology. The Yule–Simon distribution is particularly known for its application in modeling the distribution of species in ecological communities and the frequency of words in natural languages.

Definition

The probability mass function (PMF) of the Yule–Simon distribution for a random variable X taking integer values 1, 2, 3, ... is given by:

P(X = k) = ρ B(k, ρ+1)

where ρ > 0 is the shape parameter of the distribution, and B is the Beta function. The parameter ρ controls the skewness of the distribution. A higher value of ρ results in a less skewed distribution.

Properties

Mean

The mean of the Yule–Simon distribution is given by:

E[X] = 1 + 1/ρ

for ρ > 1. This indicates that the mean increases as the shape parameter ρ decreases.

Variance

The variance of the Yule–Simon distribution is:

Var(X) = ρ^2 / ((ρ-1)^2(ρ-2))

for ρ > 2. The variance increases with a decrease in the value of ρ, similar to the mean.

Applications

The Yule–Simon distribution has been applied in various fields:

• In ecology, it is used to model the number of species as a function of the number of individuals.
• In linguistics, it helps in modeling the frequency of words in natural language texts.
• In bibliometrics, it is applied to model the distribution of citations among papers.

Related Distributions

• The Yule–Simon distribution is related to the Pareto distribution in the context of continuous distributions.
• It is also related to the Zipf's law when modeling the frequency of words in natural languages.

See Also

• Probability distribution
• Beta function
• Pareto distribution
• Zipf's law