Table 1.

Useful quantities of some power series distributions

Distribution	$a_n$	$C(\theta )$	$C^{\prime }(\theta )$	$C^{\prime \prime }(\theta )$	${C(\theta )}^{-1}$	$\mathrm{\Theta }$
Poisson	$n{!}^{-1}$	$e^{\theta }-1$	$e^{\theta }$	$e^{\theta }$	$log(\theta +1)$	$\theta \in (0,\infty )$
Logarithmic	$n^{-1}$	$-log(1-\theta )$	${(1-\theta )}^{-1}$	${(1-\theta )}^{-2}$	$1-e^{-\theta }$	$\theta \in (0,1)$
Geometric	1	$\theta {(1-\theta )}^{-1}$	${(1-\theta )}^{-2}$	2${(1-\theta )}^{-3}$	${\theta (\theta +1)}^{-1}$	$\theta \in (0,1)$
Binomial	$\left(\genfrac{}{}{0pt}{}{m}{n}\right)$	${(\theta +1)}^m-1$	$m{(\theta +1)}^{m-1}$	$\frac{m(m-1)}{{(\theta +1)}^{2-m}}$	${(\theta -1)}^{\frac{1}{m}}-1$	$\theta \in (0,1)$