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| ====Probability Plotting====
| | #REDIRECT [[The_Normal_Distribution#Probability_Plotting]] |
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| 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:
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| ::<math>F(t)=\Phi \left( \frac{t-\mu }{{{\sigma }}} \right)</math>
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| or:
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| ::<math>{{\Phi }^{-1}}\left[ F(t) \right]=-\frac{\mu}{\sigma}+\frac{1}{\sigma}t</math>
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| where:
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| ::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>
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| Now, let:
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| ::<math>y={{\Phi }^{-1}}\left[ F(t) \right]</math>
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| ::<math>a=-\frac{\mu }{\sigma }</math>
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| and:
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| ::<math>b=\frac{1}{\sigma }</math>
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| which results in the linear equation of:
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| ::<math>y=a+bT</math>
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| The normal probability paper resulting from this linearized <math>cdf</math> function is shown next.
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| [[Image:normalPP.gif|thumb|center|300px| ]]
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| 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.
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| 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.
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| [[Image:68.3.gif|thumb|center|300px| ]] | |
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| 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:
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| ::<math>\widehat{\sigma }=\frac{t(Q=84.15%)-t(Q=15.85%)}{2}</math>
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| 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.
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| '''Example 1:'''
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| {{Example: Normal Distribution Probability Plot}}
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