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The probability mass function of the Yule-Simon(ρ) distribution is

for integer k ≥ 1 and real ρ > 0, where B is the beta function.

The probability mass function f has the property that for sufficiently large k we have

This means that the tail of the Yule-Simon distribution is a realization of Zipf's law: f(k) can be used to model, for example, the relative frequency of the kth most frequent word in a large collection of text, which according to Zipf's law is inversely proportional to a (typically small) power of k.

The cumulative distribution function of a Yule-Simon(ρ) distributed random variable X is

.

Furthermore, X has the following properties:

mode

1

mean

varianceThis article is about mathematics. Alternate meaning: variance (land use). In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically

skewnessIn probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. Roughly speaking, a distribution has positive skew if the positive tail is longer and negative skew if the negat

kurtosis excess

1 Generalizations

Simon also hinted at a two-parameter generalization of the Yule-Simon distribution, in which the beta function is replaced by an incomplete beta functionIn mathematics, the incomplete beta function is a generalization of the beta function that replaces the definite integral of the beta function with an indefinite integral. The situation is analogous to the incomplete gamma function being a generalization. The probability mass function of the generalized Yule-Simon(ρ, α) distribution is defined as

with 0 ≤ α < 1. For α = 0 the ordinary Yule-Simon(ρ) distribution is obtained as a special case.

2 References

• Herbert A. Simon, On a Class of Skew Distribution Functions, Biometrika 42(3/4): 425–440, December 1955.