User:BeyondNormality/Ball-Sansom distribution

The probability mass function of the Ball-Sansom distribution is given by

$$    \mathrm{BallSansom}(b,L,N,\alpha) \equiv \Pr(X = x) = \begin{cases} c (\alpha b)^{x-1} \binom{N}{x-1} & x= 1, 2, \dots,  N+1, \\ \frac{1}{L} c b^{-N+x-2} \binom{N}{-N+x-2} & x= N+2, N+3, \dots, 2N+2, \\ \end{cases} $$


 * $$ x = 1, 2, 3, \dots, 2N+2 $$
 * $$ \alpha, b \ge 0 $$
 * $$ L > 0 $$
 * $$ N \in \mathbb{N}_0  $$
 * $$ c = {\left((1+\alpha b )^N + \frac{(1+b)^N}{L}\right)}^{-1}  $$

Cumulative Distribution Function

F_X(x) = \begin{cases} c \left((\alpha b+1)^N-(\alpha  b)^x \binom{N}{x} \, _2F_1(1,x-N;x+1;-b \alpha )\right) & x= 1, 2, \dots,  N+1, \\ c \left(\frac{(b+1)^N-b^{-N+x-1} \binom{N}{-N+x-1} \, _2F_1(1,-2 N+x-1;x-N;-b)}{L}+(\alpha b+1)^N\right) & x= N+2, N+3, \dots,  2N+2, \\ \end{cases} $$



F_X(x) = \begin{cases} \frac{L \left((\alpha b+1)^N-(\alpha  b)^x \binom{N}{x} \, _2F_1(1,x-N;x+1;-b \alpha )\right)}{L (\alpha  b+1)^N+(b+1)^N} & x= 1, 2, \dots,  N+1, \\ 1-\frac{b^{-N+x-1} \binom{N}{-N+x-1} \, _2F_1(1,-2 N+x-1;x-N;-b)}{L (\alpha b+1)^N+(b+1)^N} & x= N+2, N+3, \dots,  2N+2, \\ \end{cases} $$

Recurrence Relation

\begin{cases} x \Pr (x+1)-\alpha b (N-x+1) \Pr (x)=0 & x= 1, 2, \dots,  N+1, \\ \Pr (x) (-2 b N+b x-2 b)+(-N+x-1) \Pr (x+1)=0 & x= N+2, N+3, \dots, 2N+2, \\ \end{cases} $$

Expected Value
 * $$ \mathbb E(X) = \frac{L (\alpha b+1)^{N-1} (\alpha  b (N+1)+1)+(2 b (N+1)+N+2) (b+1)^{N-1}}{L (\alpha  b+1)^N+(b+1)^N} $$

Moment Generating Function
 * $$ M_X(t)=c e^t \left(\frac{\left(e^t\right)^{N+1} \left(b e^t+1\right)^N}{L}+\left(\alpha b e^t+1\right)^N\right) $$

Characteristic Function
 * $$ \varphi_X(t)=c e^{i t} \left(\frac{\left(e^{i t}\right)^{N+1} \left(b e^{i t}+1\right)^N}{L}+\left(\alpha b e^{i t}+1\right)^N\right) $$

Probability Generating Function
 * $$ G(t) = c t \left(\frac{t^{N+1} (b t+1)^N}{L}+(\alpha b t+1)^N\right) $$