|
|
| (3 intermediate revisions by 2 users not shown) |
| Line 1: |
Line 1: |
| ==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 Eqn. (UnR fcn), <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 Eqn. (z func of parameters)
| |
| | |
| ::<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:ldaLD10.4.gif|thumb|center|400px| ]] | |
| | |
| 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.
| |