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 Ingram Olkin (July 23, 1924 – April 28, 2016) was a professor emeritus and chair of statistics and education at Stanford University and the Stanford Graduate School of Education. He is known for developing statistical analysis for evalua ...

. One of its main uses is in reliability theory, where the Marshall–Olkin copula models the dependence between random variables subjected to external shocks.

Definition

Let

\

be a set of independent,

exponentially distributed In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...

s, where

E_B

has mean

1/\lambda_B

. Let :

T_j=\min\,\ \ j=1,\ldots,b.

The joint distribution of

T=(T_1,\ldots,T_b)

is called the Marshall–Olkin exponential distribution with parameters

\.

Concrete example

Suppose ''b'' = 3. Then there are seven nonempty subsets of = ; hence seven different exponential random variables: :

E_, E_, E_, E_, E_, E_, E_

Then we have: :

\begin
T_1 & = \min\ \\
T_2 & = \min\ \\
T_3 & = \min\ \\
\end

References

* Xu M, Xu S. "An Extended Stochastic Model for Quantitative Security Analysis of Networked Systems". ''Internet Mathematics'', 2012, 8(3): 288–320. {{DEFAULTSORT:Marshall-Olkin exponential distribution Statistics articles needing expert attention Continuous distributions Exponentials Exponential family distributions