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STATLECT > Probability distributions

Poisson distribution

The Poisson distribution is related to the exponential distribution. Suppose a certain event can occur many times within a unit of time. Denote by X the total number of occurrences within a unit of time. When X is unknown, we can think of it as a random variable. If the time elapsed between two successive occurrences of the event has an exponential distribution and it is independent of previous occurrences, then X has a Poisson distribution.

Definition

The Poisson distribution is characterized as follows:

Definition_ Let X be a discrete random variable. Let its support be the set of positive integer numbers:[eq1]Let [eq2]. We say that X has a Poisson distribution with parameter $lambda $ if its probability mass function is[eq3]where $x!$ is the factorial of x.

A random variable having a Poisson distribution is also called a Poisson random variable.

Relation to the exponential distribution

The relation between the Poisson distribution and the exponential distribution is summarized by the following proposition:

Proposition_ X (the number of occurrences of an event within a unit of time) has a Poisson distribution with parameter $lambda $ if and only if the time elapsed between two successive occurrences of the event has an exponential distribution with parameter $lambda $ and it is independent of previous occurrences.

nav_button Proof

Denote by:[eq4]Since $Xgeq x$ if and only if [eq5] (convince yourself of this fact), the proposition is true if and only if:[eq6]for any $xin R_{X}$. To verify that the equality holds, we need to separately compute the two probabilities. We start with:[eq7]Denote by Z the sum of waiting times:[eq8]Since the sum of independent exponential random variables with common parameter $lambda $ is a Gamma random variable with parameters $2x$ and $ rac{x}{lambda }$, then Z is a Gamma random variable with parameters $2x$ and $ rac{x}{lambda }$, i.e. its probability density function is:[eq9]where [eq10]and the last equality stems from the fact that we are considering only integer values of x. We need to integrate the density function to compute the probability that Z is less than 1:[eq11]The last integral can be computed integrating by parts $x-1$ times:[eq12]Multiplying by $c$, we obtain:[eq13]Thus, we have obtained:[eq14]Now, we need to compute the probability that X is greater than or equal to x:[eq15]which is exactly what we needed to prove.

Expected value

The expected value of a Poisson random variable X is:[eq16]

nav_button Proof

It can be derived as follows:[eq17]

Variance

The variance of a Poisson random variable X is:[eq18]

nav_button Proof

It can be derived thanks to the usual variance formula ([eq19]):[eq20]

Moment generating function

The moment generating function of a Poisson random variable X is defined for any t in R:[eq21]

nav_button Proof

Using the definition of moment generating function:[eq22]where:[eq23]is the usual Taylor series expansion of the exponential function. Furthermore, since the series converges for any value of $t$, the moment generating function of a Poisson random variable exists for any t in R.

Characteristic function

The characteristic function of a Poisson random variable X is:[eq24]

nav_button Proof

Using the definition of characteristic function:[eq25]where:[eq26]is the usual Taylor series expansion of the exponential function (note that the series converges for any value of $t$).

Distribution function

The distribution function of a Poisson random variable X is:[eq27]where [eq28] is the floor of x, i.e. the largest integer not greater than x.

nav_button Proof

Using the definition of distribution function:[eq29]

Values of [eq30] are usually computed by computer algorithms. For example, the MATLAB command:[eq31]returns the value of the distribution function at the point x when the parameter of the distribution is equal to lambda.

Solved exercises

Below you can find some exercises with explained solutions:

  1. Exercise set 1

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