Template:Logistic probability paper: Difference between revisions

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==Logistic Distribution Probability Paper==
#REDIRECT [[The_Logistic_Distribution#Logistic_Distribution_Probability_Paper]]
The form of the Logistic probability paper is based on linearizing the  <math>cdf</math> .
From unreliability equation,  <math>z</math>  can be calculated as a function of the  <math>cdf</math>  <math>F</math>  as follows:
 
::<math>z=\ln (F)-\ln (1-F)</math>
 
or using the equation for ''z''
 
::<math>\frac{t-\mu }{\sigma }=\ln (F)-\ln (1-F)</math>
 
Then:
 
::<math>\ln (F)-\ln (1-F)=-\frac{\mu }{\sigma }+\frac{1}{\sigma }t</math>
 
Now let:
 
::<math>y=\ln (F)-\ln (1-F)</math>
 
::<math>x=t</math>
 
and:
 
::<math>a=-\frac{\mu }{\sigma }</math>
 
::<math>b=\frac{1}{\sigma }</math>
 
which results in the following linear equation:
 
::<math>y=a+bx</math>
 
The logistic probability paper resulting from this linearized  <math>cdf</math>  function is shown next.
 
[[Image:WB.14 logistic probability plot.png|center|250px| ]]  
 
Since the logistic distribution is symmetrical, the area under the  <math>pdf</math>  curve from  <math>-\infty </math>  to  <math>\mu </math>  is  <math>0.5</math> , as is the area from  <math>\mu </math>  to  <math>+\infty </math> . Consequently, the value of  <math>\mu </math>  is said to be the point where  <math>R(t)=Q(t)=50%</math> .  This means that the estimate of  <math>\mu </math>  can be read from the point where the plotted line crosses the 50% unreliability line.
 
For  <math>z=1</math> ,  <math>\sigma =t-\mu </math>  and  <math>R(t)=\tfrac{1}{1+\exp (1)}\approx 0.2689.</math>  Therefore,  <math>\sigma </math>  can be found by subtracting  <math>\mu </math>  from the time value where the plotted probability line crosses the 73.10% unreliability (26.89% reliability) horizontal line.

Latest revision as of 01:54, 15 August 2012

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