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Exponential distribution

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The exponential distribution is a continuous probability distribution used to model the time elapsed before a given event occurs.

Sometimes it is also called negative exponential distribution.

Hourglasses: the exponential distribution is primarily used to model elapsed time.

Table of Contents

How the distribution is used

The exponential distribution is frequently used to provide probabilistic answers to questions such as:

All these questions concern the time we need to wait before a given event occurs.

If this waiting time is unknown, it is often appropriate to think of it as a random variable having an exponential distribution.

Comparison: the exponential distribution is a continuous distribution used to model the time elapsed before an event occurs; the Poisson distribution is a discrete distribution used to model the number of occurrences of an event in a unit of time.

Waiting time

A waiting time X has an exponential distribution if the probability that the event occurs during a certain time interval is proportional to the length of that time interval.

More precisely, X has an exponential distribution if the conditional probability[eq1]is approximately proportional to the length $Delta t$ of the time interval comprised between the times $t$ and $t+Delta t$, for any time instant $t$.

In several practical situations this property is realistic. This is the reason why the exponential distribution can be used to model waiting times.

Definition

The exponential distribution is characterized as follows.

Definition Let X be a continuous random variable. Let its support be the set of positive real numbers:[eq2]Let [eq3]. We say that X has an exponential distribution with parameter $lambda $ if and only if its probability density function is[eq4]The parameter $lambda $ is called rate parameter.

A random variable having an exponential distribution is also called an exponential random variable.

The following is a proof that [eq5] is a legitimate probability density function.

Proof

Non-negativity is obvious. We need to prove that the integral of [eq6] over R equals 1. This is proved as follows:[eq7]

To better understand the exponential distribution, you can have a look at its density plots.

The rate parameter and its interpretation

We have mentioned that the probability that the event occurs between two dates $t$ and $t+Delta t$ is proportional to $Delta t$ (conditional on the information that it has not occurred before $t$).

The rate parameter $lambda $ is the constant of proportionality:[eq8]where [eq9] is an infinitesimal of higher order than $Delta t$ (i.e. a function of $Delta t$ that goes to zero more quickly than $Delta t$ does).

The above proportionality condition is also sufficient to completely characterize the exponential distribution.

Proposition The proportionality condition[eq10]is satisfied only if X has an exponential distribution.

Proof

The conditional probability [eq11] can be written as[eq12]Denote by [eq13] the distribution function of X, that is,[eq14]and by [eq15] its survival function:[eq16]Then,[eq17]Dividing both sides by $-Delta t$, we obtain[eq18]where $oleft( 1 ight) $ is a quantity that tends to 0 when $Delta t$ tends to 0. Taking limits on both sides, we obtain[eq19]or, by the definition of derivative:[eq20]This differential equation is easily solved by using the chain rule:[eq21]Taking the integral from 0 to x of both sides, we get[eq22]and[eq23]or[eq24]But [eq25] (because X cannot take negative values) implies[eq26]Exponentiating both sides, we obtain[eq27]Therefore,[eq28]or[eq29]But the density function is the first derivative of the distribution function:[eq30]and the rightmost term is the density of an exponential random variable. Therefore, the proportionality condition is satisfied only if X is an exponential random variable

Expected value

The expected value of an exponential random variable X is[eq31]

Proof

It can be derived as follows:[eq32]

Variance

The variance of an exponential random variable X is[eq33]

Proof

It can be derived thanks to the usual variance formula ([eq34]):[eq35]

Moment generating function

The moment generating function of an exponential random variable X is defined for any $t<lambda $:[eq36]

Proof

The definition of moment generating function gives[eq37]Of course, the above integrals converge only if [eq38], i.e. only if $t<lambda $. Therefore, the moment generating function of an exponential random variable exists for all $t<lambda $.

Characteristic function

The characteristic function of an exponential random variable X is[eq39]

Proof

By using the definition of characteristic function and the fact that [eq40]we can write[eq41]We now compute separately the two integrals. The first integral is[eq42]Therefore,[eq43]which can be rearranged to yield[eq44]or[eq45]The second integral is[eq46]Therefore,[eq47]which can be rearranged to yield[eq48]or[eq49]By putting pieces together, we get[eq50]

Distribution function

The distribution function of an exponential random variable X is[eq51]

Proof

If $x<0$, then[eq52]because X can not take on negative values. If $xgeq 0$, then[eq53]

More details

In the following subsections you can find more details about the exponential distribution.

Memoryless property

One of the most important properties of the exponential distribution is the memoryless property: [eq54]for any $xgeq 0$.

Proof

This is proved as follows:[eq55]

X is the time we need to wait before a certain event occurs. The above property says that the probability that the event happens during a time interval of length $y$ is independent of how much time has already elapsed (x) without the event happening.

The sum of exponential random variables is a Gamma random variable

Suppose that X_1, X_2, ..., X_n are n mutually independent random variables having exponential distribution with parameter $lambda $.

Define[eq56]

Then, the sum Z is a Gamma random variable with parameters $2n$ and $frac{n}{lambda }$.

Proof

This is proved using moment generating functions (remember that the moment generating function of a sum of mutually independent random variables is just the product of their moment generating functions):[eq57]The latter is the moment generating function of a Gamma distribution with parameters $2n$ and $frac{n}{lambda }$. So Z has a Gamma distribution, because two random variables have the same distribution when they have the same moment generating function.

The random variable Z is also sometimes said to have an Erlang distribution.

The Erlang distribution is just a special case of the Gamma distribution: a Gamma random variable is an Erlang random variable only when it can be written as a sum of exponential random variables.

Relation to the Poisson distribution

The exponential distribution is strictly related to the Poisson distribution.

Suppose that

  1. an event can occur more than once;

  2. the time elapsed between two successive occurrences is exponentially distributed and independent of previous occurrences.

Then, the number of occurrences of the event within a given unit of time has a Poisson distribution.

We invite the reader to see the lecture on the Poisson distribution for a more detailed explanation and an intuitive graphical representation of this fact.

Discrete counterpart

The exponential distribution is the continuous counterpart of the geometric distribution, which is instead discrete.

Density plot

The next plot shows how the density of the exponential distribution changes by changing the rate parameter:

The thin vertical lines indicate the means of the two distributions. Note that, by increasing the rate parameter, we decrease the mean of the distribution from 1 to $1/2$.

Exponential density plot 1

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Let X be an exponential random variable with parameter [eq58]. Compute the following probability:[eq59]

Solution

First of all we can write the probability as[eq60]using the fact that the probability that a continuous random variable takes on any specific value is equal to zero (see Continuous random variables and zero-probability events). Now, the probability can be written in terms of the distribution function of X as[eq61]

Exercise 2

Suppose the random variable X has an exponential distribution with parameter $lambda =1$. Compute the following probability:[eq62]

Solution

This probability can be easily computed by using the distribution function of X:[eq63]

Exercise 3

What is the probability that a random variable X is less than its expected value, if X has an exponential distribution with parameter $lambda $?

Solution

The expected value of an exponential random variable with parameter $lambda $ is[eq64]The probability above can be computed by using the distribution function of X:[eq65]

How to cite

Please cite as:

Taboga, Marco (2021). "Exponential distribution", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/probability-distributions/exponential-distribution.

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