User:BeyondNormality/Ball-Sansom distribution

Ball-Sansom

Notation	$\mathrm {BallSansom} (b,L,N,\alpha )$
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	${\displaystyle }$

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

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

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

Cumulative Distribution Function

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

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

Recurrence Relation

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

Expected Value

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

Moment Generating Function

$M_{X}(t)=ce^{t}\left({\frac {\left(e^{t}\right)^{N+1}\left(be^{t}+1\right)^{N}}{L}}+\left(\alpha be^{t}+1\right)^{N}\right)$

Characteristic Function

$\varphi _{X}(t)=ce^{it}\left({\frac {\left(e^{it}\right)^{N+1}\left(be^{it}+1\right)^{N}}{L}}+\left(\alpha be^{it}+1\right)^{N}\right)$

Probability Generating Function

$G(t)=ct\left({\frac {t^{N+1}(bt+1)^{N}}{L}}+(\alpha bt+1)^{N}\right)$

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
$\Rightarrow$	1-displaced zero-N $\left(N+1,{\frac {1}{L+1}}\right)$	$\alpha b=0$
$\Rightarrow$	1-displaced zero-one $\left({\frac {1}{L+1}}\right)$	$N=0$

References

• Ball, F., Sansom, M. (1988). Aggregated Markov processes incorporating time interval omission. Advances in Applied Probability 20, 546-572.
• Wimmer, G., Altmann. (1999). Thesaurus of univariate discrete probability distributions. Stamm; 1. ed (1999), pg 12