User:BeyondNormality/Banach distribution

Banach

Notation	$\mathrm {Banach} (n)$
Parameters	${\displaystyle }$
Support	$x=0,1,2,\dots ,n$
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 Banach distribution is

$\mathrm {Banach} (x;n)\equiv \Pr(X=x)=2^{x-2n}{\binom {2n-x}{n}}$

• $x=0,1,\dots ,n$
• $n\in \mathbb {N} _{0}$

Recurrence Relation

$2(n-x)\Pr(x)+(x-2n)\Pr(x+1)=0$

Cumulative Distribution Function

$F_{X}(x)=4^{-n}\left({\binom {2n}{n}}\,_{2}F_{1}(1,-n;-2n;2)-2^{x+1}{\binom {2n-x-1}{n}}\,_{2}F_{1}(1,-n+x+1;-2n+x+1;2)\right)$

Expected Value

$\mathbb {E} (X)=4^{-n}(2n+1){\binom {2n}{n}}-1$

Moment Generating Function

$M_{X}(t)=4^{-n}{\binom {2n}{n}}\,_{2}F_{1}\left(1,-n;-2n;2e^{t}\right)$

Characteristic Function

$\varphi _{X}(t)=4^{-n}{\binom {2n}{n}}\,_{2}F_{1}\left(1,-n;-2n;2e^{it}\right)$

Probability Generating Function

$G(t)=4^{-n}{\binom {2n}{n}}\,_{2}F_{1}(1,-n;-2n;2t)$

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
$\Leftarrow$	Kapur-hypergeometric family $\left(a_{i},b_{j},\theta ,k,r\right)$	$k=r=1\qquad \theta =2$
$\Leftarrow$	Kemp-Dacey-hypergeometric family (II/10)
$\Leftarrow$	McGregor-Banach $(n,p)$	$p=1/2$
$\Leftarrow$	reversed form of Uppuluri-Blot $(m,p)$	$m=n+1\qquad p=1/2$
$\Rightarrow$	deterministic $(0)$	$n=0$

References

• Blackwell, D., Hodges, J.J., Jr. (1957). Design for the control of selection bias. The Annals of Mathematical Statistics 28, 449-460.
• Dacey, M.F. (1972). Some properties of the link magnitude for channel networks and network patterns. Water Resources Research 8, 1106-1111.
• Feller, W. (1962) An Introduction to Probability Theory and its Applications. Vol. I. New York: Wiley
• Holst, L. (1989). A note on Banach's match box problem. Statistics & Probability Letters 8, 441-443.
• Kaucký, J. (1962). Note on the Banach's match-box problem. Matematicko-fyzikálny časopis SAV 1962, 28-35
• Kemp, A.W. (1968). A wide class of discrete distributions and the associated differential equations. Sankhyä A, 30, 401-410.
• Kotz, S., Johnson, N.L. (1982) Errors in inspection and grading: distributional aspects of screening and hierarchical screening. Communications in Statistics - Theory and Methods 11, 1997-2016.
• Moran, P.A.P. (1968). An Introduction to Probability Theory. Oxford; Clarendon
• Riordan, J. (1968). Combinatorial Identities. New York: Wiley
• Uppuluri, V.R.R., Blot, W.J. (1974). Asymptotic properties of the number of replications of a paired comparison. J. of Applied Probability 11, 43-52.
• Wimmer, G., Altmann. (1999). Thesaurus of univariate discrete probability distributions. Stamm; 1. ed (1999), pg 12