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

Weibull Distribution

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 2-parameter 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.

 Mean $$\Gamma(\frac{\gamma + 1} {\gamma})$$ where Γ is the gamma function $$\Gamma(a) = \int_{0}^{\infty} {t^{a-1}e^{-t}dt}$$ Median $$\ln(2)^{1/\gamma}$$ Mode $$(1 - \frac{1} {\gamma})^{1/\gamma} \hspace{.2in} \gamma > 1$$ $$0 \hspace{1.05in} \gamma \le 1$$ Range 0 to $$\infty$$. Standard Deviation $$\sqrt{\Gamma(\frac{\gamma + 2} {\gamma}) - (\Gamma(\frac{\gamma + 1} {\gamma}))^{2}}$$ Coefficient of Variation $$\sqrt{\frac{\Gamma(\frac{\gamma + 2} {\gamma})} {(\Gamma(\frac{\gamma + 1} {\gamma}))^{2}} - 1}$$

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.