Exponential-logarithmic distribution: Difference between revisions

Revision as of 11:45, 25 March 2010

Revision as of 11:05, 5 December 2010

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{{Wikify|date=August 2009}}

In [[probability theory]] and [[statistics]], the '''exponential-logarithmic (EL) distribution''' is a family of lifetime [[probability distribution|distributions]] with

decreasing [[failure rate]], defined on the interval&nbsp;(0,&nbsp;&infin;∞). This distribution is [[Parametric family|parameterized]] by two parameters <math>p\in(0,1)</math> and <math>\beta >0</math>.

<TABLE class="infobox bordered wikitable"

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

where <math>p\in (0,1)</math> and <math>\beta >0</math>. This function is strictly decreasing in <math>x</math> and tends to zero as <math>x\rightarrow \infty</math>. The EL distribution has [[mode|modal value]] given, at x=0, by

:<math>\frac{\beta (1-p)}{-p \ln p}</math>

The EL reduces to the [[exponential distribution]] with parameter <math>\beta</math>, as <math>p\rightarrow 1</math>.

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Let ''U'' be a [[random variate]] from the standard [[Uniform distribution (continuous)|uniform distribution]].

Then the following transformation of ''U'' has the EL distribution with

parameters ''p'' and&nbsp;''&beta;β'':

: <math> X = \frac{1}{\beta}\ln \left(\frac{1-p}{1-p^U}\right).</math>