Day 15: Mixed Poisson Process#

The Mixed Poisson Process (MPP) is an extension of the classical Homogeneous Poisson process that incorporates randomness into the rate parameter \(\lambda\). In a standard homogeneous Poisson process, the rate parameter \(\lambda\) is constant, and events occur independently over time or space. In contrast, a Mixed Poisson Process assumes \(\lambda\) is itself a random variable governed by a specified distribution.

Definition (on the real-line)#

A Mixed Poisson process (MPP) \(\{N(t), t\in \[0, \infty)\}\) is a counting process with counting distribution of the form:

\[P(N(t)= n) = \int\_0^{\infty} \frac{1}{n!} e^{-\lambda t} (\lambda t)^n d \Lambda(\lambda), \qquad n\in \mathbb{N},\]

where \(\Lambda\) is the structure distribution given by \(\Lambda(\lambda) = P(\Lambda \leq \lambda)\) with \(\Lambda(0)=0\). This type of distribution is known as mixed Poisson distribution which gives the name to the processes.

Note that if \(\Lambda\) has density function \(g\), we have

\[P(N(t)= n) = \int\_0^{\infty} \frac{1}{n!} e^{-\lambda t} (\lambda t)^n g(\lambda) d\lambda, \qquad n\in \mathbb{N}.\]

Thus, given \(\Lambda = \lambda\), \(N(t)\) follows a Poisson distribution with parameter \(\lambda t\), i.e.:

\[P\left(N(t)= n| \Lambda=\lambda\right) = \frac{1}{n!} e^{-\lambda t} (\lambda t)^n, \qquad n\in\mathbb{N}.\]

🔔 Random Facts 🔔#

  • Mixed Poisson processes were first introduced in 1938 by French mathematician J. Dubourdieu, who wanted to describe the number of claims, in fixed periods, occurring in sickness and accident insurance

  • If the intensity random variable \(\Lambda\) is degenerate at a constant \(\lambda\) (\(\lambda > 0\)), then we have the Homogeneous Poisson process. This is the only MPP that is simultaneously a renewal process.

  • If the intensity random variable \(\Lambda\) follows a Gamma distribution, then the resulting counting process \(N(t)\) follows a Pascal or negative Binomial distribution. This case is known as the Pascal or Polya process.

  • The Mixed Poisson Process is a specific case of a Cox Process, also known as a doubly stochastic Poisson process. The Cox process generalises the Poisson process even further by allowing the intensity/rate to vary over time according to an underlying stochastic process

More to Read 📚#

P.s. If you are curious about probability distributions visit the Advent Calendar 2023 ✨