1.
Exploratory Data Analysis
1.3. EDA Techniques 1.3.6. Probability Distributions 1.3.6.6. Gallery of Distributions


Probability Density Function 
The formula for the probability density
function of the general Weibull distribution is
\( f(x) = \frac{\gamma} {\alpha} (\frac{x\mu} {\alpha})^{(\gamma  1)}\exp{(((x\mu)/\alpha)^{\gamma})} \hspace{.3in} x \ge \mu; \gamma, \alpha > 0 \) where γ is the shape parameter, μ is the location parameter and α is the scale parameter. The case where μ = 0 and α = 1 is called the standard Weibull distribution. The case where μ = 0 is called the 2parameter Weibull distribution. The equation for the standard Weibull distribution reduces to \( f(x) = \gamma x^{(\gamma  1)}\exp((x^{\gamma})) \hspace{.3in} x \ge 0; \gamma > 0 \) 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. The following is the plot of the Weibull probability density function.


Cumulative Distribution Function 
The formula for the cumulative distribution
function of the Weibull distribution is
\( F(x) = 1  e^{(x^{\gamma})} \hspace{.3in} x \ge 0; \gamma > 0 \) The following is the plot of the Weibull cumulative distribution function with the same values of γ as the pdf plots above.


Percent Point Function 
The formula for the percent point
function of the Weibull distribution is
\( G(p) = (\ln(1  p))^{1/\gamma} \hspace{.3in} 0 \le p < 1; \gamma > 0 \) The following is the plot of the Weibull percent point function with the same values of γ as the pdf plots above.


Hazard Function 
The formula for the hazard
function of the Weibull distribution is
\( h(x) = \gamma x^{(\gamma  1)} \hspace{.3in} x \ge 0; \gamma > 0 \) The following is the plot of the Weibull hazard function with the same values of γ as the pdf plots above.


Cumulative Hazard Function 
The formula for the cumulative hazard
function of the Weibull distribution is
\( H(x) = x^{\gamma} \hspace{.3in} x \ge 0; \gamma > 0 \) The following is the plot of the Weibull cumulative hazard function with the same values of γ as the pdf plots above.


Survival Function 
The formula for the survival
function of the Weibull distribution is
\( S(x) = \exp{(x^{\gamma})} \hspace{.3in} x \ge 0; \gamma > 0 \) The following is the plot of the Weibull survival function with the same values of γ as the pdf plots above.


Inverse Survival Function 
The formula for the inverse
survival function of the Weibull distribution is
\( Z(p) = (\ln(p))^{1/\gamma} \hspace{.3in} 0 \le p < 1; \gamma > 0 \) The following is the plot of the Weibull inverse survival function with the same values of γ as the pdf plots above.


Common Statistics 
The formulas below are with the location parameter equal to zero
and the scale parameter equal to one.


Parameter Estimation  Maximum likelihood estimation for the Weibull distribution is discussed in the Reliability chapter (Chapter 8). It is also discussed in Chapter 21 of Johnson, Kotz, and Balakrishnan.  
Comments  The Weibull distribution is used extensively in reliability applications to model failure times.  
Software  Most general purpose statistical software programs support at least some of the probability functions for the Weibull distribution. 