Exponential Distribution

The exponential distribution describes the interval between events when the average number of events per unit of continuum has a Poisson distribution.

A good example is the load on a web site, the example below shows the interval between visits to a recipe site during the tea time rush:

This example is not statistically significant at the 10% level, but for many applications, the exponential distribution would be adequate for simulating the load on a web site.

Profile

Truncation

Unlike most distributions, where truncation significantly complicates the analysis, the left truncation of an exponential distribution, simply shifts the mean, mode and median by the amount of the truncation.

Measures of dispersion remain unchanged.

Parameters

Parameter	Description	Characteristics
mean	The mean interval between events.  Also referred to as the Scale parameter.	A float > 0

Functions

Properties

Alternative Parameter

The parameter lambda which is defined as the reciprocal of the mean is often used in the literature and software documentation, in this case the probability density and cumulative probability functions are written:

Spreadsheets

Both MS Excel and the Google Docs spreadsheet have a function called ExponDist which returns either the probability density function of the cumulative density function.

ExponDist(x,lambda,cum)

x is the value for which the function will return a value.

lamda is the reciprocal of the mean interval between events.

cum is a boolean flag, if it is false (i.e. is zero), the return value is the probability density, if it is true (i.e. non-zero), the return value is the cumulative probability.

Random Number Generation

Exponentially distributed random numbers (r) can be generated by transforming a standard uniform distribution (u):

r=g(u)

For programming purposes, the formula for g is simplified to:

This is because 1-u and u are both uniformly distributed random numbers, thus providing the opportunity to dispense with a subtract operation.

In Basic style psuedo code, the function looks like this:

r= -mean*log(rnd)

Parameter Estimation

The matching moments and maximum likelihood equations for parameter estimation are the same:

Page Updated: 13-Aug-2008

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