Hypoexponential Distribution - Characterization

Characterization

A random variable has cumulative distribution function given by,

$F(x)=1-\boldsymbol{\alpha}e^{x\Theta}\boldsymbol{1}$

and density function,

$f(x)=-\boldsymbol{\alpha}e^{x\Theta}\Theta\boldsymbol{1}\; ,$

where is a column vector of ones of the size k and is the matrix exponential of A. When for all, the density function can be written as

$f(x) = \sum_{i=1}^k \lambda_i e^{-x \lambda_i} \left(\prod_{j=1, j \ne i}^k \frac{\lambda_j}{\lambda_j - \lambda_i}\right) = \sum_{i=1}^k \ell_i(0) \lambda_i e^{-x \lambda_i}$

where are the Lagrange basis polynomials associated with the points .

The distribution has Laplace transform of

$\mathcal{L}\{f(x)\}=-\boldsymbol{\alpha}(sI-\Theta)^{-1}\Theta\boldsymbol{1}$

Which can be used to find moments,

$E=(-1)^{n}n!\boldsymbol{\alpha}\Theta^{-n}\boldsymbol{1}\; .$

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