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| ===Probability Plotting===
| | #REDIRECT [[The_Lognormal_Distribution]] |
| 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 lognormal 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 }'}}\cdot {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+b{t}'</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:lda_lognormalplot.gif|thumb|center|400px| ]]
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| The process for reading the parameter estimate values from the lognormal probability plot is very similar to the method employed for the normal distribution (see [[The Normal Distribution]] Chapter). However, since the lognormal distribution models the natural logarithms of the times-to-failure, the values of the parameter estimates must be read and calculated based on a logarithmic scale, as opposed to the linear time scale as it was done with the normal distribution. This parameter scale appears at the top of the lognormal probability plot.
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| The process of lognormal probability plotting is illustrated in the following example.
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| '''Example 1:'''
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| {{Example: Lognormal Distribution Probability Plot}}
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