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| '''One Parameter Exponential Probability Plot Example'''
| | #REDIRECT [[1P_Exponential_Example]] |
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| Six units are put on a life test and tested to failure. The failure times are 7, 12, 19, 29, 41, and 67 hours. Estimate the failure rate parameter for a one-parameter exponential distribution using the probability plotting method.
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| '''Solution'''
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| In order to plot the points for the probability plot, the appropriate reliability estimate values must be obtained. These will be equivalent to <math>100%-MR</math>, since the y-axis represents the reliability and the <math>MR</math> values represent unreliability estimates.
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| <center><math>\begin{matrix}
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| \text{Time-to-} & \text{Reliability} \\
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| \text{failure, hr} & \text{Estimate, }% \\
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| 7 & 100-10.91=89.09% \\
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| 12 & 100-26.44=73.56% \\
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| 19 & 100-42.14=57.86% \\
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| 29 & 100-57.86=42.14% \\
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| 41 & 100-73.56=26.44% \\
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| 67 & 100-89.09=10.91% \\
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| \end{matrix}</math></center>
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| Next, these points are plotted on exponential probability plotting paper. A sample of this type of plotting paper is shown next, with the sample points in place. Notice how these points describe a line with a negative slope.
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| Once the points are plotted, draw the best possible straight line through these points. The time value at which this line intersects with a horizontal line drawn at the 36.8% reliability mark is the mean life, and the reciprocal of this is the failure rate <math>\lambda </math>.
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| This is because at <math>t=m=\tfrac{1}{\lambda }</math>:
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| ::<math>\begin{align}
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| R(t)= & {{e}^{-\lambda \cdot t}} \\
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| R(t)= & {{e}^{-\lambda \cdot \tfrac{1}{\lambda }}} \\
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| R(t)= & {{e}^{-1}}=0.368=36.8%.
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| \end{align}</math>
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| These steps are shown graphically in the next pages.
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| [[Image:weibull EPP.png|center|400px|]] | |
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| As can be seen in the plot below, the best-fit line through the data points crosses the <math>R=36.8%</math> line at <math>t=33</math> hours.
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| [[Image:weibull EPP2.png|center|400px|]]
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| Since <math>\tfrac{1}{\lambda }=33</math> hours, <math>\lambda =0.0303</math> failures/hour.
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