User:BeyondNormality/Ahuja-negative binomial distribution

Also known as the distribution of the sum of independent decapitated negative binomial variables, n-fold convolution of the zero-truncated negative binomial distribution, associated Lah distribution, the probability mass function of the Ahuja-negative binomial distribution is given by

$$ \begin{align} \mathrm{AhujaNB}(x; k,n,p) \equiv \Pr(X = x) & = \sum _{r=1}^n (-1)^{n-r} q^x \binom{n}{r} \left(p^{-k}-1\right)^{-n} \binom{k r+x-1}{x} \\ & = \sum _{r=1}^n (-1)^{n-r} q^x \binom{n}{r} p^{k n} \left(1-p^k\right)^{-n} \binom{k r+x-1}{x} \end{align}$$


 * $$ x = n,n+1,n+2, \dots $$
 * $$ q=1-p $$
 * $$ N  \in \mathbb{N}  $$
 * $$ k \ne 0 $$
 * $$ 0 < p < 1 $$
 * $$ N  \in \mathbb{N}_0  $$
 * $$ k \ge 0 $$
 * $$ 0 < p \le 1 $$
 * $$ -1 \le k < 0 $$
 * $$ 0 \le p \le 1 $$

Expected Value


 * $$ \mathbb E(X) = \frac{k n q}{p \left(1-p^k\right)} $$

Variance


 * $$ \operatorname{Var}(X) = \frac{k n q \left(1-(1+ k q) p^k\right)}{p^2 \left(p^k-1\right)^2} $$

Moment Generating Function


 * $$ M_X(t)=\left(\frac{(1-q)^k \left(1-\left(1-q e^t\right)^{-k}\right)}{(1-q)^k-1}\right)^n $$

Characteristic Function


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

Probability Generating Function


 * $$ \begin{align} G(t) & = {\left(\frac{t \, _2F_1(k+1,1;2;q t)}{\, _2F_1(k+1,1;2;q)}\right)}^n \\

& = \left(\frac{1-p^k (1-q t)^{-k}}{p^k-1}\right)^n \qquad k \ne 0, 0 < p < 1 \end{align} $$