User:BeyondNormality/Aitchinson distribution

The probability mass function of the Aitchinson distribution is given by



\mathrm{Aitchinson}\left(x;a,b,\theta\right) \equiv \Pr(X = x) = e^{-a} \sum _{j=0}^{-2 k+x} \sum _{k=0}^{\left[\frac{x}{2}\right]} \frac{(b x)^{-j-2 k+x} \left(-a e^{-b} (-1+\theta )\right)^k (a \theta )^j}{j! k!  (-j-2 k+x)!} $$


 * $$ x = 0, 1, 2, \dots  $$
 * $$ a, b \ge 0 $$
 * $$ 0 \le \theta \le 1 $$
 * $$ \left[z\right] = \text{integer part of z} $$

Expected Value
 * $$   \operatorname{E}[X] = -a (b (\theta -1)+\theta -2) $$

Variance
 * $$ \operatorname{Var}(X) = a (-b (b+5) (\theta -1)-3 \theta +4) $$

Moment Generating Function
 * $$ M_X(t)=\exp \left(-a \left((\theta -1) e^{b \left(e^t-1\right)+2 t}-\theta e^t+1\right)\right) $$

Characteristic Function
 * $$ \varphi_X(t)=\exp \left(-a \left((\theta -1) e^{b \left(e^{i t}-1\right)+2 i t}-\theta e^{i t}+1\right)\right) $$

Probability Generating Function
 * $$ G(t) = \exp \left(-a \left((\theta -1) t^2 e^{b (t-1)}-\theta t+1\right)\right) $$