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Table 1 Useful quantities for some power series distributions

From: The TR {Y} power series family of probability distributions

DistributionanC(θ)C(θ)C′′(θ)C′ ′ ′(θ)C−1(θ)parameter space
Binomial\( \left(\genfrac{}{}{0pt}{}{m}{n}\right) \)(1 + θ)m − 1m(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)
Geometric1θ(1 − θ)−1(1 − θ)−22(1 − θ)−36(1 − θ)−4θ(1 + θ)−1θ (0, 1)
Logarithmicn-1− log(1 − θ)(1 − θ)−1(1 − θ)−22(1 − θ)−31 − eθθ (0, 1)
Poisson(n!)-1eθ-1eθeθeθlog(θ+1)θ (0, ∞)
  1. Source: Morais and Barreto-Souza [22]
  2. Note: In the table, m is the number of trials or replicas in the binomial experiment