Beta-Pascal
Notation	${\displaystyle \mathrm {Beta-Pascal} ()}$
Parameters	${\displaystyle }$
Support	${\displaystyle }$
PMF	${\displaystyle }$
CDF	${\displaystyle }$
Mean	${\displaystyle }$
Median	${\displaystyle }$
Mode	${\displaystyle }$
Variance	${\displaystyle }$
Skewness	${\displaystyle }$
Excess kurtosis	${\displaystyle }$
Entropy	${\displaystyle }$
MGF	${\displaystyle }$
CF	${\displaystyle }$
PGF	${\displaystyle }$

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

${\displaystyle {\begin{aligned}\mathrm {BetaPascal} (x;n,k,m)\equiv \Pr(X=x)&={\frac {\Gamma (k+m)\Gamma (m+n)(k)_{x}(n)_{x}}{x!\Gamma (m)\Gamma (k+m+n)(k+m+n)_{x}}}\\&={\frac {\Gamma (k+m)\Gamma (m+n)(k)_{x}(n)_{x}}{x!\Gamma (m)\Gamma (k+m+n)(k+m+n)_{x}}}\end{aligned}}}$

• ${\displaystyle x=0,1,2,\dots }$
• ${\displaystyle k,n\geq 0}$
• ${\displaystyle m>0}$

Cumulative Distribution Function

${\displaystyle F_{X}(x)=1-{\frac {\Gamma (n+x+1)B(m+n,k+x+1)\,_{3}F_{2}(1,k+x+1,n+x+1;x+2,k+m+n+x+1;1)}{\Gamma (n)\Gamma (x+2)B(m,k)}}}$

Recurrence relation

${\displaystyle x\Pr(x)(k+m+n+x-1)+(-k-x+1)(n+x-1)\Pr(x-1)=0}$

Expected Value

${\displaystyle \mathbb {E} (X)={\begin{cases}{\frac {kn}{m-1}}&m>1\\\infty &{\text{True}}\end{cases}}}$

Variance

${\displaystyle \operatorname {Var} (X)={\begin{cases}{\frac {kn(k+m-1)(m+n-1)}{(m-2)(m-1)^{2}}}&m>2\\\infty &{\text{True}}\end{cases}}}$

Moment Generating Function

${\displaystyle M_{X}(t)={\frac {\,_{2}F_{1}\left(k,n;k+m+n;e^{t}\right)}{\,_{2}F_{1}(k,n;k+m+n;1)}}}$

Characteristic Function

${\displaystyle \varphi _{X}(t)={\frac {\,_{2}F_{1}\left(k,n;k+m+n;e^{it}\right)}{\,_{2}F_{1}(k,n;k+m+n;1)}}}$

Probability Generating Function

${\displaystyle G(t)={\frac {\,_{2}F_{1}(k,n;k+m+n;t)}{\,_{2}F_{1}(k,n;k+m+n;1)}}}$

Symbol	Meaning
${\displaystyle \sim }$	${\displaystyle X\sim Y}$: the random variable X is distributed as the random variable Y
${\displaystyle \equiv }$	the distribution in the title is identical with this distribution
${\displaystyle \Leftarrow }$	the distribution in title is a special case of this distribution
${\displaystyle \Rightarrow }$	this distribution is a special case of the distribution in the title
${\displaystyle \leftarrow }$	this distribution converges to the distribution in the title
${\displaystyle \rightarrow }$	the distribution in the title converges to this distribution

Relationship	Distribution	When
${\displaystyle \equiv }$	confluent hypergeometric ${\displaystyle (a,k+m+n,k)\quad {\underset {a}{\bigwedge }}\quad f(a)={\frac {a^{n-1}\,_{1}F_{1}(m+n;k+m+n;-a)}{\Gamma (n)\,_{2}F_{1}(k,n;k+m+n;1)}}}$
${\displaystyle \equiv }$	negative binomial (k, p) ${\displaystyle \quad {\underset {p}{\bigwedge }}\quad }$ beta (m,n)	${\displaystyle m,n>0}$
${\displaystyle \equiv }$	Poisson ${\displaystyle (\lambda )\quad {\underset {\lambda }{\bigwedge }}\quad {\begin{aligned}f(\lambda )&={\frac {e^{\lambda /2}\Gamma (k+m)\lambda ^{{\frac {1}{2}}(k+n-3)}\Gamma (m+n)W_{{\frac {1}{2}}(-k-n+1)-m,{\frac {n-k}{2}}}(\lambda )}{\Gamma (k)\Gamma (m)\Gamma (n)}}\\&={\frac {\lambda ^{n-1}\Gamma (k+m)\Gamma (m+n)U(m+n,-k+n+1,\lambda )}{\Gamma (k)\Gamma (m)\Gamma (n)}}\end{aligned}}}$	${\displaystyle k,m,n>0}$
${\displaystyle \equiv }$	[Poisson (aj) ${\displaystyle {\underset {a}{\bigwedge }}}$ gamma (k,1) ] ${\displaystyle {\underset {j}{\bigwedge }}f(j)={\frac {j^{m-1}}{B(m,n){(1+j)}^{m+n}}}}$	${\displaystyle j\geq 0}$
${\displaystyle \Leftarrow }$	Holla-negative binomial ${\displaystyle (a,b,k,\beta )}$	${\displaystyle a=m\qquad b=n\qquad \beta =0}$
${\displaystyle \Leftarrow }$	inverse-Pólya ${\displaystyle (k,m,n,r)}$	${\displaystyle r=1}$
${\displaystyle \Leftarrow }$	Kemp-Dacey-hypergeometric family ${\displaystyle (I/8)}$
${\displaystyle \Leftarrow }$	Ord family ${\displaystyle (a,b_{0},b_{1},b_{2})}$	${\displaystyle a={\frac {(k-1)(n-1)}{(m+1)}}\qquad b_{0}=0\qquad b_{1}={\frac {k+m+n}{m+1}}\qquad b_{2}={\frac {1}{m+1}}}$
${\displaystyle \Rightarrow }$	deterministic (0)	${\displaystyle n=0}$
${\displaystyle \Rightarrow }$	1-shifted Feller-Shreve	${\displaystyle {\begin{cases}n=1\qquad k=m={\frac {1}{2}}\\k=1\qquad m=n={\frac {1}{2}}\\\end{cases}}}$
${\displaystyle \Rightarrow }$	1-shifted Johnson-Kotz (a)	${\displaystyle {\begin{cases}n=m=1\qquad k=a\\k=m=1\qquad n=a\\\end{cases}}}$
${\displaystyle \Rightarrow }$	Kemp family Type IV ${\displaystyle (a,b,n')}$	${\displaystyle k=-a\qquad m=a+b+1\qquad n=n'\qquad (k,m,n)>0}$
${\displaystyle \Rightarrow }$	Marlow ${\displaystyle (m',n')}$	${\displaystyle k=m'-n'+1\qquad m=n'-1\qquad n=1\qquad {k,m}\in \mathbb {N} }$
${\displaystyle \Rightarrow }$	1-shifted Miller ${\displaystyle (m')}$	${\displaystyle k=1\qquad m=m'+1\qquad n=1}$
${\displaystyle \Rightarrow }$	1-shifted Prasad ${\displaystyle (k)}$	${\displaystyle n=1\qquad m=2}$
${\displaystyle \Rightarrow }$	Salvia-Bolinger ${\displaystyle (\alpha )}$	${\displaystyle k=1\qquad m=\alpha \qquad n=1-\alpha }$
${\displaystyle \Rightarrow }$	1-shifted Schwarz-Tversky 1	${\displaystyle k=n=1\qquad m=2}$
${\displaystyle \Rightarrow }$	1-shifted Simon	${\displaystyle k=1\qquad m=1\qquad n=1}$
${\displaystyle \Rightarrow }$	Waring ${\displaystyle (b,n)}$	${\displaystyle k=1\qquad m=b}$
${\displaystyle \Rightarrow }$	Yule ${\displaystyle (m)}$	${\displaystyle k=1\qquad n=1}$
${\displaystyle \rightarrow }$	2-shifted Flory	${\displaystyle k=2\qquad m\rightarrow \infty \qquad n\rightarrow \infty \qquad {\frac {m}{m+n}}\rightarrow {\frac {1}{2}}}$
${\displaystyle \rightarrow }$	geometric (q)	${\displaystyle n=1\qquad k\rightarrow \infty \qquad m\rightarrow \infty \qquad {\frac {m}{m+n}}\rightarrow p}$
${\displaystyle \rightarrow }$	negative binomial (k,p)	${\displaystyle m\rightarrow \infty \qquad n\rightarrow \infty \qquad {\frac {m}{m+n}}\rightarrow p}$
${\displaystyle \rightarrow }$	Poisson (a)	${\displaystyle m\rightarrow \infty \qquad n\rightarrow \infty \qquad k\rightarrow \infty \qquad {\frac {n}{m+n}}\rightarrow 0\qquad {\frac {kn}{m+n}}\rightarrow a}$