Exponential-logarithmic distribution: Difference between revisions

Revision as of 22:39, 27 March 2013

Revision as of 21:53, 1 May 2013

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{{Infobox probability distribution

| name = <CAPTION>Exponential-Logarithmic distribution (EL)</CAPTION>

| type = continuous

| pdf_image = <TD colSpan=2>[[File:Pdf EL.png|thumb|center|300px|Probability density function]]</TD></TR>

| cdf_image =

| notation =

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

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

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

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

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

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

| median <TD> = <math>\frac{\ln(1+\sqrt{p})}{\beta}</math></TD></TR>

| mode = 0

| variance <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>

| skewness =

| kurtosis =

| entropy =

| mgf <TD> = <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></TD></TR>

| cf =

| pgf =

| fisher =

}}

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;∞). 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"

style="FONT-SIZE: 95%; MARGIN-BOTTOM: 0.5em; MARGIN-LEFT: 1em; WIDTH: 320px">

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

<TR style="TEXT-ALIGN: center">

<TD colSpan=2>[[File:Pdf EL.png|thumb|center|300px|Probability density function]]</TD></TR>

<TR style="TEXT-ALIGN: center">

<TD colSpan=2>[[File:Hazard EL.png|thumb|center|300px|Hazard function]]</TD></TR>

<TR vAlign=top>

<TH>Parameters</TH>

<TD><SPAN><math>p\in (0,1)</math></SPAN><BR><SPAN> <math>\beta >0</math></SPAN></TD></TR>

<TR>

<TH>Support</TH>

<TD><math>x\in[0,\infty)</math></TD></TR>

<TR>

<TH>Probability density function (pdf)</TH>

<TD><math>\frac{1}{-\ln p} \times \frac{\beta(1-p) e^{-\beta

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

<TR>

<TH>Cumulative distribution function (cdf)</TH>

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

<TR>

<TH>Mean</TH>

<TD><math>-\frac{\text{polylog}(2,1-p)}{\beta\ln p}</math></TD></TR>

<TR>

<TH>Median</TH>

<TD><math>\frac{\ln(1+\sqrt{p})}{\beta}</math></TD></TR>

<TR>

<TH>Mode</TH>

<TD>0</TD></TR>

<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>

<TD>&nbsp;</TD></TR>

<TR>

<TH>Excess kurtosis</TH>

<TD>&nbsp;</TD></TR>

<TR>

<TH>Moment-generating function (mgf)</TH>

<TD><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></TD></TR>

<TR>

<TH>Characteristic function</TH>

<TD>&nbsp;</TD></TR>

</TABLE>

== Introduction ==

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=== The survival, hazard and mean residual life functions ===

<TD colSpan=2>[[File:Hazard EL.png|thumb|center|300px|Hazard function]]</TD></TR>

The [[survival function]] (also known as the reliability

function) and [[hazard function]] (also known as the failure rate