Exponential-logarithmic distribution: Difference between revisions

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The survival function (also known as reliability

function) and hazard function (also known as failure rate

function) of the EL distribution are given, respectively, by<br />

<math>s(x)=\frac{\ln(1-(1-p)e^{-\beta x})}{\ln p},</math>

<math>h(x)=\frac{-\beta(1-p)e^{-\beta x}}{(1-(1-p)e^{-\beta x})\ln(1-(1-p)e^{-\beta x})}.</math>

: <math>s(x)=\frac{\ln(1-(1-p)e^{-\beta x})}{\ln p},</math>

The mean residual lifetime

of the EL distribution is given by<br/>

: <math>mh(x_0;p,\betax)=E(X\frac{-x_0|X\geq x_0;\beta,(1-p)=e^{-\fracbeta x}}{dilog(1-(1-p)e^{-\beta x_0x})}{\beta \ln(1-(1-p)e^{-\beta x_0x})}.</math>

<br/>

The mean residual lifetime of the EL distribution is given by

where <math>dilog(.)</math> is dilogarithm function and it is defined as

follows:<br/>

: <math>dilog m(ax_0;p,\beta)=E(X-x_0|X\int_1^a{geq x_0;\beta,p)=-\frac{\lnoperatorname{dilog}(x1-(1-p)e^{-\beta x_0})}{\beta \ln(1-x(1-p)e^{-\beta x_0}dx)}.</math>

where <math>dilog(.)</math> is [[dilogarithm]] function and it is defined as follows:

: <math>\operatorname{dilog}(a)=\int_1^a \frac{\ln(x)}{1-x} \, dx.</math>

=== Random number generation ===