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The Pareto distribution is a continuous probability distribution. [[Zipf's law]], also sometimes called the [[zeta distribution]], is a discrete distribution, separating the values into a simple ranking. Both are a simple power law with a negative exponent, scaled so that their cumulative distributions equal 1.  Zipf's can be derived from the Pareto distribution if the <math>x</math> values (incomes) are binned into <math>N</math> ranks so that the number of people in each bin follows a 1/rank pattern. The distribution is normalized by defining <math>x_m</math> so that <math>\alpha x_\mathrm{m}^\alpha = \frac{1}{H(N,\alpha-1)}</math> where <math>H(N,\alpha-1)</math> is the [[Harmonic number#Generalized harmonic numbers|generalized harmonic number]]. This makes Zipf's probability density function derivable from Pareto's.

The Pareto distribution is a continuous probability distribution. [[Zipf's law]], also sometimes called the [[zeta distribution]], is a discrete distribution, separating the values into a simple ranking. Both are a simple power law with a negative exponent, scaled so that their cumulative distributions equal 1.  Zipf's can be derived from the Pareto distribution if the <math>x</math> values (incomes) are binned into <math>N</math> ranks so that the number of people in each bin follows a 1/rank pattern. The distribution is normalized by defining <math>x_m</math> so that <math>\alpha x_\mathrm{m}^\alpha = \frac{1}{H(N,\alpha-1)}</math> where <math>H(N,\alpha-1)</math> is the [[Harmonic number#Generalized harmonic numbers|generalized harmonic number]]. This makes Zipf's probability density function derivable from Pareto's.

帕累托分布是一个连续的概率分布。[[Zipf定律]]，有时也称为[[zeta分布]]，是一个离散分布，将值分成一个简单的排名。它们都是一个简单的幂律，具有负指数，缩放后它们的累积分布等于1。如果将<math>x</math>值（收入）组合到<math>N</math>等级中，那么Zipf可以从Pareto分布中得出，这样每个箱子中的人数遵循1/rank模式。通过定义<math>xum</math>使<math>\alpha x\\mathrm{m}\alpha=\frac{1}{H（N，alpha-1）}</math>其中<math>H（N，alpha-1）</math>是[[调和数#广义调和数|广义调和数]]。齐夫函数的概率可由齐夫函数导出。

帕累托分布是一个连续的概率分布。[[Zipf定律]]，有时也称为[[zeta分布]]，是一个离散分布，将值分成一个简单的排名。它们都是一个简单的幂律，具有负指数，缩放后它们的累积分布等于1。如果将<math>x</math>值（收入）组合到<math>N</math>等级中，那么Zipf可以从Pareto分布中得出，这样每个箱子中的人数遵循1/rank模式。通过定义<math>xum</math>使<math>\alpha x\\mathrm{m}\alpha=\frac{1}{H（N，alpha-1）}</math>其中<math>H（N，alpha-1）</math>是[[调和数#广义调和数|广义调和数]]。这使得Zipf的概率密度函数可以从Pareto得到。

: <math>f(x) = \frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}} = \frac{1}{x^s H(N,s)}</math>

: <math>f(x) = \frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}} = \frac{1}{x^s H(N,s)}</math>