Lindley distribution

Lindley
Parameters	scale: θ > 0 {\displaystyle \theta >0}
Support	x ∈ [ 0 , ∞ ) {\displaystyle x\in [0,\infty )}
PDF	θ 2 θ + 1 ( 1 + x ) e − θ x {\displaystyle {\frac {\theta ^{2}}{\theta +1}}(1+x)e^{-\theta x}}
CDF	1 − θ + 1 + θ x θ + 1 e − θ x {\displaystyle 1-{\frac {\theta +1+\theta x}{\theta +1}}e^{-\theta x}}
Mean	θ + 2 θ ( θ + 1 ) {\displaystyle {\frac {\theta +2}{\theta (\theta +1)}}}
Variance	2 ( θ + 3 ) θ 2 ( θ + 1 ) {\displaystyle {\frac {2(\theta +3)}{\theta ^{2}(\theta +1)}}}
Skewness	6 ( θ + 4 ) θ 3 ( θ + 1 ) {\displaystyle {\frac {6(\theta +4)}{\theta ^{3}(\theta +1)}}}
Excess kurtosis	24 ( θ + 5 ) θ 4 ( θ + 1 ) {\displaystyle {\frac {24(\theta +5)}{\theta ^{4}(\theta +1)}}}
CF	θ 2 ( θ + 1 − i x ) ( θ + 1 ) ( θ − i x ) 2 {\displaystyle {\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.^[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 {\displaystyle k=2} ) and an exponential distribution.

The probability density function of the Lindley distribution is:

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

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

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

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

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