Distribution

The probability density function (pdf) of the EL distribution is given by Tahmasbi and Rezaei (2008)^[1]

${\displaystyle f(x;p,\beta ):=\left({\frac {1}{-\ln p}}\right){\frac {\beta (1-p)e^{-\beta x}}{1-(1-p)e^{-\beta x}}}}$

where ${\displaystyle p\in (0,1)}$ and ${\displaystyle \beta >0}$. This function is strictly decreasing in ${\displaystyle x}$ and tends to zero as ${\displaystyle x\rightarrow \infty }$. The EL distribution has its modal value of the density at x=0, given by

${\displaystyle {\frac {\beta (1-p)}{-p\ln p}}}$

The EL reduces to the exponential distribution with rate parameter ${\displaystyle \beta }$, as ${\displaystyle p\rightarrow 1}$.

The cumulative distribution function is given by

${\displaystyle F(x;p,\beta )=1-{\frac {\ln(1-(1-p)e^{-\beta x})}{\ln p}},}$

and hence, the median is given by

${\displaystyle x_{\text{median}}={\frac {\ln(1+{\sqrt {p}})}{\beta }}}$.

Moments

The moment generating function of ${\displaystyle X}$ can be determined from the pdf by direct integration and is given by

${\displaystyle M_{X}(t)=E(e^{tX})=-{\frac {\beta (1-p)}{\ln p(\beta -t)}}F_{2,1}\left(\left[1,{\frac {\beta -t}{\beta }}\right],\left[{\frac {2\beta -t}{\beta }}\right],1-p\right),}$

where ${\displaystyle F_{2,1}}$ is a hypergeometric function. This function is also known as Barnes's extended hypergeometric function. The definition of ${\displaystyle F_{N,D}({n,d},z)}$ is

${\displaystyle F_{N,D}(n,d,z):=\sum _{k=0}^{\infty }{\frac {z^{k}\prod _{i=1}^{p}\Gamma (n_{i}+k)\Gamma ^{-1}(n_{i})}{\Gamma (k+1)\prod _{i=1}^{q}\Gamma (d_{i}+k)\Gamma ^{-1}(d_{i})}}}$

where ${\displaystyle n=[n_{1},n_{2},\dots ,n_{N}]}$ and ${\displaystyle {d}=[d_{1},d_{2},\dots ,d_{D}]}$.

The moments of ${\displaystyle X}$ can be derived from ${\displaystyle M_{X}(t)}$. For ${\displaystyle r\in \mathbb {N} }$, the raw moments are given by

${\displaystyle E(X^{r};p,\beta )=-r!{\frac {\operatorname {Li} _{r+1}(1-p)}{\beta ^{r}\ln p}},}$

where ${\displaystyle \operatorname {Li} _{a}(z)}$ is the polylogarithm function which is defined as follows:^[2]

${\displaystyle \operatorname {Li} _{a}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k^{a}}}.}$

Hence the mean and variance of the EL distribution are given, respectively, by

${\displaystyle E(X)=-{\frac {\operatorname {Li} _{2}(1-p)}{\beta \ln p}},}$

${\displaystyle \operatorname {Var} (X)=-{\frac {2\operatorname {Li} _{3}(1-p)}{\beta ^{2}\ln p}}-\left({\frac {\operatorname {Li} _{2}(1-p)}{\beta \ln p}}\right)^{2}.}$

The survival, hazard and mean residual life functions

Hazard function

The survival function (also known as the reliability function) and hazard function (also known as the failure rate function) of the EL distribution are given, respectively, by

${\displaystyle s(x)={\frac {\ln(1-(1-p)e^{-\beta x})}{\ln p}},}$

${\displaystyle h(x)={\frac {-\beta (1-p)e^{-\beta x}}{(1-(1-p)e^{-\beta x})\ln(1-(1-p)e^{-\beta x})}}.}$

The mean residual lifetime of the EL distribution is given by

${\displaystyle m(x_{0};p,\beta )=E(X-x_{0}|X\geq x_{0};\beta ,p)=-{\frac {\operatorname {Li} _{2}(1-(1-p)e^{-\beta x_{0}})}{\beta \ln(1-(1-p)e^{-\beta x_{0}})}}}$

where ${\displaystyle \operatorname {Li} _{2}}$ is the dilogarithm function

Random number generation

Let U be a random variate from the standard uniform distribution. Then the following transformation of U has the EL distribution with parameters p and β:

${\displaystyle X={\frac {1}{\beta }}\ln \left({\frac {1-p}{1-p^{U}}}\right).}$