Английская Википедия:Delaporte distribution

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Шаблон:Infobox probability distribution{\Gamma(\alpha)i!(1+\beta)^{\alpha+i}(k-i)!}</math> | cdf = <math>\sum_{j=0}^k\sum_{i=0}^j\frac{\Gamma(\alpha + i)\beta^i\lambda^{j-i}e^{-\lambda}}{\Gamma(\alpha)i!(1+\beta)^{\alpha+i}(j-i)!}</math> | mean = <math>\lambda + \alpha\beta</math> | median = | mode = <math>\begin{cases}z, z+1 & \{z \in \mathbb{Z}\}:\; z = (\alpha-1)\beta+\lambda\\ \lfloor z \rfloor & \textrm{otherwise}\end{cases}</math> | variance = <math>\lambda + \alpha\beta(1+\beta)</math> | skewness = See #Properties | kurtosis = See #Properties | entropy = | mgf = <math>\frac{e^{\lambda(e^{t}-1)}}{(1-\beta(e^{t}-1))^\alpha}</math> | cf = | pgf = | fisher = }} The Delaporte distribution is a discrete probability distribution that has received attention in actuarial science. It can be defined using the convolution of a negative binomial distribution with a Poisson distribution. 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 <math>\lambda</math> parameter, and a gamma-distributed variable component, which has the <math>\alpha</math> and <math>\beta</math> parameters. The distribution is named for Pierre Delaporte, who analyzed it in relation to automobile accident claim counts in 1959, although it appeared in a different form as early as 1934 in a paper by Rolf von Lüders, where it was called the Formel II distribution.

Properties

The skewness of the Delaporte distribution is:

<math> \frac{\lambda + \alpha\beta(1+3\beta+2\beta^2)}{\left(\lambda + \alpha\beta(1+\beta)\right)^{\frac{3}{2}}} </math>

The excess kurtosis of the distribution is:

<math> \frac{\lambda+3\lambda^2+\alpha\beta(1+6\lambda+6\lambda\beta+7\beta+12\beta^2+6\beta^3+3\alpha\beta+6\alpha\beta^2+3\alpha\beta^3)}{\left(\lambda + \alpha\beta(1+\beta)\right)^2} </math>

Further reading

• Шаблон:Cite journal