Delaporte distribution

{{Infobox probability distribution

| name = Delaporte

| type = discrete

| pdf_image = File:DelaportePMF.svg
When $\backslash alpha$ and $\backslash beta$ are 0, the distribution is the Poisson.
When $\backslash lambda$ is 0, the distribution is the negative binomial.

| cdf_image = File:DelaporteCDF.svg
When $\backslash alpha$ and $\backslash beta$ are 0, the distribution is the Poisson.
When $\backslash lambda$ is 0, the distribution is the negative binomial.

| notation =

| parameters = $\backslash lambda\; >\; 0$ (fixed mean)

$\backslash alpha,\; \backslash beta\; >\; 0$ (parameters of variable mean)

| support = $k\; \backslash in\; \backslash \{0,\; 1,\; 2,\; \backslash ldots\backslash \}$

| pdf = $\backslash sum\_\{i=0\}^k\backslash frac\{\backslash Gamma(\backslash alpha\; +\; i)\backslash beta^i\backslash lambda^\{k-i\}e^\{-\backslash lambda\}\}\{\backslash Gamma(\backslash alpha)i!(1+\backslash beta)^\{\backslash alpha+i\}(k-i)!\}$

| cdf = $\backslash sum\_\{j=0\}^k\backslash sum\_\{i=0\}^j\backslash frac\{\backslash Gamma(\backslash alpha\; +\; i)\backslash beta^i\backslash lambda^\{j-i\}e^\{-\backslash lambda\}\}\{\backslash Gamma(\backslash alpha)i!(1+\backslash beta)^\{\backslash alpha+i\}(j-i)!\}$

| mean = $\backslash lambda\; +\; \backslash alpha\backslash beta$

| median =

| mode =

| variance = $\backslash lambda\; +\; \backslash alpha\backslash beta(1+\backslash beta)$

| skewness = See #Properties

| kurtosis = See #Properties

| entropy =

| mgf = $\backslash frac\{e^\{\backslash lambda(e^\{t\}-1)\}\}\{(1-\backslash beta(e^\{t\}-1))^\backslash alpha\}$

| cf =

| pgf =

| fisher = }}

The Delaporte distribution is a discrete probability distribution that has received attention in actuarial science.{{cite encyclopedia

| last = Panjer

| author-link = Harry Panjer

| first = Harry H.

| editor1-last = Teugels

| editor1-first = Jozef L.

| editor2-first = Bjørn

| editor2-last = Sundt

| encyclopedia = Encyclopedia of Actuarial Science

| title = Discrete Parametric Distributions

| year = 2006

| publisher = John Wiley & Sons

| isbn = 978-0-470-01250-5

| doi = 10.1002/9780470012505.tad027

}}

It can be defined using the convolution of a negative binomial distribution with a Poisson distribution.{{cite book

| last1 = Johnson

| first1 = Norman Lloyd

| author1-link = Norman Lloyd Johnson

| last2 = Kemp

| first2 = Adrienne W.

| last3 = Kotz

| first3 = Samuel

| author3-link = Samuel Kotz

| title = Univariate discrete distributions

| edition = Third

| year = 2005

| publisher = John Wiley & Sons

| isbn = 978-0-471-27246-5

| pages = 241–242

}}

Just as the negative binomial distribution can be viewed as a Poisson distribution where the mean parameter is itself a random variable with a gamma distribution, the Delaporte distribution can be viewed as a compound distribution based on a Poisson distribution, where there are two components to the mean parameter: a fixed component, which has the $\backslash lambda$ parameter, and a gamma-distributed variable component, which has the $\backslash alpha$ and $\backslash beta$ parameters.{{cite book

| last1 = Vose

| first1 = David

| title = Risk analysis: a quantitative guide

| edition = Third, illustrated

| year = 2008

| publisher = John Wiley & Sons

| isbn = 978-0-470-51284-5

| lccn = 2007041696

| pages = 618–619

}}

The distribution is named for Pierre Delaporte, who analyzed it in relation to automobile accident claim counts in 1959,{{cite journal

| last1 = Delaporte

| first1 = Pierre J.

| year = 1960

| title = Quelques problèmes de statistiques mathématiques poses par l'Assurance Automobile et le Bonus pour non sinistre

|trans-title= Some problems of mathematical statistics as related to automobile insurance and no-claims bonus

| journal = Bulletin Trimestriel de l'Institut des Actuaires Français

| volume = 227

| pages = 87–102

| language = French

}}

although it appeared in a different form as early as 1934 in a paper by Rolf von Lüders,{{cite journal

| last1 = von Lüders

| first1 = Rolf

| year = 1934

| title = Die Statistik der seltenen Ereignisse

|trans-title= The statistics of rare events

| journal = Biometrika

| volume = 26

| issue = 1–2

| pages = 108–128

| language = German

| doi=10.1093/biomet/26.1-2.108

| jstor=2332055

}}

where it was called the Formel II distribution.