The simplest model for events in time is a homogeneous Poisson process in which events occur singly at a constant rate over time, λ, with events in non-overlapping time periods independent.
The number of events in a Poisson process over a fixed period of time has a Poisson distribution. It can be found as a limit of binomial distributions.
The Poisson distribution's probability function is given and some properties are described.
A Poisson distribution's mean and variance both equal λ. The distribution's shape is close to a normal distribution when λ is large.
A few probabilities relating to Poisson distributions are calculated here.
The maximum likelihood estimate of the Poisson parameter, λ, is the sample mean.
The standard error of the maximum likelihood estimator is derived and a confidence interval for the parameter, λ, is found from it.
The time until the first event of a homogeneous Poisson process has an exponential distribution. Its pdf is derived.
The exponential distribution has a "memoryless" property — knowing that there were no events up to time t gives no information about when events will happen in the future.
This page describes some properties of gamma functions, a generalisation of factorials. Gamma functions will be used later in the e-book.
The exponential distribution's mean and variance are derived.
The maximum likelihood estimate of the exponential distribution's parameter, λ, is the inverse of the sample mean.
An approximate formula for the standard error of the MLE of λ is derived and used to find a confidence interval.
If X is the lifetime of an item, its survivor function is S(x) = P(X > x). Its hazard function, h(x), describes the momentary risk of dying, conditional on having survived until time x.
If an item's lifetime corresponds to the first event in a homogeneous Poisson process, its lifetime distribution is exponential and its hazard rate is constant.
The time until the k'th event in a homogeneous Poisson process has an Erlang distribution; its probability density function is derived.
The sum of k independent exponential random variables has an Erlang distribution. This is used to derive the distribution's mean and variance.
The Erlang distribution is a special case of the Gamma distribution. The way to find probabilities for it will be described for the Gamma distribution in the next chapter.