Mixed Poisson Process

Poisson Process with many possible rates $\lambda$

The rate occurs probabistically, independent of time

Definition

Let $\{N(t):t\ge 0\}$ be a Counting Process.

We say that $N(t)$ is a Mixed Poisson Process if, conditional on a positive random variable $L=\lambda$, the process $N(t)$ is a Poisson process having rate $\lambda$.

If the PDF of $L$ is $g(\lambda )$, then: $P\{N(t+s)-N(s)=n\}={\int }_0^{\infty }\frac{e^{-\lambda t}(\lambda t{)}^n}{n!}g(\lambda )d\lambda$

Properties

1. Stationary increments
2. not have Independent Increments

Property 2 is because knowing how many events occur in an interval gives information about the possible value of $L$, which affects the distribution of the number of events in any other interval.

Example

Scenario: Customers arrive at a store, but the rate varies unpredictably by day. The rate $\Lambda$ is modeled as random: $\Lambda =3$ with probability $0.4$ (slow day), $\Lambda =8$ with probability $0.6$ (busy day).

Given $\Lambda =\lambda$, arrivals follow a Poisson process with fixed rate $\lambda$.

Marginal distribution of $N(1)$ (number of arrivals in 1 hour):

$P(N(1)=n)=0.4\cdot P(\text{Po}(3)=n)+0.6\cdot P(\text{Po}(8)=n)$

For $n=5$:

$P(N(1)=5)=0.4\cdot \frac{e^{-3}{3}^5}{5!}+0.6\cdot \frac{e^{-8}{8}^5}{5!}\approx 0.4\times 0.101+0.6\times 0.092\approx 0.095$

Why independent increments fail: Observing many arrivals early suggests $\Lambda =8$ (busy day), which increases predictions for later intervals. The past informs the future via the shared $\Lambda$.

Contrast with regular Poisson: A fixed-rate Poisson process has rate $\lambda =5$ every day. The mixed Poisson captures rate uncertainty.

Common Case: Gamma Mixing

If $\Lambda \sim \text{Gamma}(\alpha ,\beta )$, then $N(t)$ marginally follows a Negative Binomial distribution. This is the Gamma-Poisson mixture — a standard result used in Bayesian inference and overdispersed count data.