Marshall–Olkin exponential distribution

In applied statistics, the Marshall–Olkin exponential distribution is any member of a certain family of continuous multivariate probability distributions with positive-valued components. It was introduced by Albert W. Marshall and Ingram Olkin.^[1] One of its main uses is in reliability theory, where the Marshall–Olkin copula models the dependence between random variables subjected to external shocks. ^{[2] [3]}

Let ${\displaystyle \{E_{B}:\varnothing \neq B\subset \{1,2,\ldots ,b\}\}}${\displaystyle \{E_{B}:\varnothing \neq B\subset \{1,2,\ldots ,b\}\}} be a set of independent, exponentially distributed random variables, where ${\displaystyle E_{B}}${\displaystyle E_{B}} has mean ${\displaystyle 1/\lambda _{B}}${\displaystyle 1/\lambda _{B}}. Let

${\displaystyle T_{j}=\min\{E_{B}:j\in B\},\ \ j=1,\ldots ,b.}${\displaystyle T_{j}=\min\{E_{B}:j\in B\},\ \ j=1,\ldots ,b.}

The joint distribution of ${\displaystyle T=(T_{1},\ldots ,T_{b})}${\displaystyle T=(T_{1},\ldots ,T_{b})} is called the Marshall–Olkin exponential distribution with parameters ${\displaystyle \{\lambda _{B},B\subset \{1,2,\ldots ,b\}\}.}${\displaystyle \{\lambda _{B},B\subset \{1,2,\ldots ,b\}\}.}

Suppose b = 3. Then there are seven nonempty subsets of { 1, ..., b } = { 1, 2, 3 }; hence seven different exponential random variables:

${\displaystyle E_{\{1\}},E_{\{2\}},E_{\{3\}},E_{\{1,2\}},E_{\{1,3\}},E_{\{2,3\}},E_{\{1,2,3\}}}${\displaystyle E_{\{1\}},E_{\{2\}},E_{\{3\}},E_{\{1,2\}},E_{\{1,3\}},E_{\{2,3\}},E_{\{1,2,3\}}}

Then we have:

${\displaystyle {\begin{aligned}T_{1}&=\min\{E_{\{1\}},E_{\{1,2\}},E_{\{1,3\}},E_{\{1,2,3\}}\}\\T_{2}&=\min\{E_{\{2\}},E_{\{1,2\}},E_{\{2,3\}},E_{\{1,2,3\}}\}\\T_{3}&=\min\{E_{\{3\}},E_{\{1,3\}},E_{\{2,3\}},E_{\{1,2,3\}}\}\\\end{aligned}}}${\displaystyle {\begin{aligned}T_{1}&=\min\{E_{\{1\}},E_{\{1,2\}},E_{\{1,3\}},E_{\{1,2,3\}}\}\\T_{2}&=\min\{E_{\{2\}},E_{\{1,2\}},E_{\{2,3\}},E_{\{1,2,3\}}\}\\T_{3}&=\min\{E_{\{3\}},E_{\{1,3\}},E_{\{2,3\}},E_{\{1,2,3\}}\}\\\end{aligned}}}

• Xu M, Xu S. "An Extended Stochastic Model for Quantitative Security Analysis of Networked Systems". Internet Mathematics, 2012, 8(3): 288–320.

• Continuous distributions
• Exponentials
• Exponential family distributions

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