User:BeyondNormality/Beta-Pascal distribution

Also known as the beta-negative binomial distribution, generalized Waring distribution, inverse Markov-Pólya distribution, Kemp and Kemp's Type IV distribution, negative binomial beta distribution, Ord distribution Type VI, Pascal beta distribution, the probability mass function of the Beta-Pascal distribution is given by

$$ \begin{align} \mathrm{BetaPascal}(x;n,k,m) \equiv \Pr(X = x) & = \frac{\Gamma (k+m) \Gamma (m+n) (k)_x (n)_x}{x! \Gamma (m) \Gamma (k+m+n) (k+m+n)_x} \\ & = \frac{\Gamma (k+m) \Gamma (m+n) (k)_x (n)_x}{x! \Gamma (m) \Gamma (k+m+n) (k+m+n)_x} \end{align}$$
 * $$ x = 0,1,2, \dots $$
 * $$ k, n \ge 0 $$
 * $$ m > 0 $$

Cumulative Distribution Function

F_X(x) = 1-\frac{\Gamma (n+x+1) B(m+n,k+x+1) \, _3F_2(1,k+x+1,n+x+1;x+2,k+m+n+x+1;1)}{\Gamma (n) \Gamma (x+2) B(m,k)} $$

Recurrence relation
 * $$ x \Pr (x) (k+m+n+x-1)+(-k-x+1) (n+x-1) \Pr (x-1)=0 $$

Expected Value
 * $$ \mathbb E(X) = \begin{cases}

\frac{k n}{m-1} & m>1 \\ \infty & \text{True} \end{cases} $$

Variance
 * $$ \operatorname{Var}(X) = \begin{cases}

\frac{k n (k+m-1) (m+n-1)}{(m-2) (m-1)^2} & m>2 \\ \infty & \text{True} \end{cases} $$

Moment Generating Function
 * $$ M_X(t)=\frac{\, _2F_1\left(k,n;k+m+n;e^t\right)}{\, _2F_1(k,n;k+m+n;1)} $$

Characteristic Function
 * $$ \varphi_X(t)=\frac{\, _2F_1\left(k,n;k+m+n;e^{i t}\right)}{\, _2F_1(k,n;k+m+n;1)} $$

Probability Generating Function

G(t) = \frac{\, _2F_1(k,n;k+m+n;t)}{\, _2F_1(k,n;k+m+n;1)} $$