Distribution The
probability density function (pdf) of the EL distribution is given by Tahmasbi and Rezaei (2008) :\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 :E(X)=-\frac{\operatorname{Li}_2(1-p)}{\beta\ln p}, :\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 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 : s(x)=\frac{\ln(1-(1-p)e^{-\beta x})}{\ln p}, : 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 : 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 \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: : X = \frac{1}{\beta}\ln \left(\frac{1-p}{1-p^U}\right). == Estimation of the parameters ==