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Zipf's Law

Applies to frequency table of word in corpus of language:

\text{word frequency} \propto \frac{1}{\text{word rank}}

Empirically:

the most common word occurs approximately twice as often as the next common one, three times as often as the third most common, and so on.

also known in Zipf-Mandelbrot’s law:

\begin{aligned} \text{frequency} &\propto \frac{1}{(\text{rank} + b)^a} \\[8pt] &\because a,b: \text{fitted parameters with } a \approx 1 \text{ and } b \approx 2.7 \end{aligned}

definition

Zipf distribution

the distribution on $N$ elements assign to element of rank $k$ (counting from 1) the probability:
$\begin{aligned} f(k;N) &= \begin{cases} \frac{1}{H_N} \frac{1}{k}, & \text{if } 1 \leq k \leq N, \\ 0, & \text{if } k < 1 \text{ or } N < k. \end{cases} \\[12pt] &\because H_N \equiv \sum_{k=1}^{N} \frac{1}{k}. (\text{normalisation constant}) \end{aligned}$