- 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}}}}