Seismologists often formulate the occurrences of earthquakes as Poisson processes. A Poisson process assumes that the occurrences of earthquakes in time are independent. For any assumed magnitude range, there is a constant parameter (\(\lambda\)) that approximates the probability of occurrence of one or more events, the probability of more than one event is assumed to be negligible, in an infinitesimal time (\(\Delta t\)) interval as \(\lambda *\Delta t\). The parameter \(\lambda\) is typically interpreted as the average rate of occurrence. The Poisson process is often called a memory-less and time-independent process and plays an important role in formulating the time-independent PSHA.
For an event with a Poisson distribution with an average rate of \(\lambda\), the probability of occurrence of one or more events within a time window \(T_w\) can be estimated as:
$$P=1-e^{-\lambda* T_w}$$