Yule-Simon Distribution
X ~ YuleSimon(ρ)
Heavy-Tailed / Preferential-Attachment Family • Published July 22, 2026
The rich-get-richer distribution: history from Udny Yule's 1925 model of species-per-genus growth and Herbert Simon's 1955 rediscovery via word-frequency preferential attachment, a genuinely infinite-support power-law tail whose exponent is derived from the generative process itself (unlike the already-published Zipf sheet's assumed finite-support power law), moments that only exist up to a point set by ρ, a digamma-based maximum-likelihood equation with no closed form, and simulation via an exact Beta-Geometric mixture representation.
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About This Reference Sheet
The rich-get-richer distribution: history from Udny Yule's 1925 model of species-per-genus growth and Herbert Simon's 1955 rediscovery via word-frequency preferential attachment, a genuinely infinite-support power-law tail whose exponent is derived from the generative process itself (unlike the already-published Zipf sheet's assumed finite-support power law), moments that only exist up to a point set by ρ, a digamma-based maximum-likelihood equation with no closed form, and simulation via an exact Beta-Geometric mixture representation.
Support
x ∈ {1, 2, 3, ...}
Parameters
ρ > 0 (tail-decay parameter)