Lindley distribution

\theta >0

x\in [0,\infty )

{\frac {\theta ^{2}}{\theta +1}}(1+x)e^{-\theta x}

1-{\frac {\theta +1+\theta x}{\theta +1}}e^{-\theta x}

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]

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 $k=2$ ) and an exponential distribution.

References

Lindley
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	${\frac {6(\theta +4)}{\theta ^{3}(\theta +1)}}$
Excess kurtosis	${\frac {24(\theta +5)}{\theta ^{4}(\theta +1)}}$
CF	${\frac {\theta ^{2}(\theta +1-ix)}{(\theta +1)(\theta -ix)^{2}}}$