Poisson Process

Poisson Process

The Poisson process is a continuous time Markov chain that can be used to model the process of waiting for events to happen. As an example this random variable could be used to model the number of radioactive decays a sample will undergo during a time window of a particular length. As anywhere between 0 and an infinite number of events can occur in any given time window this particular Markov chain has an infinite number of states. The jump rate matrix is thus the infinity by infinity square matrix shown below: $$ \mathbf{Q} = \left( \begin{matrix} -\lambda & \lambda & 0 & 0 & 0 & \dots \\ 0 & -\lambda & \lambda & 0 & 0 & \dots \\ 0 & 0 & -\lambda & \lambda & 0 & \dots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{matrix} \right) $$ It is possible to show by induction that the solution to the resulting family of differential equations for the probabilities of having $n$ events during a time window of length $t$ is given by: $$ p_n(t) = \frac{(\lambda t)^n}{n!} e^{-\lambda t} $$

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