Английская Википедия:Displaced Poisson distribution

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

In statistics, the displaced Poisson, also known as the hyper-Poisson distribution, is a generalization of the Poisson distribution.

Definitions

Probability mass function

The probability mass function is

<math>

P(X=n) = \begin{cases}

   e^{-\lambda}\dfrac{\lambda^{n+r}}{\left(n+r\right)!}\cdot\dfrac{1}{I\left(r,  \lambda\right)}, \quad n=0,1,2,\ldots &\text{if } r\geq 0\\[10pt]
   e^{-\lambda}\dfrac{\lambda^{n+r}}{\left(n+r\right)!}\cdot\dfrac{1}{I\left(r+s,\lambda\right)},\quad n=s,s+1,s+2,\ldots &\text{otherwise}
 \end{cases}

</math>

where <math>\lambda>0</math> and r is a new parameter; the Poisson distribution is recovered at r = 0. Here <math>I\left(r,\lambda\right)</math> is the Pearson's incomplete gamma function:

<math>

I(r,\lambda)=\sum^\infty_{y=r}\frac{e^{-\lambda} \lambda^y}{y!}, </math> where s is the integral part of r. The motivation given by Staff is that the ratio of successive probabilities in the Poisson distribution (that is <math>P(X=n)/P(X=n-1)</math>) is given by <math>\lambda/n</math> for <math>n>0</math> and the displaced Poisson generalizes this ratio to <math>\lambda/\left(n+r\right)</math>.

Examples

One of the limitations of the Poisson distribution is that it assumes equidispersion – the mean and variance of the variable are equal. The displaced Poisson distribution may be useful to model underdispersed or overdispersed data, such as:

• the distribution of insect populations in crop fields;
• the number of flowers on plants;
• motor vehicle crash counts; and
• word or sentence lengths in writing.

Properties

Descriptive Statistics

• For a displaced Poisson-distributed random variable, the mean is equal to <math>\lambda - r</math> and the variance is equal to <math>\lambda</math>.
• The mode of a displaced Poisson-distributed random variable are the integer values bounded by <math>\lambda - r - 1</math> and <math>\lambda - r</math> when <math>\lambda \geq r+1</math>. When <math>\lambda < r+1</math>, there is a single mode at <math>x=0</math>.
• The first cumulant <math>\kappa_{1}</math> is equal to <math>\lambda - r</math> and all subsequent cumulants <math>\kappa_{n}, n \geq 2</math> are equal to <math>\lambda</math>.

References

Шаблон:Stat-stub