ExponentialDistribution
✖
ExponentialDistribution
represents an exponential distribution with scale inversely proportional to parameter λ.
Details
- The probability density for value in an exponential distribution is proportional to for , and is zero for . »
- ExponentialDistribution allows λ to be any positive real number.
- ExponentialDistribution allows λ to be a quantity of any unit dimension. »
- ExponentialDistribution can be used with such functions as Mean, CDF, and RandomVariate. »
Background & Context
- ExponentialDistribution[λ] represents a continuous statistical distribution defined over the interval and parametrized by a positive real number λ. The probability density function (PDF) of an exponential distribution is monotonically decreasing. In addition, the tails of the PDF are "thin", in the sense that the PDF decreases exponentially for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The exponential distribution is sometimes referred to as the negative exponential distribution, the one-parameter exponential distribution, or the antilogarithmic distribution.
- Historically, the exponential distribution has been used most widely to describe events recurring "at random in time", i.e. in circumstances for which the future lifetime of an individual has the same distribution regardless of its present state. Use of the exponential distribution has increased significantly over the last 75 years, due in part to considerable research within the field of order statistics beginning in the early to mid-1950s. Since then, the exponential distribution has been used to model various phenomena over intervals of approximately constant rate, e.g. the number of phone calls placed in a specific time interval each day. In stochastic processes, the exponential distribution describes the lengths of interarrival times in homogeneous Poisson processes (i.e. continuous-time counting processes whose increments are independent, stationary, and Poisson-distributed—implemented as PoissonProcess). The exponential distribution is also used in credit risk modeling, queueing theory, reliability theory, physics, and hydrology.
- RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from an exponential distribution. Distributed[x,ExponentialDistribution[λ]], written more concisely as xExponentialDistribution[λ], can be used to assert that a random variable x is distributed according to an exponential distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
- The probability density and cumulative distribution functions may be given using PDF[ExponentialDistribution[λ],x] and CDF[ExponentialDistribution[λ],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
- DistributionFitTest can be used to test if a given dataset is consistent with an exponential distribution, EstimatedDistribution to estimate an exponential parametric distribution from given data, and FindDistributionParameters to fit data to an exponential distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic exponential distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic exponential distribution.
- TransformedDistribution can be used to represent a transformed exponential distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain an exponential distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving exponential distributions.
- The exponential distribution is related to a large number of other distributions. Its occurrence relative to a PoissonProcess induces a relationship with PoissonDistribution and CompoundPoissonDistribution. ExponentialDistribution can be thought of as a basis for the extreme value distributions family, due to the fact that ExponentialDistribution[1] can be transformed into each of ExtremeValueDistribution, GumbelDistribution, FrechetDistribution, and WeibullDistribution, while ExponentialDistribution can be obtained from each of GammaDistribution, LaplaceDistribution, BenktanderWeibullDistribution, LogisticDistribution, ParetoDistribution, PearsonDistribution, PowerDistribution, and RayleighDistribution by way of TransformedDistribution and/or TruncatedDistribution. Several distributions can be derived by combining various other distributions with ExponentialDistribution, e.g. GeometricDistribution (by combining ExponentialDistribution with PoissonDistribution), KDistribution (by combining with GammaDistribution), HoytDistribution (by combining with ArcSinDistribution), and ParetoDistribution (by combining with either ErlangDistribution or GammaDistribution).
Examples
open allclose allBasic Examples (4)Summary of the most common use cases
https://wolfram.com/xid/0fq56t4nb5rw6-hphar7
https://wolfram.com/xid/0fq56t4nb5rw6-161kd7
Cumulative distribution function:
https://wolfram.com/xid/0fq56t4nb5rw6-3ca6vk
https://wolfram.com/xid/0fq56t4nb5rw6-q0ch0f
https://wolfram.com/xid/0fq56t4nb5rw6-cwk
https://wolfram.com/xid/0fq56t4nb5rw6-pfz
https://wolfram.com/xid/0fq56t4nb5rw6-iv7hna
Scope (7)Survey of the scope of standard use cases
Generate a sample of pseudorandom numbers from an exponential distribution:
https://wolfram.com/xid/0fq56t4nb5rw6-qhtk5j
Compare its histogram to the PDF:
https://wolfram.com/xid/0fq56t4nb5rw6-03mwaz
Distribution parameters estimation:
https://wolfram.com/xid/0fq56t4nb5rw6-45b7g2
Estimate the distribution parameters from sample data:
https://wolfram.com/xid/0fq56t4nb5rw6-epi747
Compare the density histogram of the sample with the PDF of the estimated distribution:
https://wolfram.com/xid/0fq56t4nb5rw6-f8ui5o
Skewness and kurtosis are constant:
https://wolfram.com/xid/0fq56t4nb5rw6-ptk
https://wolfram.com/xid/0fq56t4nb5rw6-ogw
Different moments with closed forms as functions of parameters:
https://wolfram.com/xid/0fq56t4nb5rw6-js043h
https://wolfram.com/xid/0fq56t4nb5rw6-rx074o
Closed form for symbolic order:
https://wolfram.com/xid/0fq56t4nb5rw6-hf4lli
https://wolfram.com/xid/0fq56t4nb5rw6-pknsqa
Closed form for symbolic order:
https://wolfram.com/xid/0fq56t4nb5rw6-ubq525
https://wolfram.com/xid/0fq56t4nb5rw6-zg9ct4
https://wolfram.com/xid/0fq56t4nb5rw6-9gzmth
Closed form for symbolic order:
https://wolfram.com/xid/0fq56t4nb5rw6-5f9ua1
Hazard function of exponential distribution is constant and depends on the parameter λ:
https://wolfram.com/xid/0fq56t4nb5rw6-kh3ltq
https://wolfram.com/xid/0fq56t4nb5rw6-qag282
https://wolfram.com/xid/0fq56t4nb5rw6-iih
Consistent use of Quantity in parameters yields QuantityDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-bc5g3d
https://wolfram.com/xid/0fq56t4nb5rw6-ep3kcu
https://wolfram.com/xid/0fq56t4nb5rw6-hvx1sm
Applications (9)Sample problems that can be solved with this function
A battery has a lifespan that is exponentially distributed with rate parameter per hour. Find the probability that a random battery has a lifespan of less than 2500 hours:
https://wolfram.com/xid/0fq56t4nb5rw6-t10o7o
Compute directly using the CDF:
https://wolfram.com/xid/0fq56t4nb5rw6-npzl38
A relay has an exponentially distributed lifetime with a failure rate of failures per year. In order to estimate warranty costs, estimate the number of relays out of 10000 that will fail in the first six months of use. The failure rate is also known as the hazard rate:
https://wolfram.com/xid/0fq56t4nb5rw6-pnjbpb
Hence the lifetime distribution (in years) for relays is:
https://wolfram.com/xid/0fq56t4nb5rw6-8wp7se
The probability of failure within the first six months:
https://wolfram.com/xid/0fq56t4nb5rw6-39zuw5
The expected number of failures within the first six months in the batch of 10000 relays:
https://wolfram.com/xid/0fq56t4nb5rw6-m63gxj
A product has a time to failure that is exponentially distributed with parameter . Find the product's reliability at 1, 2, and 3 years. Reliability is another name for SurvivalFunction:
https://wolfram.com/xid/0fq56t4nb5rw6-l5544w
https://wolfram.com/xid/0fq56t4nb5rw6-l89h22
https://wolfram.com/xid/0fq56t4nb5rw6-w94dgj
Suppose the lifetime of an appliance has an exponential distribution with average lifetime of 10 years. Find the appliance lifetime distribution:
https://wolfram.com/xid/0fq56t4nb5rw6-b1ot60
https://wolfram.com/xid/0fq56t4nb5rw6-xfaka9
Find the probability that a used appliance with years of use will not fail in the next 5 years:
https://wolfram.com/xid/0fq56t4nb5rw6-ljw5vl
Using the memoryless property of ExponentialDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-zlcifs
Assume the waiting time a customer spends in a restaurant is exponentially distributed with an average wait time of 5 minutes. Find the probability that the customer will have to wait more than 10 minutes:
https://wolfram.com/xid/0fq56t4nb5rw6-dlbgup
https://wolfram.com/xid/0fq56t4nb5rw6-cxj43t
Find the probability that the customer has to wait an additional 10 minutes, given that he or she has already been waiting for at least 10 minutes (the past does not matter):
https://wolfram.com/xid/0fq56t4nb5rw6-10jjl4
The data contains waiting times in days between serious (magnitude at least 7.5 or over 1000 fatalities) earthquakes worldwide, recorded from 12/16/1902 to 3/4/1977:
https://wolfram.com/xid/0fq56t4nb5rw6-nvx1zv
Fit ExponentialDistribution to the data:
https://wolfram.com/xid/0fq56t4nb5rw6-v3m962
Compare the histogram of the data with the PDF of the estimated distribution:
https://wolfram.com/xid/0fq56t4nb5rw6-nogujz
Find the average number of days between major earthquakes:
https://wolfram.com/xid/0fq56t4nb5rw6-mqc6z9
Find the probability that two serious earthquakes occur within 100 days:
https://wolfram.com/xid/0fq56t4nb5rw6-oiqjn8
Simulate times between the next 30 serious earthquakes occurring worldwide:
https://wolfram.com/xid/0fq56t4nb5rw6-7e8a88
Waiting times at a receiver for signals coming from four independent transmitters are exponentially distributed with parameters , , , and , respectively. Find the probability that the signal from the third transmitter arrives first to the receiver:
https://wolfram.com/xid/0fq56t4nb5rw6-s8lno5
Find the distribution of the waiting time for any signal at the receiver:
https://wolfram.com/xid/0fq56t4nb5rw6-o652xe
Find the average waiting time for any signal at the receiver:
https://wolfram.com/xid/0fq56t4nb5rw6-mvi6ra
Simulate the waiting time between signals arriving at the receiver for , , , and :
https://wolfram.com/xid/0fq56t4nb5rw6-ncmerr
A system is composed of 4 independent components, each with lifespan exponentially distributed with parameter per hour. Find the probability that no component fails before 500 hours:
https://wolfram.com/xid/0fq56t4nb5rw6-7ff699
https://wolfram.com/xid/0fq56t4nb5rw6-c5eynp
Directly use SurvivalFunction:
https://wolfram.com/xid/0fq56t4nb5rw6-w5i35l
Find the probability that exactly one component will fail in the first 1200 hours:
https://wolfram.com/xid/0fq56t4nb5rw6-wpn666
Directly use CDF and SurvivalFunction:
https://wolfram.com/xid/0fq56t4nb5rw6-re8jnb
https://wolfram.com/xid/0fq56t4nb5rw6-u352jx
By using BooleanCountingFunction, you can also define the logical condition:
https://wolfram.com/xid/0fq56t4nb5rw6-wgknfy
https://wolfram.com/xid/0fq56t4nb5rw6-z73bgz
https://wolfram.com/xid/0fq56t4nb5rw6-kvzav3
https://wolfram.com/xid/0fq56t4nb5rw6-4x5qsy
In an optical communication system, transmitted light generates current at the receiver. The number of electrons follows the parametric mixture of Poisson distribution and other distributions, depending on the type of light. If the source uses coherent laser light of intensity λ, then the electron count distribution is Poisson:
https://wolfram.com/xid/0fq56t4nb5rw6-x9m11
https://wolfram.com/xid/0fq56t4nb5rw6-f4xu5d
https://wolfram.com/xid/0fq56t4nb5rw6-k39jqj
https://wolfram.com/xid/0fq56t4nb5rw6-dl5uo7
Which is PoissonDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-gt6wb3
If the source uses thermal illumination, then the Poisson parameter follows ExponentialDistribution with parameter and the electron count distribution is:
https://wolfram.com/xid/0fq56t4nb5rw6-moe5a3
These two distributions are distinguishable and allow the type of source to be determined:
https://wolfram.com/xid/0fq56t4nb5rw6-gvixwl
Properties & Relations (31)Properties of the function, and connections to other functions
Exponential distribution is closed under scaling by a positive factor:
https://wolfram.com/xid/0fq56t4nb5rw6-kyu0f
The variance is the square of the mean:
https://wolfram.com/xid/0fq56t4nb5rw6-zf6911
https://wolfram.com/xid/0fq56t4nb5rw6-6g8erk
The minimum of exponential distributions is exponentially distributed:
https://wolfram.com/xid/0fq56t4nb5rw6-1yweu4
The minimum of identically distributed variables:
https://wolfram.com/xid/0fq56t4nb5rw6-xn6nke
The exponential distribution is memoryless (the past does not matter):
https://wolfram.com/xid/0fq56t4nb5rw6-djpjt8
https://wolfram.com/xid/0fq56t4nb5rw6-d8xms4
https://wolfram.com/xid/0fq56t4nb5rw6-ve4q3m
Relationships to other distributions:
BenktanderWeibullDistribution reduces to a truncated ExponentialDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-gp51az
https://wolfram.com/xid/0fq56t4nb5rw6-ek5h2w
https://wolfram.com/xid/0fq56t4nb5rw6-enr0m0
Shifted ExponentialDistribution is a BenktanderWeibullDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-sc22s4
Exponential distribution is a limit of a scaled BetaDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-d5fhku
https://wolfram.com/xid/0fq56t4nb5rw6-don187
https://wolfram.com/xid/0fq56t4nb5rw6-itl7pb
https://wolfram.com/xid/0fq56t4nb5rw6-8b3vgg
https://wolfram.com/xid/0fq56t4nb5rw6-th75rv
PowerDistribution is a transformation of an exponential distribution:
https://wolfram.com/xid/0fq56t4nb5rw6-civn05
Exponential distribution can be obtained from PowerDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-yrvjnu
Exponential distribution can be obtained from BetaDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-z7usgj
Sum of independent exponentially distributed random variables follows ErlangDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-6ukdjm
For an arbitrary number of variables:
https://wolfram.com/xid/0fq56t4nb5rw6-jxz6or
https://wolfram.com/xid/0fq56t4nb5rw6-pklxrb
https://wolfram.com/xid/0fq56t4nb5rw6-q12prz
ExponentialDistribution[1] can be transformed into an extreme value distributions family:
https://wolfram.com/xid/0fq56t4nb5rw6-45gvso
https://wolfram.com/xid/0fq56t4nb5rw6-okn5oy
https://wolfram.com/xid/0fq56t4nb5rw6-5j60pp
https://wolfram.com/xid/0fq56t4nb5rw6-pmzfwi
ExponentialDistribution is a special case of WeibullDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-j7wg2g
https://wolfram.com/xid/0fq56t4nb5rw6-nnqn6a
https://wolfram.com/xid/0fq56t4nb5rw6-068o56
ExponentialDistribution is a special case of GammaDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-kny
https://wolfram.com/xid/0fq56t4nb5rw6-0pguz2
https://wolfram.com/xid/0fq56t4nb5rw6-k44c2s
The difference of two variates from the same exponential distribution follows LaplaceDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-wgk9p1
The difference of two different exponential distributions follows VarianceGammaDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-hlt6fs
Exponential distribution is a transformation of LaplaceDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-vgeon0
LogisticDistribution is a transformation from exponential distribution:
https://wolfram.com/xid/0fq56t4nb5rw6-mo4jpo
https://wolfram.com/xid/0fq56t4nb5rw6-4i4lsp
https://wolfram.com/xid/0fq56t4nb5rw6-qv4j4l
https://wolfram.com/xid/0fq56t4nb5rw6-4prdhg
LogisticDistribution is a transformation of exponential distribution:
https://wolfram.com/xid/0fq56t4nb5rw6-8029ht
https://wolfram.com/xid/0fq56t4nb5rw6-hd1bdx
https://wolfram.com/xid/0fq56t4nb5rw6-rvy6ob
https://wolfram.com/xid/0fq56t4nb5rw6-rcm5e5
ParetoDistribution is a transformation of exponential distribution:
https://wolfram.com/xid/0fq56t4nb5rw6-d7hyvh
Transformation of a ParetoDistribution yields an exponential distribution:
https://wolfram.com/xid/0fq56t4nb5rw6-89vvl3
Exponential distribution is a special case of type 3 PearsonDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-q9ld0b
https://wolfram.com/xid/0fq56t4nb5rw6-3cr68n
https://wolfram.com/xid/0fq56t4nb5rw6-3sm5lo
PowerDistribution is a transformation of exponential distribution:
https://wolfram.com/xid/0fq56t4nb5rw6-gccrh9
https://wolfram.com/xid/0fq56t4nb5rw6-c9yfv7
https://wolfram.com/xid/0fq56t4nb5rw6-8pgvkf
https://wolfram.com/xid/0fq56t4nb5rw6-m1dhjn
Exponential distribution can be obtained from RayleighDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-dk9v3h
https://wolfram.com/xid/0fq56t4nb5rw6-vgjcxj
https://wolfram.com/xid/0fq56t4nb5rw6-2pvw3f
https://wolfram.com/xid/0fq56t4nb5rw6-vjk7h9
Exponential distribution is the limiting distribution of the where has UniformDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-1m36d8
https://wolfram.com/xid/0fq56t4nb5rw6-u9bfo1
https://wolfram.com/xid/0fq56t4nb5rw6-kc1s8j
https://wolfram.com/xid/0fq56t4nb5rw6-1st0we
https://wolfram.com/xid/0fq56t4nb5rw6-9e134y
The parametric mixture of PoissonDistribution and exponential distribution follows GeometricDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-3ph7o2
KDistribution can be obtained from ExponentialDistribution and GammaDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-b00ug9
HoytDistribution can be obtained from ExponentialDistribution and ArcSinDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-3l50h9
https://wolfram.com/xid/0fq56t4nb5rw6-hk4j5l
ParetoDistribution can be obtained as a quotient of ExponentialDistribution and ErlangDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-oixbsl
https://wolfram.com/xid/0fq56t4nb5rw6-2u6ttl
https://wolfram.com/xid/0fq56t4nb5rw6-duq95h
https://wolfram.com/xid/0fq56t4nb5rw6-ln82as
ParetoDistribution can be obtained as a quotient of ExponentialDistribution and GammaDistribution:
https://wolfram.com/xid/0fq56t4nb5rw6-gruq01
https://wolfram.com/xid/0fq56t4nb5rw6-lvqs29
https://wolfram.com/xid/0fq56t4nb5rw6-eh4z3p
https://wolfram.com/xid/0fq56t4nb5rw6-7r8hq7
Possible Issues (2)Common pitfalls and unexpected behavior
ExponentialDistribution is not defined when λ is not a positive real number:
https://wolfram.com/xid/0fq56t4nb5rw6-jd0
Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:
https://wolfram.com/xid/0fq56t4nb5rw6-t70
Wolfram Research (2007), ExponentialDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/ExponentialDistribution.html (updated 2016).
Text
Wolfram Research (2007), ExponentialDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/ExponentialDistribution.html (updated 2016).
Wolfram Research (2007), ExponentialDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/ExponentialDistribution.html (updated 2016).
CMS
Wolfram Language. 2007. "ExponentialDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/ExponentialDistribution.html.
Wolfram Language. 2007. "ExponentialDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/ExponentialDistribution.html.
APA
Wolfram Language. (2007). ExponentialDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ExponentialDistribution.html
Wolfram Language. (2007). ExponentialDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ExponentialDistribution.html
BibTeX
@misc{reference.wolfram_2024_exponentialdistribution, author="Wolfram Research", title="{ExponentialDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/ExponentialDistribution.html}", note=[Accessed: 11-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_exponentialdistribution, organization={Wolfram Research}, title={ExponentialDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/ExponentialDistribution.html}, note=[Accessed: 11-January-2025
]}