the Exponential Distribution | | DJS

the Exponential Distribution

Quotes below are sourced from, an article in the Encylopaedia of Mathematics by B.A. Sevast'yanov.. See also Wikipedia 's article – there is much good material there these days.

Exponential distribution

A continuous distribution of a random variable  X defined by the density

f(x) = λe−µx for x ≥0 and zero otherwise.                                                                (1)

The probability density function f(x) is dependent on the positive scale parameter µ. The expectation equals  1/µ and the variance equals 1/µ². The letter lambda λ is sometimes used in textbooks. I have used unicode characters to try to immunise these pages against poor display. Here I am using alpha as  𝛂  or 𝛼.

The cumulative probability function F(x) = 1- e−µx  .[the integral from 0 to infinity of f(x) dx]

It is quite common to replace µ with its reciprocal 1/ß.  ß is then called the survival parameter, because now we have E(X) = ß, the expected duration of survival of the system is β units of time. Once more, if  X is the duration of time that a given biological or mechanical system manages to survive and X ~ Exp(β) then E(X) = ß. Note the opportunity for confusion in using Exp(ß) rather than µ.

Conversely, if you can model the time between events as an exponential distribution, you will have yourself one of these distributions.

The exponential distribution belongs to the family of gamma-distributions (cf. Gamma-distribution) which are defined by the densities

p(x) = µx-1 e−µx  / (Gamma function (å))             for å >0 and non-negative x

Sevast’yanov’s article¹ says “the n-fold convolution of the density shown in (1) is equal to the gamma-density with the same parameter µ and with  å= n.”  Yeah, right.

The exponential distribution is the unique distribution having the property of no after-effect:

For any x>0, y>0,  one has P(X>x+y | X>y) =P(X>x). 

This also called the lack-of-memory property. There is a sense in which every x is independent of y. You can say that the exponential describes the time between consecutive rare random events in a process with no memory. CIE exam papers frequently explore this. I think the distribution is easily recognised from the function, but where you are kept from seeing that, this property may well be the giveaway clue.

Sevast’yanov continues: “In a homogeneous Poisson process, the distances between successive events have an exponential distribution. Conversely, a renewal process with exponential lifetime (1) is a Poisson process. An exponential distribution often arises as a limit process on the superposition or extension of renewal processes, as well as in high-level intersection problems in various random-path schemes, in critical branching processes, etc.

The features explained above show why the exponential distribution is widely used in calculating various systems in queueing theory and reliability theory. One assumes that the lifetimes of the devices are independent random variables with exponential distributions, and then the property (2) enables one to examine a queueing system by means of finite or denumerable Markov chains with continuous time. Similarly, one uses Markov chains in reliability theory, where the fault-free operating times of the individual devices can often be taken as independent and as having exponential distributions.

DJS 20090417



There is a discussion of maths similar to this in Growing  Up and Dying Away, at a simpler level than this.

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