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Chapter 6   Events in Time

6.1   Poisson distribution

6.1.1   Poisson process

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.

6.1.2   Events in fixed time

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.

6.1.3   Poisson probability function

The Poisson distribution's probability function is given and some properties are described.

6.1.4   Mean and variance

A Poisson distribution's mean and variance both equal λ. The distribution's shape is close to a normal distribution when λ is large.

6.1.5   Example

A few probabilities relating to Poisson distributions are calculated here.

6.1.6   Maximum likelihood

The maximum likelihood estimate of the Poisson parameter, λ, is the sample mean.

6.1.7   Confidence interval for rate

The standard error of the maximum likelihood estimator is derived and a confidence interval for the parameter, λ, is found from it.

6.2   Exponential distribution

6.2.1   Time until first event

The time until the first event of a homogeneous Poisson process has an exponential distribution. Its pdf is derived.

6.2.2   Other exponential probabilities

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.

6.2.3   Gamma functions

This page describes some properties of gamma functions, a generalisation of factorials. Gamma functions will be used later in the e-book.

6.2.4   Mean and variance

The exponential distribution's mean and variance are derived.

6.2.5   Maximum likelihood

The maximum likelihood estimate of the exponential distribution's parameter, λ, is the inverse of the sample mean.

6.2.6   Confidence intervals

An approximate formula for the standard error of the MLE of λ is derived and used to find a confidence interval.

6.3   Lifetimes

6.3.1   Survivor and hazard functions

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.

6.3.2   Relationship to Poisson process

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.

6.4   Time until k'th event

6.4.1   Erlang distribution

The time until the k'th event in a homogeneous Poisson process has an Erlang distribution; its probability density function is derived.

6.4.2   Mean and variance

The sum of k independent exponential random variables has an Erlang distribution. This is used to derive the distribution's mean and variance.

6.4.3   Probabilities

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.