User:BeyondNormality/Additive binomial distribution

Also known as the additive generalization of the binomial distribution and correlated binomial distribution, the probability mass function of the additive binomial distribution is given by



\mathrm{AdditiveBinomial}(x; a,n,p) \equiv \Pr(X = x) = \binom{n}{x} p^x q^{n-x} \left(1+ \frac{a}{2}  \left( \frac{(x-1) x}{p} + \frac{(n-x) (n-x-1)}{q}-n (n-1) \right)\right) $$


 * $$ x = 0, 1, 2, \dots ,n $$
 * $$ n \in \mathbb{N}_0  $$
 * $$ 0 < p < 1 $$
 * $$ q = 1 - p $$
 * $$ \begin{cases}

-\min \left( \frac{p}{q}, \frac{q}{p} \right) \le a \le 1 & n = 2, \\ \frac{-2}{n(n-1)} \min \left( \frac{p}{q}, \frac{q}{p} \right) \le a \le 2 \left(n+\frac{(q-p)^2}{4 p q}\right)^{-1} & n \in \mathbb{N}-\{1,2\} \end{cases} $$

Expected Value

$$ \mathbb E(X) = n p $$

Variance

$$ \operatorname{Var}(X) = \frac{a n (1-p)^n \, _4F_3\left(2,2,2,1-n;1,1,1;\frac{p}{p-1}\right)+n (p-1) \left(a ((n-1) p (p (n ((n-5) p+6)+4 (p-2))+5)+1)-2 (p-1)^2 p\right)}{2 (p-1)^2} $$

Recurrence Relation

$$ \left\{p (x-n) \left(a n^2 p^2-a n p^2-2 a n p x-2 a n p+2 a p x+2 a p+a x^2+a x-2 p^2+2 p\right) \Pr (x) -(p-1) (x+1) \left(a n^2 p^2-a n p^2-2 a n p x+2 a p x+a  x^2-a x-2 p^2+2 p\right) \Pr (x+1)  =0\right\} $$

Moment Generating Function


 * $$ M_X(t)=\left(p e^t+q\right)^{n-2} \left(\frac{1}{2} a (n-1) n p q \left(e^t-1\right)^2+\left(p e^t+q\right)^2\right) $$

Characteristic Function


 * $$ \varphi_X(t)=\left(p e^{i t}+q\right)^{n-2} \left(\frac{1}{2} a (n-1) n p q \left(e^{i t}-1\right)^2+\left(p e^{i t}+q\right)^2\right) $$

Probability Generating Function


 * $$ G(t) = (q+p t)^{n-2} \left((q+p t)^2+\frac{1}{2} p q (1-t)^2 (a n (n-1))\right) $$