Forward equations of the Poisson process
Section of Poisson processes, part of Stochastic processes
Forward equations of the Poisson process
| Type | Section |
| Taught in | Poisson processes |
| Part | Stochastic processes |
Let p_n(t)=\mathbb{P}\{N(t)=n\}. In a short interval of length h the probability of one event is \lambda h+o(h), and the probability of two or more is o(h). Conditioning on the count at time t and sending h\to 0 gives the forward equations
p_0'(t)=-\lambda p_0(t), \qquad p_n'(t)=\lambda p_{n-1}(t)-\lambda p_n(t)\quad(n\ge 1),
with p_0(0)=1 and p_n(0)=0 for n\ge 1. The Poisson probabilities solve this system. For n=0 the solution is e^{-\lambda t}. If p_{n-1} is the Poisson probability with index n-1, the equation for p_n is a linear ODE whose solution is the Poisson probability with index n.
A second construction divides [0,t] into m slots and puts an event in each slot with probability \lambda t/m, independently. The count is binomial. As m\to\infty that binomial law converges to \mathrm{Poisson}(\lambda t), which is the theorem linked above. The limit is a reason to consider the model. It does not check, in a real event stream, that disjoint intervals are independent or that the rate is constant.
The path N(t) is a right-continuous step function. It starts at 0 and jumps by 1 at each arrival. The gaps between jumps are the interarrival times.
Used in
See also
Homogeneous Poisson process · Nonhomogeneous Poisson process · Superposition and thinning · Spatial Poisson process · Intensity function · Interarrival time