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probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

the hypoexponential distribution or the generalized

Erlang distribution The Erlang distribution is a two-parameter family of continuous probability distributions with Support (mathematics), support x \in

is a continuous distribution">, \infty). The two parameters are: * a positive integer k, the "shape", and * a positive real number \lambda, ...

is a continuous distribution, that has found use in the same fields as the Erlang distribution, such as queueing theory, teletraffic engineering and more generally in stochastic processes. It is called the hypoexponetial distribution as it has a coefficient of variation less than one, compared to the hyper-exponential distribution which has coefficient of variation greater than one and the

exponential distribution In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuousl ...

which has coefficient of variation of one.

Overview

The

Erlang distribution The Erlang distribution is a two-parameter family of continuous probability distributions with Support (mathematics), support x \in [0, \infty). The two parameters are: * a positive integer k, the "shape", and * a positive real number \lambda, ...

is a series of ''k'' exponential distributions all with rate

\lambda

. The hypoexponential is a series of ''k'' exponential distributions each with their own rate

\lambda_

, the rate of the

i^

exponential distribution. If we have ''k'' independently distributed exponential random variables

\boldsymbol_

, then the random variable, :

\boldsymbol=\sum^_\boldsymbol_

is hypoexponentially distributed. The hypoexponential has a minimum coefficient of variation of

1/k

Relation to the phase-type distribution

As a result of the definition it is easier to consider this distribution as a special case of the phase-type distribution. The phase-type distribution is the time to absorption of a finite state Markov process. If we have a ''k+1'' state process, where the first ''k'' states are transient and the state ''k+1'' is an absorbing state, then the distribution of time from the start of the process until the absorbing state is reached is phase-type distributed. This becomes the hypoexponential if we start in the first 1 and move skip-free from state ''i'' to ''i+1'' with rate

\lambda_

until state ''k'' transitions with rate

\lambda_

to the absorbing state ''k+1''. This can be written in the form of a subgenerator matrix, :

\left[\begin-\lambda_&\lambda_&0&\dots&0&0\\
                    0&-\lambda_&\lambda_&\ddots&0&0\\
                    \vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\
                    0&0&\ddots&-\lambda_&\lambda_&0\\
                    0&0&\dots&0&-\lambda_&\lambda_\\
                    0&0&\dots&0&0&-\lambda_
\end\right]\; .

For simplicity denote the above matrix

\Theta\equiv\Theta(\lambda_,\dots,\lambda_)

. If the probability of starting in each of the ''k'' states is :

\boldsymbol=(1,0,\dots,0)

then

Hypo(\lambda_,\dots,\lambda_)=PH(\boldsymbol,\Theta).

Two parameter case

Where the distribution has two parameters (

\lambda_1 \neq \lambda_2

) the explicit forms of the probability functions and the associated statistics are: CDF:

F(x) = 1 - \frace^ - \frace^

PDF:

f(x) = \frac( e^ - e^ )

Mean:

\frac+\frac

Variance:

\frac+\frac

Coefficient of variation:

\frac

The coefficient of variation is always less than 1. Given the sample mean (

\bar

) and sample coefficient of variation (

c

), the parameters

\lambda_1

and

\lambda_2

can be estimated as follows:

\lambda_1= \frac \left 1 + \sqrt \right

\lambda_2 = \frac \left 1 - \sqrt \right

These estimators can be derived from the methods of moments by setting

\frac+\frac=\bar x

and

\frac=c

. The resulting parameters

\lambda_1

and

\lambda_2

are real values if

c^2\in .5,1 /math>.

Characterization

A random variable

\boldsymbol\sim Hypo(\lambda_,\dots,\lambda_)

has

cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...

given by, :

F(x)=1-\boldsymbole^\boldsymbol

and

density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...

, :

f(x)=-\boldsymbole^\Theta\boldsymbol\; ,

where

\boldsymbol

is a

column vector In linear algebra, a column vector with elements is an m \times 1 matrix consisting of a single column of entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some , c ...

of ones of the size ''k'' and

e^

is the

matrix exponential In mathematics, the matrix exponential is a matrix function on square matrix, square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exp ...

of ''A''. When

\lambda_ \ne \lambda_

for all

i \ne j

, the

can be written as :

f(x) = \sum_^k \lambda_i e^ \left(\prod_^k \frac\right) = \sum_^k \ell_i(0) \lambda_i e^

where

\ell_1(x), \dots, \ell_k(x)

are the Lagrange basis polynomials associated with the points

\lambda_1,\dots,\lambda_k

. The distribution has

Laplace transform In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a f ...

of :

\mathcal\=-\boldsymbol(sI-\Theta)^\Theta\boldsymbol

Which can be used to find moments, :

(-1)^n!\boldsymbol\Theta^\boldsymbol\; .

General case

In the general case where there are

a

distinct sums of exponential distributions with rates

\lambda_1,\lambda_2,\cdots,\lambda_a

and a number of terms in each sum equals to

r_1,r_2,\cdots,r_a

respectively. The cumulative distribution function for

t\geq0

is given by :

F(t)
= 1 - \left(\prod_^a \lambda_j^ \right)
\sum_^a \sum_^
\frac
 ,

with :

\Psi_(x)
= -\frac
\left(\prod_^a \left(\lambda_j+x\right)^ \right) .

with the additional convention

\lambda_0 = 0, r_0 = 1

Uses

This distribution has been used in population genetics, cell biology, and queuing theory.

Overview

Relation to the phase-type distribution

Two parameter case

Characterization

General case

Uses

See also

References

Further reading