Lindley distribution

{{Probability distribution|

name =Lindley|

type =density|

parameters =scale: $\theta>0$ |

support = $x\in [0,\infty)$ |

pdf = $\frac{\theta^2}{\theta+1} (1+x)e^{-\theta x}$ |

cdf = $1 - \frac{\theta+1+\theta x}{\theta+1}e^{-\theta x}$ |

mean = $\frac{\theta+2}{\theta(\theta+1)}$ |

variance = $\frac{2(\theta+3)}{\theta^2(\theta+1)}$ |

skewness = $\frac{6(\theta+4)}{\theta^3(\theta+1)}$ |

kurtosis = $\frac{24(\theta+5)}{\theta^4(\theta+1)}$ |

char = $\frac{\theta^2(\theta+1-ix)}{(\theta+1)(\theta-ix)^2}$ |

}}

In probability theory and statistics, the Lindley distribution is a continuous probability distribution for nonnegative-valued random variables.

The distribution is named after Dennis Lindley."Fiducial distributions and Bayes’ theorem", Journal of the Royal Statistical Society B 1958 vol.20 p.102-107

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 (with $k=2$ ) and an exponential distributon.

Definition

The probability density function of the Lindley distribution is:

: $f(x;\theta) = \frac{\theta^2}{\theta+1} (1+x)e^{-\theta x} \quad \theta, x \geq 0,$

where $\theta$ is the scale parameter of the distribution. The cumulative distribution function is:

: $F(x;\theta) = 1 - \frac{\theta+1+\theta x}{\theta+1}e^{-\theta x}$

for $x \in [0,\infty).$

References

Category:Continuous distributions

Category:Exponential family distributions

Category:Stable distributions