View source for Exponential distribution

{{MSC|60E99}}

[[Category:Distribution theory]]

A continuous distribution of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036900/e0369001.png" /> defined by the density

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036900/e0369002.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

The density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036900/e0369003.png" /> is dependent on the positive scale parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036900/e0369004.png" />. The formula for the moments is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036900/e0369005.png" />, and, in particular, the expectation equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036900/e0369006.png" /> and the variance equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036900/e0369007.png" />; the characteristic function is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036900/e0369008.png" />.

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036900/e0369009.png" /></td> </tr></table>

the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036900/e03690010.png" />-fold convolution of the density (1) is equal to the gamma-density with the same parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036900/e03690011.png" /> and with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036900/e03690012.png" />.

The exponential distribution is the unique distribution having the property of no after-effect: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036900/e03690013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036900/e03690014.png" /> one has

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036900/e03690015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036900/e03690016.png" /> is the conditional probability of the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036900/e03690017.png" /> subject to the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036900/e03690018.png" />. Property (2) is also called the lack-of-memory property.

In a homogeneous [[Poisson process|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 above features explain 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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]] , '''2''' , Wiley (1971) </TD></TR></table>

====Comments====
The lack-of-memory property is related to the [[Markov property|Markov property]] in Poisson processes.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S.M. Ross, "Stochastic processes" , Wiley (1983) {{MR|0683455}} {{ZBL|0555.60002}} {{ZBL|0568.60079}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A.D. [A.D. Solov'ev] Solovyev, "Mathematical methods of reliability theory" , Acad. Press (1969) (Translated from Russian) {{MR|0345234}} {{ZBL|0169.50501}} </TD></TR></table>