User:BeyondNormality/Abakuks distribution

The probability mass function of the Abakuks distribution is



\mathrm{Abakuks}(x; a,r,n) \equiv \Pr(X = x) = \frac{1}{x (a n)^{x} (n-x)! T}  $$


 * $$ x = r-1,r,r+1,...,n $$
 * $$ n \in \mathbb{N}^+ - \{ 1 \}  $$
 * $$ r \in \{ 2, 3, ,..., n\} $$
 * $$ a > 0 $$
 * $$ T = \sum_{j \mathop =r-1}^n \frac{1}{j (a n)^{j} (n-j)!}  $$

Recurrence relation


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

Expected Value


 * $$   \operatorname{E}[X] = \frac{a n \left(\frac{1}{a n}\right)^r \, _2F_0\left(1,-n+r-1;;-\frac{1}{a n}\right)}{T \Gamma (n-r+2)} $$

Moment Generating Function


 * $$ M_X(t)=\frac{\Gamma (r-1) \left(\frac{e^t}{a n}\right)^{r-1} \, _3\tilde{F}_1\left(1,r-1,-n+r-1;r;-\frac{e^t}{a n}\right)}{T \Gamma (n-r+2)} $$

Characteristic Function


 * $$ \varphi_X(t)=\frac{\Gamma (r-1) \left(\frac{e^{i t}}{a n}\right)^{r-1} \, _3\tilde{F}_1\left(1,r-1,-n+r-1;r;-\frac{e^{i t}}{a n}\right)}{T \Gamma (n-r+2)} $$

Probability Generating Function


 * $$ G(t) = \frac{\left(\frac{t}{a n}\right)^{r-1} \, _3F_1\left(1,r-1,-n+r-1;r;-\frac{t}{a n}\right)}{(r-1) T (n-r+1)!} $$