IntensityFunction

An intensity function is a mathematical function that describes the instantaneous rate at which events occur in a stochastic process. It is denoted by $\lambda(t)$, where $t$ is the time variable.

The concept of an intensity function is particularly important in the study of counting processes, which are stochastic processes that count the number of events that occur over time. In this context, the intensity function describes the probability that an event occurs in a small time interval around time $t$, given that no event has occurred up to time $t$.

Formally, the intensity function is defined as the derivative of the cumulative distribution function (CDF) of the waiting time between events, with respect to time. That is:

The intensity function is often used to model real-world phenomena in which events occur randomly and at unpredictable times. For example, it can be used to model

1. the arrival of customers in a queue,
2. the occurrence of earthquakes in a certain region, or
3. the failure of components in a mechanical system.

One of the advantages of using an intensity function to model such phenomena is that it allows us to predict the distribution of the waiting times between events, and to estimate the probability of observing a certain number of events in a given time interval.