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====Probability Plotting====
#REDIRECT [[The_Normal_Distribution#Probability_Plotting]]
 
As described before, probability plotting involves plotting the failure times and associated unreliability estimates on specially constructed probability plotting paper. The form of this paper is based on a linearization of the  <math>cdf</math>  of the specific distribution. For the normal distribution, the cumulative density function can be written as:
 
::<math>F(t)=\Phi \left( \frac{t-\mu }{{{\sigma }}} \right)</math>
 
or:
 
::<math>{{\Phi }^{-1}}\left[ F(t) \right]=-\frac{\mu}{\sigma}+\frac{1}{\sigma}t</math>
 
where:
 
::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>
 
Now, let:
 
::<math>y={{\Phi }^{-1}}\left[ F(t) \right]</math>
 
::<math>a=-\frac{\mu }{\sigma }</math>
 
and:
 
::<math>b=\frac{1}{\sigma }</math>
 
which results in the linear equation of:
 
::<math>y=a+bT</math>
 
The normal probability paper resulting from this linearized  <math>cdf</math>  function is shown next.
 
[[Image:normalPP.gif|thumb|center|300px| ]]
 
Since the normal 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.
 
To determine the value of  <math>\sigma </math>  from the probability plot, it is first necessary to understand that the area under the  <math>pdf</math>  curve that lies between one standard deviation in either direction from the mean (or two standard deviations total) represents 68.3% of the area under the curve.  This is represented graphically in the following figure.
 
[[Image:68.3.gif|thumb|center|300px| ]]
 
Consequently,  the interval between  <math>Q(t)=84.15%</math>  and  <math>Q(t)=15.85%</math>  represents two standard deviations, since this is an interval of 68.3% ( <math>84.15-15.85=68.3</math> ), and is centered on the mean at 50%.  As a result, the standard deviation can be estimated from:
 
::<math>\widehat{\sigma }=\frac{t(Q=84.15%)-t(Q=15.85%)}{2}</math>
 
That is: the value of  <math>\widehat{\sigma }</math>  is obtained by subtracting the time value where the plotted line crosses the 84.15% unreliability line from the time value where the plotted line crosses the 15.85% unreliability line and dividing the result by two.  This process is illustrated in the following example.
 
 
'''Example 1:'''
{{Example: Normal Distribution Probability Plot}}

Latest revision as of 04:12, 13 August 2012

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