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Weibull distribution

Weibull Distribution

Probability Density FunctionThe formula for the probability density function of the general Weibull distribution isf(x) = (gamma/alpha)*((x-mu)/alpha)**(gamma-1)* EXP(-((x-mu)/alpha)**gamma) for x >= mu
where gamma is the shape parameter, mu is the location parameter and alpha is thescale parameter. The case where mu = 0 and alpha = 1 is called the standard Weibull distribution. The case where mu = 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)) for x >= 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.
plot of the Weibull probability density function
Cumulative Distribution FunctionThe formula for the cumulative distribution function of the Weibull distribution isF(x) = 1 - EXP(-(x**gamma)) for x >= 0; gamma > 0
The following is the plot of the Weibull cumulative distribution function with the same values of gamma as the pdf plots above.
plot of the Weibull cumulative distribution function with the same values of gamma as the pdf plots above
Percent Point FunctionThe formula for the percent point function of the Weibull distribution isG(p) = (-LN(1-p))**(1/gamma) 0 <= p < 1; gamma > 0
The following is the plot of the Weibull percent point function with the same values of gamma as the pdf plots above.
plot of the Weibull percent point function with the same values of gamma as the pdf plots above
Hazard FunctionThe formula for the hazard function of the Weibull distribution isgamma*x**(gamma-1) for x >= 0; gamma > 0
The following is the plot of the Weibull hazard function with the same values of gamma as the pdf plots above.
plot of the Weibull hazard function with the same values of gamma as the pdf plots above
Cumulative Hazard FunctionThe formula for the cumulative hazard function of the Weibull distribution isH(x) = x**gamma for x >= 0; gamma > 0
The following is the plot of the Weibull cumulative hazard function with the same values of gamma as the pdf plots above.
plot of the Weibull cumulative hazard function with the same values of gamma as the pdf plots above
Survival FunctionThe formula for the survival function of the Weibull distribution isS(x) = EXP(-(x**gamma)) for x >= 0
The following is the plot of the Weibull survival function with the same values of gamma as the pdf plots above.
plot of the Weibull survival function with the same values of gamma as the pdf plots above
Inverse Survival FunctionThe formula for the inverse survival function of the Weibull distribution isZ(p) = (-LN(p))**(1/gamma)
The following is the plot of the Weibull inverse survival function with the same values of gamma as the pdf plots above.
plot of the Weibull inverse survival function with the same values of gamma as the pdf plots above
Common StatisticsThe formulas below are with the location parameter equal to zero and the scale parameter equal to one.
MeanGAMMA((gamma+1)/gamma)where GAMMA is the gamma function
GAMMA(a) = INTEGRAL[t**(a-1)*EXP(-t)dt] where the integration is from 0 to infinity
MedianLOG(2)**(1/gamma)
Mode(1 - 1/gamma)**(1/gamma) gamma > 1
0 gamma <= 1
RangeZero to positive infinity.
Standard DeviationSQRT[GAMMA((gamma+2)/gamma) - (GAMMA((gamma+1)/gamma))**2]
Coefficient of VariationSQRT((GAMMA((gamma+2)/gamma)/GAMMA((gamma+1)/gamma)**2) - 1)
Parameter EstimationMaximum likelihood estimation for the Weibull distribution is discussed in theReliability chapter (Chapter 8). It is also discussed in Chapter 21 of Johnson, Kotz, and Balakrishnan.
CommentsThe Weibull distribution is used extensively in reliability applications to model failure times.
SoftwareMost general purpose statistical software programs support at least some of the probability functions for the Weibull distribution.

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