Editing Exponential distribution (section)

===Occurrence of events===
The exponential distribution occurs naturally when describing the lengths of the inter-arrival times in a homogeneous [[Poisson process]].

The exponential distribution may be viewed as a continuous counterpart of the [[geometric distribution]], which describes the number of [[Bernoulli trial]]s necessary for a ''discrete'' process to change state. In contrast, the exponential distribution describes the time for a continuous process to change state.

In real-world scenarios, the assumption of a constant rate (or probability per unit time) is rarely satisfied. For example, the rate of incoming phone calls differs according to the time of day. But if we focus on a time interval during which the rate is roughly constant, such as from 2 to 4 p.m. during work days, the exponential distribution can be used as a good approximate model for the time until the next phone call arrives. Similar caveats apply to the following examples which yield approximately exponentially distributed variables:
* The time until a radioactive [[particle decay]]s, or the time between clicks of a [[Geiger counter]]
* The time between receiving one telephone call and the next
* The time until default (on payment to company debt holders) in reduced-form credit risk modeling

Exponential variables can also be used to model situations where certain events occur with a constant probability per unit length, such as the distance between [[mutation]]s on a [[DNA]] strand, or between [[roadkill]]s on a given road.

In [[queuing theory]], the service times of agents in a system (e.g. how long it takes for a bank teller etc. to serve a customer) are often modeled as exponentially distributed variables.  (The arrival of customers for instance is also modeled by the [[Poisson distribution]] if the arrivals are independent and distributed identically.)  The length of a process that can be thought of as a sequence of several independent tasks follows the [[Erlang distribution]] (which is the distribution of the sum of several independent exponentially distributed variables).
[[Reliability theory]] and [[reliability engineering]] also make extensive use of the exponential distribution. Because of the memoryless property of this distribution, it is well-suited to model the constant [[hazard rate]] portion of the [[bathtub curve]] used in reliability theory. It is also very convenient because it is so easy to add [[failure rate]]s in a reliability model. The exponential distribution is however not appropriate to model the overall lifetime of organisms or technical devices, because the "failure rates" here are not constant: more failures occur for very young and for very old systems.
[[File:FitExponDistr.tif|thumb|260px|Fitted cumulative exponential distribution to annually maximum 1-day rainfalls using [[CumFreq]]<ref>{{cite web |url=http://www.waterlog.info/cumfreq.htm| title=Cumfreq, a free computer program for cumulative frequency analysis}}</ref>]]

In [[physics]], if you observe a [[gas]] at a fixed [[temperature]] and [[pressure]] in a uniform [[gravitational field]], the heights of the various molecules also follow an approximate exponential distribution, known as the [[Barometric formula]]. This is a consequence of the entropy property mentioned below.

In [[hydrology]], the exponential distribution is used to analyze extreme values of such variables as monthly and annual maximum values of daily rainfall and river discharge volumes.<ref>{{cite book|editor-last=Ritzema|editor-first=H.P.|title=Frequency and Regression Analysis|year=1994|publisher=Chapter 6 in: Drainage Principles and Applications, Publication 16, International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands|pages=[https://archive.org/details/drainageprincipl0000unse/page/175 175–224]|url=https://archive.org/details/drainageprincipl0000unse/page/175| isbn=90-70754-33-9}}</ref>

:The blue picture illustrates an example of fitting the exponential distribution to ranked annually maximum one-day rainfalls showing also the 90% [[confidence belt]] based on the [[binomial distribution]]. The rainfall data are represented by [[plotting position]]s as part of the [[cumulative frequency analysis]].
In operating-rooms management, the distribution of surgery duration for a category of surgeries with [[Predictive methods for surgery duration|no typical work-content]] (like in an emergency room, encompassing all types of surgeries).