Exponential-logarithmic distribution: Difference between revisions

Revision as of 12:30, 17 March 2011

Revision as of 12:36, 17 March 2011

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

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<CAPTION>Exponential-Logarithmic distribution (EL)</CAPTION>

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=== Distribution ===

The [[probability density function]] (pdf) of the EL distribution is given by{{cnCitation needed|date=March 2011}}

:<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>

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: <math>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)}</math>

where <math>n=[n_1, n_2,\dots , n_N]</math> and <math>{d}=[d_1, d_2, \dots , d_D]</math>.

The moments of <math>X</math> can be derived from <math>M_X(t)</math>. For