Chinese restaurant table distribution

In probability theory and statistics, the Chinese restaurant table distribution (CRT) is the distribution on the number of tables in the Chinese restaurant process.^[1] It can be understood as the sum of n independent random variables, each with a different Bernoulli distribution:

${\displaystyle \begin{array}{rl}L& ={\displaystyle \sum _{n=1}^m}{b}_n\\ b_n& \sim \mathrm{Bernoulli}\left(\frac{\theta }{n-1+\theta }\right)\end{array}}$

The probability mass function of L is given by ^[2]

${\displaystyle f(\ell )=\frac{\mathrm{\Gamma }(\theta )}{\mathrm{\Gamma }(m+\theta )}|s(m,\ell )|{\theta }^{\ell }}$

where s denotes Stirling numbers of the first kind.

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