Exponential-logarithmic distribution: Difference between revisions

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{{Short description|Family of lifetime distributions with decreasing failure rate}}

{{Infobox probability distribution

| name = Exponential-Logarithmic distribution (EL)

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== Introduction ==

The study of lengths of the lives 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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==Related distributions==

The EL distribution has been generalized to form the Weibull-logarithmic distribution.<ref>Ciumara1Ciumara, Roxana; Preda2Preda, Vasile (2009) [httphttps://www.vgtuproquest.ltcom/leidiniaiopenview/leidykla7f1efa684243ce36231867620f09373a/ASMDA_2009/PDF/16_sec_081_Ciumara_The_Weibull.pdf1 "The Weibull-logarithmic distribution in lifetime analysis and its properties"]. In: L. Sakalauskas, C. Skiadas and

E. K. Zavadskas (Eds.) [http://www.vgtu.lt/leidiniai/leidykla/ASMDA_2009/ ''Applied Stochastic Models and Data Analysis''] {{Webarchive|url=https://web.archive.org/web/20110518043330/http://www.vgtu.lt/leidiniai/leidykla/ASMDA_2009/ |date=2011-05-18 }}, The XIII International Conference, Selected papers. Vilnius, 2009 {{ISBN |978-9955-28-463-5}}</ref>

If ''X'' is defined to be the [[random variable]] which is the minimum of ''N'' independent realisations from an [[exponential distribution]] with rate paramerterparameter ''&beta;'', and if ''N'' is a realisation from a [[logarithmic distribution]] (where the parameter ''p'' in the usual parameterisation is replaced by {{nowrap|1=(1&nbsp;&minus;&nbsp;''p'')}}), then ''X'' has the exponential-logarithmic distribution in the parameterisation used above.

==References==