The Weibull Distribution
The Weibull distribution is used extensively in reliability applications to model failure times.
Formula
The formula for the probability density function of the general Weibull distribution is
\[ f(x)=\gamma \alpha(x-\mu\alpha)(\gamma-1)exp(-((x-\mu)/\alpha)\gamma)x=\mu;\gamma,\alpha>0\]
where is the shape parameter, is the location parameter and is the scale parameter.
The case where = 0 and \(\alpha = 1\) is called the standard Weibull distribution.
| The case where = 0 is called the 2-parameter Weibull distribution. The equation for the standard Weibull distribution reduces to |
| \[ f(x)=\gamma x(\gamma-1)exp(-(x\gamma))x=0;\gamma>0 \] |
| Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function. |
Standard parameterization
The probability density function of a Weibull random variable is:
\[{\displaystyle f(x;\lambda ,k)={\begin{cases}{\frac {k}{\lambda }}\left({\frac {x}{\lambda }}\right)^{k-1}e^{-(x/\lambda )^{k}}&x\geq 0,\\0&x<0,\end{cases}}}\]
where \(k > 0\) is the shape parameter and \(\lambda > 0\) is the scale parameter of the distribution. Its complementary cumulative distribution function is a stretched exponential function.
The Weibull distribution is related to a number of other probability distributions; in particular, it interpolates between the exponential distribution (\(k = 1\)) and the Rayleigh distribution (\(k = 2\)) and \[{\displaystyle \lambda ={\sqrt {2}}\sigma }\].
