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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> .
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| From unreliability equation, <math>z</math> can be calculated as a function of the <math>cdf</math> <math>F</math> as follows:
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| ::<math>z=\ln (F)-\ln (1-F)</math>
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| or using the equation for ''z''
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| ::<math>\frac{t-\mu }{\sigma }=\ln (F)-\ln (1-F)</math>
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| Then:
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| ::<math>\ln (F)-\ln (1-F)=-\frac{\mu }{\sigma }+\frac{1}{\sigma }t</math>
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| Now let:
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| ::<math>y=\ln (F)-\ln (1-F)</math>
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| ::<math>x=t</math>
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| and:
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| ::<math>a=-\frac{\mu }{\sigma }</math>
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| ::<math>b=\frac{1}{\sigma }</math>
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| which results in the following linear equation:
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| ::<math>y=a+bx</math>
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| The logistic probability paper resulting from this linearized <math>cdf</math> function is shown next.
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| [[Image:WB.14 logistic probability plot.png|center|250px| ]] | |
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| 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.
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| 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.
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