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:

${\displaystyle {\begin{aligned}L&=\sum _{n=1}^{m}b_{n}\\[4pt]b_{n}&\sim \operatorname {Bernoulli} \left({\frac {\theta }{n-1+\theta }}\right)\end{aligned}}}$

Chinese restaurant table

Parameters	${\displaystyle \theta >0}$ ${\displaystyle m\in \{0,1,2,\ldots \}}$
Support	${\displaystyle L\in \{0,1,2,\ldots ,m\}}$
pmf	${\displaystyle {\frac {\Gamma (\theta )}{\Gamma (m+\theta )}}|s(m,\ell )|\theta ^{\ell }}$
Mean	${\displaystyle \theta (\psi (\theta +m)-\psi (\theta ))}$ (see digamma function)

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

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

where s denotes Stirling numbers of the first kind.