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 ...

and

statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

, the Lindley distribution is a

continuous probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...

for nonnegative-valued

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...

s. The distribution is named after

Dennis Lindley Dennis Victor Lindley (25 July 1923 – 14 December 2013) was an English statistician, decision theorist and leading advocate of Bayesian statistics. Biography Lindley grew up in the south-west London suburb of Surbiton. He was an only child a ...

. The Lindley distribution is used to describe the lifetime of processes and devices."Lindley distribution and its application", ''Mathematics and computers in simulation'' 2008 vol.78(4) p.493-506 In engineering, it has been used to model system reliability. The distribution can be viewed as a mixture of 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, ...

(with

k=2

) and an exponential distributon.

Definition

The probability density function of the Lindley distribution is: :

f(x;\theta) = \frac (1+x)e^ \quad \theta, x \geq 0,

where

\theta

is the scale parameter of the distribution. The cumulative distribution function is: :

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

for

x \in,\infty).

References

Continuous distributions Exponential family distributions">Continuous distributions">,\infty).

References

Continuous distributions Exponential family distributions Stable distributions {{probability-stub