# HypoexponentialDistribution

HypoexponentialDistribution[{λ1,,λm}]

represents an m-phase hypoexponential distribution with rates λ1, , λm.

# Examples

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## Basic Examples(4)

Probability density function:

Cumulative distribution function:

Mean and variance:

Median can be found exactly in some cases:

But the general closed form does not exist:

## Scope(8)

Generate a sample of pseudorandom numbers from a hypoexponential 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:

Skewness:

The limiting values:

When both parameters go to simultaneously:

Kurtosis:

The limiting values:

When both parameters go to simultaneously:

Different moments with closed forms as functions of parameters:

Closed form for symbolic order:

Hazard function:

Quantile function:

Consistent use of Quantity in parameters yields QuantityDistribution:

Convert to days:

Find the median service time:

## Applications(1)

A process is composed of three independent consecutive steps, each with mean time exponentially distributed with parameters 0.003 hr., .002 hr., and 0.01 hr. respectively. Find the probability that the process takes less than 500 hours:

Find the mean duration of the process:

Find the median duration of the process:

Simulate the duration of the process in 50 consecutive runs:

## Properties & Relations(11)

The variation coefficient of hypoexponential distribution is always less than the variation coefficient of ExponentialDistribution:

There are no valid parameters such that the variation coefficient of the hypoexponential distribution is greater than or equal to the variation coefficient of an exponential distribution:

Theoretically there is no limit on the length of the vector of s:

Hypoexponential distribution is invariant under any permutation of the rates vector:

HypoexponentialDistribution is closed under scaling by a positive factor:

Relationships to other distributions:

Sum of independent variables following ExponentialDistribution has hypoexponential distribution:

Hypoexponential distribution with single rate reduces to ExponentialDistribution:

HypoexponentialDistribution is a special case of CoxianDistribution:

Hypoexponential distribution with all rates equal is ErlangDistribution:

Hypoexponential distribution for all rates equal is GammaDistribution:

## Neat Examples(1)

PDFs for different μ values with CDF contours:

Wolfram Research (2012), HypoexponentialDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/HypoexponentialDistribution.html (updated 2016).

#### Text

Wolfram Research (2012), HypoexponentialDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/HypoexponentialDistribution.html (updated 2016).

#### CMS

Wolfram Language. 2012. "HypoexponentialDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/HypoexponentialDistribution.html.

#### APA

Wolfram Language. (2012). HypoexponentialDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HypoexponentialDistribution.html

#### BibTeX

@misc{reference.wolfram_2024_hypoexponentialdistribution, author="Wolfram Research", title="{HypoexponentialDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/HypoexponentialDistribution.html}", note=[Accessed: 19-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_hypoexponentialdistribution, organization={Wolfram Research}, title={HypoexponentialDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/HypoexponentialDistribution.html}, note=[Accessed: 19-July-2024 ]}