User:BeyondNormality/Beta-Pascal distribution

Beta-Pascal

Notation	$\mathrm {Beta-Pascal} ()$
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

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

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

Cumulative Distribution Function

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

Recurrence relation

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

Expected Value

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

Variance

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

Moment Generating Function

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

Characteristic Function

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

Probability Generating Function

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

Interrelations

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

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