Exponential-logarithmic distribution: Difference between revisions

Revision as of 21:53, 1 May 2013

Latest revision as of 01:36, 6 April 2024

(13 intermediate revisions by 12 users not shown)

Line 1:

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

{{Infobox probability distribution

| name = Exponential-Logarithmic distribution (EL)

| type = continuous

| pdf_image = [[File:Pdf EL.png|300px|Probability density function]]

| cdf_image =

| notation =

| parameters = <math>p\in (0,1)</math></SPANbr><BR><SPAN> <math>\beta >0</math>

| support = <math>x\in[0,\infty)</math>

| pdf = <math>\frac{1}{-\ln p} \times \frac{\beta(1-p) e^{-\beta x}}{1-(1-p) e^{-\beta x}}</math>

x}}{1-(1-p) e^{-\beta x}}</math>

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

| mean = <math>-\frac{\text{polylog}(2,1-p)}{\beta\ln p}</math>

Line 14:

| mode = 0

| variance = <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>

| skewness =

| kurtosis =

| entropy =

| mgf = <math>-\frac{\beta(1-p)}{\ln p (\beta-t)} \text{hypergeom}_{2,1} </math><br> <math>([1,\frac{\beta-t}{\beta}],[\frac{2\beta-t}{\beta}],1-p)</math>

| cf =

| pgf =

| fisher =

}}

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

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>.

== 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).

Line 105:

==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==

{{Reflist}}

{{ProbDistributions|continuous-semi-infinite}}

[[Category:Continuous distributions]]