Zipf Distribution
X ~ Zipf(N, s)
Power-Law / Rank-Frequency Family • Published July 20, 2026
The rank-frequency power-law distribution behind Zipf's law: history from Pareto, Auerbach, and Estoup's independent early observations through George Zipf's 1949 'least effort' account, the infinite-support zeta-distribution limit, why the mode is always rank 1, the exponential-family structure that makes Fisher information exactly the variance of log-rank under the model itself, a maximum-likelihood estimator verified two independent ways, and simulation via inverse-transform sampling on the discrete CDF.
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About This Reference Sheet
The rank-frequency power-law distribution behind Zipf's law: history from Pareto, Auerbach, and Estoup's independent early observations through George Zipf's 1949 'least effort' account, the infinite-support zeta-distribution limit, why the mode is always rank 1, the exponential-family structure that makes Fisher information exactly the variance of log-rank under the model itself, a maximum-likelihood estimator verified two independent ways, and simulation via inverse-transform sampling on the discrete CDF.
Support
x ∈ {1, 2, ..., N}
Parameters
N ∈ ℕ (ranked items), s > 0 (exponent)