represent an m-phase Coxian distribution with phase probabilities αi and rates λi.


  • An m-phase Coxian distribution can be interpreted as m sequential service phases with rates λi, where one continues to service phase i+1 with probability αi and finishes with probability 1-αi.
  • The probability density for value and distinct rates is a linear combination of exponentials for and zero for .
  • CoxianDistribution allows αi to be any positive number not greater than 1 and λi to be any positive real numbers.
  • CoxianDistribution allows λi to be any quantities of the same unit dimensions and αi to be dimensionless quantities. »
  • CoxianDistribution can be used with such functions as Mean, CDF, and RandomVariate.

Background & Context

  • CoxianDistribution[{α1,,αm-1},{λ1,,λm}] represents a continuous statistical distribution defined over the interval , parameterized by two vectors (α1,,αm-1) and (λ1,,λm), and known as an -phase Coxian distribution. The parameters αi are called "phase probabilities" and have values in the interval , while the parameters λi are called "phase rates" and have positive real values. Together, these parameters determine the overall shape of the probability density function (PDF) and, depending on their values, the PDF may be monotonic decreasing or unimodal. In addition, the tails of the PDF are "thin" in the sense that the PDF decreases exponentially rather than decreasing algebraically for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) Random variables satisfying XCoxianDistribution[{α1,,αm-1},{λ1,,λm}] are sometimes said to have a Coxian distribution of order .
  • While the foundations of Coxian distributions originate with the work of mathematician D. R. Cox in the 1950s, much of the current corpus of knowledge was established through work on generalizations of hyperexponential distributions dating from the 1980s. To be mathematically precise, a random variable has a Coxian distribution of order if it starts in phase 1 and goes through no more than exponential phases where, for the ^(th) phase (which has mean length equal to ), continues to phase i+1 with probability αi and finishes with probability 1-αi. A number of real-world phenomena behave in a way naturally modeled by a Coxian distribution, including teletraffic in mobile cellular networks, durations of stay among patients in geriatric facilities, and queueing systems of various types.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Coxian distribution. Distributed[x,CoxianDistribution[{α1,,αm-1},{λ1,,λm}]], written more concisely as xCoxianDistribution[{α1,,αm-1},{λ1,,λm}], can be used to assert that a random variable x is distributed according to a Coxian 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[CoxianDistribution[{α1,,αm-1},{λ1,,λm}],x] and CDF[CoxianDistribution[{α1,,αm-1},{λ1,,λm}],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 a Coxian distribution, EstimatedDistribution to estimate a Coxian parametric distribution from given data, and FindDistributionParameters to fit data to a Coxian distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Coxian distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Coxian distribution.
  • TransformedDistribution can be used to represent a transformed Coxian 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 a Coxian distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Coxian distributions.
  • The Coxian distribution is related to a number of other distributions. For example, CoxianDistribution is related to HyperexponentialDistribution both in its derivation and in the sense that the PDF of -phase distribution CoxianDistribution[{1,,1},{λ1,,λm}] is the same as that of ExponentialDistribution[{λ1,,λm}]. This creates a link between CoxianDistribution and ExponentialDistribution as well, and this link can be made precise by noting that the -phase CoxianDistribution[{0,,αm-1},{λ1,,λm}] has the same PDF as ExponentialDistribution[λ1]. Finally, for any λ, the PDF of the -phase distribution CoxianDistribution[{1,,1},{λ,,λ}] is the same as that of ErlangDistribution[{m,λ}].


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Basic Examples  (4)

Probability density function:

Cumulative distribution function:

Mean and variance:

Median can be found numerically:

Scope  (8)

Generate a sample of pseudorandom numbers from a Coxian distribution:

Compare the histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare a density histogram of the sample with the PDF of the estimated distribution:



Different moments with closed forms as functions of parameters:





Hazard function:

Quantile function:

Consistent use of Quantity in parameters yields QuantityDistribution:

Find the median time:

Applications  (2)

A customer enters a feed-forward queueing system with two service locations that have exponential service times with rates of 25 and 28 customers per hour, respectively. After being serviced at the first location, the customer prematurely leaves the system with probability of ; otherwise, the customer proceeds to the next service location. Find the probability that the customer is in the system for more than 5 minutes:

The first passage time of the continuous Markov chain into the only absorbing state, having started in a transient state, is generally described by a mixture of Coxian distributions:

The first time until this system reaches the absorbing state follows a Coxian distribution:

Properties & Relations  (5)

CoxianDistribution is closed under scaling by a positive factor:

Relationships to other distributions:

Coxian distribution with all phase probabilities equal to 1 is HypoexponentialDistribution:

Coxian distribution with identical rates and phase probabilities 1 is ErlangDistribution:

Coxian distribution with first phase probability zero is ExponentialDistribution:

Neat Examples  (1)

PDFs for different λ values with CDF contours:

Wolfram Research (2012), CoxianDistribution, Wolfram Language function, (updated 2016).


Wolfram Research (2012), CoxianDistribution, Wolfram Language function, (updated 2016).


Wolfram Language. 2012. "CoxianDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016.


Wolfram Language. (2012). CoxianDistribution. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_coxiandistribution, author="Wolfram Research", title="{CoxianDistribution}", year="2016", howpublished="\url{}", note=[Accessed: 17-July-2024 ]}


@online{reference.wolfram_2024_coxiandistribution, organization={Wolfram Research}, title={CoxianDistribution}, year={2016}, url={}, note=[Accessed: 17-July-2024 ]}