Weibull distribution, Weibull probability paper

　・In Japanese
＜Premise knowledge＞
　・probability density function
　・logarithm

■What is Weibull distribution?

The Weibull distribution is a failure occurrence probability distribution proposed by Swedish scholar W. Weibull, and is expressed by the following formula as follows: "The failure of a component with a complex mechanism follows the failure of the weakest part." The chain breaking was caused by breaking the weakest part of the ring, also called the weakest link theory.

＜The probability density function＞
The probability density function is expressed by the following formula. m varies depending on the failure mode of the part.

・m<1：Early failure
・m=1：Intrinsic failure
・m>1：Wear-out failure

＜cumulative distribution function:CDF＞

The cumulative distribution function is the cumulative value of probabilities within a given range, with a cumulative value of 1 over the entire range. The horizontal axis is period. It is used to estimate the failure rate for a given period after the product is released, and is sometimes called, for example, the 3-year CDF.



＜failure rate＞
The failure rate λ(t) is shown in the graph below using the above Weibull distribution formula.



These failure rates are classified into early failure period, intrinsic failure period, and wear-out failure period, and fluctuate over time. A single curve representing this is called a bathtub curve.

■What is Weibull probability paper?

When examining the failure rate of a component, it may be difficult to determine whether it follows the Weibull distribution from the probability density function and cumulative distribution function shown above. Therefore, the Weibull probability paper is a tool for judging the suitability to the distribution by expressing the Weibull distribution on a straight line. A sample of Weibull probability paper is as follows.



The vertical axis is the cumulative distribution function expressed as a percentage, and the scale is unevenly spaced. It looks logarithmic but it is not. This is because the values are relative to the function shown on the right vertical axis, and the method of deriving that function is as follows.



Note that the following relationship holds.

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