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<TABLE class="infobox bordered wikitable"
style="FONT-SIZE: 95%; MARGIN-BOTTOM: 0.5em; MARGIN-LEFT: 1em; WIDTH:
<CAPTION>Exponential-Logarithmic distribution (EL)</CAPTION>
<TR style="TEXT-ALIGN: center">
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<TR>
<TH>Variance</TH>
<TD><math>-\frac{2 \text{polylog}(3,1-p)}{\beta^2\ln p}</math><br> <math>-\frac{ \text{polylog}^2(2,1-p)}{\beta^2\ln^2 p}</math></TD></TR>
<TR>
<TH>Skewness</TH>
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<TR>
<TH>Moment-generating function (mgf)</TH>
<TD><math>-\frac{\beta(1-p)}{\ln p (\beta-t)}
<TR>
<TH>Characteristic function</TH>
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The study of lengths of organisms, devices, materials, etc., is of major importance in the [[biological]] and [[engineering]] sciences. In general, the lifetime of a device is expected to exhibit decreasing failure rate (DFR) when its behavior over time is characterized by 'work-hardening' (in engineering terms) or 'immunity' (in biological terms).
The exponential-logarithmic model, together with its various properties, are studied by Tahmasbi and Rezaei (2008)<ref name="tahmasbi2008">Tahmasbi, R., Rezaei, S., (2008), "A two-parameter lifetime distribution with decreasing failure rate", ''Computational Statistics and Data Analysis'', 52 (8), 3889-3901. {{doi|10.1016/j.csda.2007.12.002}} </ref>
This model is obtained under the concept of population heterogeneity (through the process of
compounding).
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=== Distribution ===
The [[probability density function]] (pdf) of the EL distribution is given by
:<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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== Estimation of the parameters ==
To estimate the parameters, the [[Expectation-maximization algorithm|EM algorithm]] is used. This method is discussed by Tahmasbi and Rezaei (2008)<ref name="tahmasbi2008"/>. The EM iteration is given by
: <math>\beta^{(h+1)} = n \left( \sum_{i=1}^n\frac{x_i}{1-(1-p^{(h)})e^{-\beta^{(h)}x_i}} \right)^{-1},</math>
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