Zipf's Law Applies to frequency table of word in corpus of
word frequency ∝ 1 word rank \text{word frequency} \propto \frac{1}{\text{word rank}} word frequency ∝ word rank 1 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:
frequency ∝ 1 ( rank + b ) a ∵ a , b : fitted parameters with a ≈ 1 and b ≈ 2.7 \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} frequency ∝ ( rank + b ) a 1 ∵ a , b : fitted parameters with a ≈ 1 and b ≈ 2.7 definition
the distribution on N N N k k k
f ( k ; N ) = { 1 H N 1 k , if 1 ≤ k ≤ N , 0 , if k < 1 or N < k . ∵ H N ≡ ∑ k = 1 N 1 k . ( normalisation constant ) \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} f ( k ; N ) = { H N 1 k 1 , 0 , if 1 ≤ k ≤ N , if k < 1 or N < k . ∵ H N ≡ k = 1 ∑ N k 1 . ( normalisation constant )