21.4.7.4. Weibull Distribution

The Type-III asymptotic distribution of the minimum value, developed by Weibull for studying the fatigue and fracture of materials, is known as the Weibull distribution. The two-parameter form is widely used in the life estimation of mechanical and electronic components. It can be represented by setting ε to be zero in the Type-III minimum value distribution is Extreme Value Distribution Value. The CDF of the two-parameter Weibull distribution is

\({{F}_{Z}}(z)=1-\exp \left[ -{{\left( \frac{z}{v} \right)}^{m}} \right];z\ge 0,m>0,v>0\)

The corresponding PDF is

\({{f}_{z}}(z)=\frac{m}{v}{{\left( \frac{z}{v} \right)}^{m-1}}\exp \left[ -{{\left( \frac{z}{v} \right)}^{m}} \right];z\ge 0\)

For the two-parameter Weibull distribution, the mean and the coefficient of variance are related to the parameters \(m\) and \(v\) as follows:

\({{\mu }_{z}}=v\Gamma \left( 1+\frac{1}{m} \right)\)

and

\({{\delta }_{z}}={{\left[ \frac{\Gamma \left( 1+\frac{2}{m} \right)}{{{\Gamma }^{2}}\left( 1+\frac{1}{m} \right)}-1 \right]}^{\frac{1}{2}}}\)

If the mean and coefficient of variance are known, the parameters \(m\) and \(v\) can be approximated for practical applications:

\(m={{\left( {{\delta }_{z}} \right)}^{-1.08}}\)

and

\(v=\frac{{{\mu }_{z}}}{\Gamma \left( 1+\frac{1}{m} \right)}\)