Lindley distribution

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.^[1]

Lindley
Parameters	scale: ${\displaystyle \theta >0}$
Support	${\displaystyle x\in [0,\infty )}$
PDF	${\displaystyle {\frac {\theta ^{2}}{\theta +1}}(1+x)e^{-\theta x}}$
CDF	${\displaystyle 1-{\frac {\theta +1+\theta x}{\theta +1}}e^{-\theta x}}$
Mean	${\displaystyle {\frac {\theta +2}{\theta (\theta +1)}}}$
Variance	${\displaystyle {\frac {2(\theta +3)}{\theta ^{2}(\theta +1)}}}$
Skewness	${\displaystyle {\frac {6(\theta +4)}{\theta ^{3}(\theta +1)}}}$
Excess kurtosis	${\displaystyle {\frac {24(\theta +5)}{\theta ^{4}(\theta +1)}}}$
CF	${\displaystyle {\frac {\theta ^{2}(\theta +1-ix)}{(\theta +1)(\theta -ix)^{2}}}}$

The Lindley distribution is used to describe the lifetime of processes and devices.^[2] In engineering, it has been used to model system reliability.

The distribution can be viewed as a mixture of the Erlang distribution (with ${\displaystyle k=2}$) and an exponential distribution.

Definition

The probability density function of the Lindley distribution is:

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

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

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

for ${\displaystyle x\in [0,\infty ).}$