# Exponential distribution

The **exponential distribution** is a continuous probability distribution of a non-negative random variable with probability density function \( f(x) = \lambda e^{-\lambda x} \) for \( x \ge 0 \), where the rate parameter \( \lambda > 0 \). It describes the waiting time until a single event occurs when events happen independently at a constant average rate, and it is the only continuous distribution on \( [0, \infty) \) with the memoryless property.<sup>[1](https://encyclopediaofmath.org/wiki/Exponential_distribution)</sup><sup> • </sup><sup>[2](https://continuous-time-mcs.quantecon.org/memoryless.html)</sup>

| Key fact | Detail |
|---|---|
| Density | \( f(x) = \lambda e^{-\lambda x} \), \( x \ge 0 \); equivalently \( f(x) = (1/\beta) e^{-x/\beta} \) with scale \( \beta = 1/\lambda \)<sup>[1](https://encyclopediaofmath.org/wiki/Exponential_distribution)</sup><sup> • </sup><sup>[3](https://itl.nist.gov/div898/handbook/eda/section3/eda3667.htm)</sup> |
| CDF and survival | \( F(x) = 1 - e^{-\lambda x} \); survival \( S(x) = e^{-\lambda x} \)<sup>[3](https://itl.nist.gov/div898/handbook/eda/section3/eda3667.htm)</sup> |
| Mean, median, variance | Mean \( 1/\lambda \) (also called MTTF in reliability), median \( \ln 2 / \lambda \approx 0.693/\lambda \), variance \( 1/\lambda^2 \)<sup>[4](https://www.itl.nist.gov/div898/handbook/apr/section1/apr161.htm)</sup> |
| Hazard (failure) rate | \( h(x) = \lambda \), constant for all \( x \); the exponential is the only distribution with a constant failure rate<sup>[4](https://www.itl.nist.gov/div898/handbook/apr/section1/apr161.htm)</sup> |
| Memoryless property | \( P(X > s + t \mid X > s) = P(X > t) \) for all \( s, t > 0 \); unique among distributions supported on \( [0, \infty) \)<sup>[2](https://continuous-time-mcs.quantecon.org/memoryless.html)</sup> |
| Poisson link | In a homogeneous Poisson process, gaps between successive events are exponentially distributed; a renewal process with exponential lifetimes is a Poisson process<sup>[1](https://encyclopediaofmath.org/wiki/Exponential_distribution)</sup> |
| Estimation | The maximum likelihood estimator of \( \lambda \) is the number of failures divided by total unit test time; for a full sample, the MLE of the scale parameter is the sample mean<sup>[5](https://www.itl.nist.gov/div898/handbook/apr/section4/apr412.htm)</sup><sup> • </sup><sup>[3](https://itl.nist.gov/div898/handbook/eda/section3/eda3667.htm)</sup> |

## Definition and basic functions

With rate \( \lambda > 0 \), the density is \( f(x) = \lambda e^{-\lambda x} \) for \( x \ge 0 \) and zero otherwise. The cumulative distribution function is \( F(x) = 1 - e^{-\lambda x} \), the survival function is \( S(x) = e^{-\lambda x} \), and the percent point (quantile) function is \( G(p) = -\beta \ln(1 - p) \) in the scale parameterization \( \beta = 1/\lambda \).<sup>[3](https://itl.nist.gov/div898/handbook/eda/section3/eda3667.htm)</sup><sup> • </sup><sup>[1](https://encyclopediaofmath.org/wiki/Exponential_distribution)</sup> The NIST handbook also gives the general three-parameter form \( f(x) = (1/\beta) e^{-(x-\mu)/\beta} \) for \( x \ge \mu \), with location \( \mu \) and scale \( \beta \); the case \( \mu = 0 \), \( \beta = 1 \) is the standard exponential distribution.<sup>[3](https://itl.nist.gov/div898/handbook/eda/section3/eda3667.htm)</sup>

The moments follow a simple pattern: \( E[X^n] = n!/\lambda^n \), so the mean is \( 1/\lambda \) and the variance is \( 1/\lambda^2 \), and the standard deviation equals the mean.<sup>[1](https://encyclopediaofmath.org/wiki/Exponential_distribution)</sup><sup> • </sup><sup>[4](https://www.itl.nist.gov/div898/handbook/apr/section1/apr161.htm)</sup> The median is about 69 percent of the mean, a consequence of the distribution's strong right skew (skewness 2, kurtosis 9).<sup>[4](https://www.itl.nist.gov/div898/handbook/apr/section1/apr161.htm)</sup><sup> • </sup><sup>[3](https://itl.nist.gov/div898/handbook/eda/section3/eda3667.htm)</sup> Small values are common and large values rare: in the OpenStax business statistics example, a postal clerk spends an average of 4 minutes with a customer, giving a decay parameter \( m = 1/4 = 0.25 \) per minute, and the probability of a longer-than-average visit falls quickly with time.<sup>[6](https://openstax.org/books/introductory-business-statistics/pages/5-3-the-exponential-distribution)</sup>

## Memoryless property

The defining property is that elapsed time carries no information about the remaining wait: for all \( s, t > 0 \),

\[ P(X > s + t \mid X > s) = P(X > t). \]

The proof is one line from the survival function: \( e^{-\lambda(s+t)} / e^{-\lambda s} = e^{-\lambda t} \).<sup>[2](https://continuous-time-mcs.quantecon.org/memoryless.html)</sup> The converse also holds: a random variable supported on \( [0, \infty) \) is memoryless if and only if it is exponential, so the property characterizes the distribution.<sup>[2](https://continuous-time-mcs.quantecon.org/memoryless.html)</sup><sup> • </sup><sup>[1](https://encyclopediaofmath.org/wiki/Exponential_distribution)</sup> In reliability language, a component whose life is exponential has the same failure outlook whether it is new or has already run for years; the same mathematics applies to the geometric distribution, its discrete counterpart.<sup>[7](https://ocw.mit.edu/courses/2-854-introduction-to-manufacturing-systems-fall-2016/pages/lecture-notes/memorylessness/)</sup>

The exponential also arises as the limit of the geometric distribution: if an arrival can occur at each tick of a grid with probability \( \lambda h \), then as the spacing \( h \to 0 \) the waiting time to the first arrival converges to an exponential random variable.<sup>[2](https://continuous-time-mcs.quantecon.org/memoryless.html)</sup>

## Relation to the Poisson process

The exponential and Poisson distributions describe the same constant-rate mechanism from two sides. When events occur singly, independently, and at a constant average rate \( \lambda \), the count of events in a fixed window is Poisson, while the gap until the next event is exponential with rate \( \lambda \).<sup>[8](https://vofti.com/learn/aqa-a-level-further-maths/statistics-exponential-distribution/)</sup> In a homogeneous Poisson process the distances between successive events are exponentially distributed, and conversely a [renewal process](https://www.edgepedia.org/renewal-process) with exponential lifetimes is a Poisson process.<sup>[1](https://encyclopediaofmath.org/wiki/Exponential_distribution)</sup> This duality is why the exponential fits interarrival times of [Geiger counter](https://www.edgepedia.org/geiger-counter) clicks, call arrivals, and radioactive decay, where the [Poisson distribution](https://www.edgepedia.org/poisson-distribution) handles the counts.<sup>[4](https://www.itl.nist.gov/div898/handbook/apr/section1/apr161.htm)</sup>

Two structural consequences matter in queueing and simulation. First, if \( X_1, \ldots, X_n \) are independent exponential variables with rates \( \mu_1, \ldots, \mu_n \), the minimum \( \min(X_1, \ldots, X_n) \) is again exponential with rate \( \mu_1 + \cdots + \mu_n \), and the probability that \( X_i \) is the smallest is \( \mu_i / (\mu_1 + \cdots + \mu_n) \).<sup>[9](https://www.cs.emory.edu/~cheung/Courses/MyInterests/QueueingTheory/IvoAdan/h1.pdf)</sup> Second, the sum of \( k \) independent exponential variables with common rate \( \mu \) has the Erlang-\( k \) distribution with mean \( k/\mu \); more generally the exponential is the gamma distribution with shape parameter \( \alpha = 1 \), and its \( n \)-fold convolution is the gamma density with \( \alpha = n \).<sup>[9](https://www.cs.emory.edu/~cheung/Courses/MyInterests/QueueingTheory/IvoAdan/h1.pdf)</sup><sup> • </sup><sup>[1](https://encyclopediaofmath.org/wiki/Exponential_distribution)</sup>

## Reliability applications

In reliability engineering the exponential is the simplest life distribution model, with a single unknown parameter.<sup>[4](https://www.itl.nist.gov/div898/handbook/apr/section1/apr161.htm)</sup> Its constant failure rate \( h(t) = \lambda \) makes it a natural model for the long flat "intrinsic failure" portion of the bathtub curve, where most systems spend most of their lives and neither early failures nor wear-out dominate.<sup>[4](https://www.itl.nist.gov/div898/handbook/apr/section1/apr161.htm)</sup> The mean \( 1/\lambda \) is called the Mean Time To Fail (MTTF), and the cumulative hazard is linear, \( H(t) = \lambda t \), so a cumulative hazard plot of exponential data should be a straight line through the origin with slope \( \lambda \).<sup>[4](https://www.itl.nist.gov/div898/handbook/apr/section1/apr161.htm)</sup><sup> • </sup><sup>[10](https://www.itl.nist.gov/div898/handbook/apr/section2/apr222.htm)</sup>

The [Weibull distribution](https://www.edgepedia.org/weibull-distribution) generalizes it: with shape parameter \( \gamma = 1 \) the Weibull reduces exactly to the exponential, with scale \( \alpha = 1/\lambda \) equal to the MTTF, while \( \gamma < 1 \) gives a decreasing and \( \gamma > 1 \) an increasing failure rate.<sup>[11](https://www.itl.nist.gov/div898/handbook/apr/section1/apr162.htm)</sup> In one NIST worked example, a least-squares fit of cumulative hazard data gave a Weibull slope of 1.27, close to the exponential slope of 1, though with only 10 observations and 6 failures the data alone could not settle the choice of model.<sup>[10](https://www.itl.nist.gov/div898/handbook/apr/section2/apr222.htm)</sup>

Estimation is straightforward. For a full sample, the maximum likelihood estimator of the scale parameter is the sample mean.<sup>[3](https://itl.nist.gov/div898/handbook/eda/section3/eda3667.htm)</sup> With Type I censored data, \( n \) units tested for a fixed time \( T \) with \( r \) observed failures at times \( t_1, \ldots, t_r \), the MLE is

\[ \hat{\lambda} = \frac{r}{\sum_{i=1}^{r} t_i + (n-r)T}, \]

that is, the total number of failures divided by the total unit test time.<sup>[5](https://www.itl.nist.gov/div898/handbook/apr/section4/apr412.htm)</sup>

## History

[Karl Pearson](https://www.edgepedia.org/karl-pearson), the English mathematician and founder of mathematical statistics, referred to the "negative exponential curve" in his 1895 paper *Contributions to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous Material*, where the exponential appears as a special case of the gamma (Pearson type III) family.<sup>[12](https://mathshistory.st-andrews.ac.uk//Miller/mathword/e/)</sup> The Danish mathematician [Agner Krarup Erlang](https://www.edgepedia.org/agner-krarup-erlang) (1878–1929), founder of traffic engineering and queueing theory, was perhaps the first to use the exponential as the universal waiting-time distribution, in his 1909 work *The Theory of Probabilities and Telephone Conversations*; the memoryless property made it the central distribution of queueing theory and insurance mathematics.<sup>[13](https://lipas.uwasa.fi/~tsottine/psp/psp.pdf)</sup> A 1911 paper by Marsden and Barratt on the radioactive disintegration of thorium gave a frequency distribution following the exponential law, connecting the model to physics.<sup>[14](https://digitalcommons.usu.edu/gradreports/1135/)</sup>

## Where the model fits and fails

The exponential suits phenomena with genuinely constant rates: radioactive decay of individual atoms, arrivals of cosmic ray particles, and telephone call durations are standard examples.<sup>[4](https://www.itl.nist.gov/div898/handbook/apr/section1/apr161.htm)</sup><sup> • </sup><sup>[6](https://openstax.org/books/introductory-business-statistics/pages/5-3-the-exponential-distribution)</sup> It is also used in biology, computer programming, and telecommunications.<sup>[15](https://catalogimages.wiley.com/images/db/pdf/9781786309044.excerpt.pdf)</sup> The assumption should be checked, because many real processes are not memoryless. Empirical studies of high-frequency financial data show that trades do not follow a Poisson process and waiting times between price changes are not exponentially distributed.<sup>[16](https://arxiv.org/abs/physics/0606005)</sup> As the MIT manufacturing systems lecture notes put it, models are judged by the accuracy of the predictions they are needed for: for inventory decisions dominated by the variance of tool life, the small deviation of actual lifetimes from the exponential model may not matter, while other purposes demand a more accurate non-exponential fit.<sup>[7](https://ocw.mit.edu/courses/2-854-introduction-to-manufacturing-systems-fall-2016/pages/lecture-notes/memorylessness/)</sup>

## References

1. [Exponential distribution, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Exponential_distribution)
2. [Memoryless Distributions, Continuous Time Markov Chains (QuantEcon)](https://continuous-time-mcs.quantecon.org/memoryless.html)
3. [1.3.6.6.7. Exponential Distribution, NIST/SEMATECH Engineering Statistics Handbook](https://itl.nist.gov/div898/handbook/eda/section3/eda3667.htm)
4. [8.1.6.1. Exponential, NIST/SEMATECH Engineering Statistics Handbook](https://www.itl.nist.gov/div898/handbook/apr/section1/apr161.htm)
5. [8.4.1.2. Maximum likelihood estimation, NIST/SEMATECH Engineering Statistics Handbook](https://www.itl.nist.gov/div898/handbook/apr/section4/apr412.htm)
6. [5.3 The Exponential Distribution, Introductory Business Statistics, OpenStax](https://openstax.org/books/introductory-business-statistics/pages/5-3-the-exponential-distribution)
7. [Memorylessness in exponential and geometric random variables, MIT OpenCourseWare 2.854](https://ocw.mit.edu/courses/2-854-introduction-to-manufacturing-systems-fall-2016/pages/lecture-notes/memorylessness/)
8. [Exponential distribution, AQA A-Level Further Maths (vofti.com)](https://vofti.com/learn/aqa-a-level-further-maths/statistics-exponential-distribution/)
9. [Queueing theory notes, Ivo Adan (Emory University course PDF)](https://www.cs.emory.edu/~cheung/Courses/MyInterests/QueueingTheory/IvoAdan/h1.pdf)
10. [8.2.2.2. Hazard and cumulative hazard plotting, NIST/SEMATECH Engineering Statistics Handbook](https://www.itl.nist.gov/div898/handbook/apr/section2/apr222.htm)
11. [8.1.6.2. Weibull, NIST/SEMATECH Engineering Statistics Handbook](https://www.itl.nist.gov/div898/handbook/apr/section1/apr162.htm)
12. [Earliest Known Uses of Some of the Words of Mathematics (E), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk//Miller/mathword/e/)
13. [Lecture 7: Exponential Distribution, T. Sottinen lecture notes](https://lipas.uwasa.fi/~tsottine/psp/psp.pdf)
14. [M. C. Wang (1969). A Study of the Exponential Distributions and their Applications, Utah State University](https://digitalcommons.usu.edu/gradreports/1135/)
15. [Queues Applied to Telecoms, Wiley excerpt](https://catalogimages.wiley.com/images/db/pdf/9781786309044.excerpt.pdf)
16. [On the gap between an empirical distribution and an exponential distribution of waiting times for price changes in a financial market, arXiv:physics/0606005](https://arxiv.org/abs/physics/0606005)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Probability theory › Probability distributions › Distribution families and classification › Continuous univariate distribution families*

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