User:BeyondNormality/Bartlett distribution

Barlett

Notation	$\mathrm {Barlett} (a,q)$
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	${\displaystyle }$

The probability mass function of the Barlett distribution is given by

${\begin{aligned}\mathrm {Barlett} (x;a,q)\equiv \Pr(X=x)&=\sum _{j=0}^{x}{\frac {a^{-j+x}e^{-a}q^{j}p}{(-j+x)!}}\\&={\frac {pe^{\frac {ap}{q}}q^{x}\Gamma \left(x+1,{\frac {a}{q}}\right)}{x!}}\end{aligned}}$

• $x=0,1,2,\dots$
• $a\geq 0$
• $0<p\leq 1$
• $q=1-p$

Expected Value

$\operatorname {E} [X]=a+{\frac {1}{p}}-1$

Variance

$\operatorname {Var} (X)=a+{\frac {q}{p^{2}}}$

Moment Generating Function

$M_{X}(t)={\frac {pe^{a\left(e^{t}-1\right)}}{1-qe^{t}}}$

Characteristic Function

$\varphi _{X}(t)={\frac {pe^{a\left(e^{it}-1\right)}}{1-qe^{it}}}$

Probability Generating Function

$G(t)={\frac {pe^{a(t-1)}}{1-qt}}$

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
$\equiv$	Poisson $(a)*$ geometric $(q)$
$\equiv$	Poisson $(a-\log(p))\vee$ generalized Poisson family $(a',H(.))$	$a'=a-\log(p)\qquad H(t)={\frac {at-\log(1-qt)}{a-\log(p)}}$
$\Leftarrow$	Charlier $(a',b,m,\beta )$	$a'=m=1\qquad b=a\qquad \beta =q$
$\Leftarrow$	Lüders $(a',b,c)$	$a'-{\frac {b}{c}}=a\qquad c=q\qquad b=q^{2}$
$\Leftarrow$	Poisson-negative binomial convolution $(a,k,p)$	$k=1$
$\leftarrow$	multiple Poisson $(n,a_{i})$	$a_{1}=a+q\qquad a_{i}={\frac {q^{i}}{i}}\qquad n\rightarrow \infty$
$\Rightarrow$	geometric $(q)$	$a=0$
$\Rightarrow$	Poisson $(q)$	$p=1$

References

• Bartlett, M.S. (1969). Distributions associated with cell populations. Biometrika 56, 391-400.
• Berg, S., Jaworski, J. (1988). Modified binomial and Poisson distributions with applications in random mapping theory. J. of Statistical Planning and Inference 18, 313-322.
• Samaniego, F.J. (1976). A characterization of convoluted Poisson distributions with applications to estimation. J. of the American Statistical Association 71, 475-479.
• Wimmer, G., Altmann. (1996a). The multiple Poisson distribution, its characteristics and a variety of forms. Biometrical J. 8, 995-1011
• Wimmer, G., Altmann. (1999). Thesaurus of univariate discrete probability distributions. Stamm; 1. ed (1999), pg 14