Zipf's law

Key facts

Parameters

s ≥ 0 {\displaystyle s\geq 0\,} ( real ) N ∈ { 1 , 2 , 3 … } {\displaystyle N\in \{1,2,3\ldots \}} ( integer )

Support

k ∈ { 1 , 2 , … , N } {\displaystyle k\in \{1,2,\ldots ,N\}}

Pmf

1 / k s H N , s {\displaystyle {\frac {1/k^{s}}{H_{N,s}}}} where H N,s is the N th generalized harmonic number

Cdf

H k , s H N , s {\displaystyle {\frac {H_{k,s}}{H_{N,s}}}}

Mean

H N , s − 1 H N , s {\displaystyle {\frac {H_{N,s-1}}{H_{N,s}}}}

Mode

1 {\displaystyle 1\,}

Variance

H N , s − 2 H N , s − H N , s − 1 2 H N , s 2 {\displaystyle {\frac {H_{N,s-2}}{H_{N,s}}}-{\frac {H_{N,s-1}^{2}}{H_{N,s}^{2}}}}

Entropy

s H N , s ∑ k = 1 N ln ⁡ ( k ) k s + ln ⁡ ( H N , s ) {\displaystyle {\frac {s}{H_{N,s}}}\sum \limits _{k=1}^{N}{\frac {\ln(k)}{k^{s}}}+\ln(H_{N,s})}

Mgf

1 H N , s ∑ n = 1 N e n t n s {\displaystyle {\frac {1}{H_{N,s}}}\sum \limits _{n=1}^{N}{\frac {e^{nt}}{n^{s}}}}

Cf

1 H N , s ∑ n = 1 N e i n t n s {\displaystyle {\frac {1}{H_{N,s}}}\sum \limits _{n=1}^{N}{\frac {e^{int}}{n^{s}}}}

via Wikipedia infobox