Template:Logistic probability paper

Logistic Distribution Probability Paper

The form of the Logistic probability paper is based on linearizing the [math]\displaystyle{ cdf }[/math] . From unreliability equation, [math]\displaystyle{ z }[/math] can be calculated as a function of the [math]\displaystyle{ cdf }[/math] [math]\displaystyle{ F }[/math] as follows:

[math]\displaystyle{ z=\ln (F)-\ln (1-F) }[/math]

or using the equation for z

[math]\displaystyle{ \frac{t-\mu }{\sigma }=\ln (F)-\ln (1-F) }[/math]

Then:

[math]\displaystyle{ \ln (F)-\ln (1-F)=-\frac{\mu }{\sigma }+\frac{1}{\sigma }t }[/math]

Now let:

[math]\displaystyle{ y=\ln (F)-\ln (1-F) }[/math]

[math]\displaystyle{ x=t }[/math]

and:

[math]\displaystyle{ a=-\frac{\mu }{\sigma } }[/math]

[math]\displaystyle{ b=\frac{1}{\sigma } }[/math]

which results in the following linear equation:

[math]\displaystyle{ y=a+bx }[/math]

The logistic probability paper resulting from this linearized [math]\displaystyle{ cdf }[/math] function is shown next.

Since the logistic distribution is symmetrical, the area under the [math]\displaystyle{ pdf }[/math] curve from [math]\displaystyle{ -\infty }[/math] to [math]\displaystyle{ \mu }[/math] is [math]\displaystyle{ 0.5 }[/math] , as is the area from [math]\displaystyle{ \mu }[/math] to [math]\displaystyle{ +\infty }[/math] . Consequently, the value of [math]\displaystyle{ \mu }[/math] is said to be the point where [math]\displaystyle{ R(t)=Q(t)=50% }[/math] . This means that the estimate of [math]\displaystyle{ \mu }[/math] can be read from the point where the plotted line crosses the 50% unreliability line.

For [math]\displaystyle{ z=1 }[/math] , [math]\displaystyle{ \sigma =t-\mu }[/math] and [math]\displaystyle{ R(t)=\tfrac{1}{1+\exp (1)}\approx 0.2689. }[/math] Therefore, [math]\displaystyle{ \sigma }[/math] can be found by subtracting [math]\displaystyle{ \mu }[/math] from the time value where the plotted probability line crosses the 73.10% unreliability (26.89% reliability) horizontal line.