Exponential-logarithmic 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

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

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

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

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

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

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:

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