Inhomogenous Poisson Process in the Time Domain

Inhomogenous Poisson Process in the Time Domain

In an Inhomogeneous Poisson Process we define the Poisson Rate as

$$\Lambda(t_0,t) = \int_{t_0}^t\lambda(u)du$$

where $\lambda(u)$ is a locally integrable positive function.

We define $N(t)$ as the number of events in $(\ 0,t\ ]$, the Poisson probability mass function extended to the inhomogeneous case states that the probability of observing exactly $k$ event in the time interval $(t_0,t]$ is equal to:

$$Pr(N(t)-N(t_0) = k) \\ = \frac{\Lambda(t_0,t)^k}{k!}e^{-\Lambda(t_0,t)}\\ = \frac{\left[\int_{t_0}^{t}\lambda(u)du\right]^k}{k!}e^{-\int_{t_0}^{t}\lambda(u)du}$$

In order to build the inter-arrival intervals pdf we want to define the probability that an event occur in a given time interval $(t_0,t]$.

We firstly define the probability that the next event do not occur in the time interval $(t_0,t]$, i.e. the next event arrival time $t_{\text{next}}$ is greater than $t$ :

$$Pr(t_{\text{next}}> t)\\ =Pr(\ (N(t)-N(t_0)\ ) = 0)\\ = \frac{\left[{\int_{t_0}^{t}}\lambda(u|H_u)du\right]^0}{0!}e^{-\int_{t_0}^{t}\lambda(u)du}\\ = e^{-\int_{t_0}^{t}\lambda(u)du}$$

From here we know that

$$Pr(t_0 < t_{\text{next}} < t) = 1-e^{-\int_{t_0}^{t}\lambda(u)du}$$

which represents the cumulative distribution function (CDF) up to time $t$, the probability density function $p(t)$ of the next event time is given by the derivative of the CDF

$$p(t) = \frac{\part}{\part t}\left(1-e^{-\int_{t_0}^{t}\lambda(u)du}\right)$$

$$p(t) = \lambda(t) e^{-\int_{t_0}^{t}\lambda(u)du}$$