Table 1 Useful quantities for some power series distributions

Distribution

a_n

C(θ)

C^′(θ)

C^′′(θ)

C^{′ ′ ′}(θ)

C⁻¹(θ)

parameter space

Binomial

\( \left(\genfrac{}{}{0pt}{}{m}{n}\right) \)

(1 + θ)^m − 1

m(1 + θ)^m − 1

\( \frac{m\left(m-1\right)}{{\left(1+\theta \right)}^{2-m}} \)

\( \frac{m\left(m-1\right)\left(m-2\right)}{{\left(1+\theta \right)}^{3-m}} \)

\( {\left(\theta -1\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$m$}\right.}-1 \)

θ ∈ (0, 1)

Geometric

1

θ(1 − θ)⁻¹

(1 − θ)⁻²

2(1 − θ)⁻³

6(1 − θ)⁻⁴

θ(1 + θ)⁻¹

θ ∈ (0, 1)

Logarithmic

n^-1

− log(1 − θ)

(1 − θ)⁻¹

(1 − θ)⁻²

2(1 − θ)⁻³

1 − e^−θ

θ ∈ (0, 1)

Poisson

(n!)^-1

e^θ-1

e^θ

e^θ

e^θ

log(θ+1)

θ ∈ (0, ∞)