Weibull and Reliability Analysis - Weibull CDF, Reliability, and Hazard Functions

Note: the Survival Analysis module uses a slightly different parameterization for the Weibull distribution; see the Survival Analysis module for details.

Density function

The Weibull distribution (Weibull, 1939, 1951; see also Lieblein, 1955) has density function (for positive parameters b, c, and θ):

f(x) = c/b*[(x-θ)/b]^c-1 * e^{-[(x-θ)/b]^c}

θ < x, b > 0, c > 0

where

b	is the scale parameter of the distribution
c	is the shape parameter of the distribution
θ	is the location parameter of the distribution
e	is the base of the natural logarithm, sometimes called Euler's e (2.71...)

Cumulative distribution function (CDF)

The Weibull distribution has the cumulative distribution function (for positive parameters b, c, and θ):

F(x) = 1 - exp{-[(x-θ)/b]^c}

using the same notation and symbols as described above for the density function.

Reliability function

The Weibull reliability function is the

R(x) = 1 - F(x)

Hazard function

The hazard function describes the probability of failure during a very small time increment, assuming that no failures have occurred prior to that time. The Weibull distribution has the hazard function (for positive parameters b, c, and θ):

h(t) = f(t)/R(t) = [c*(x-θ)^c-1] / b^c

using the same notation and symbols as described above for the density and reliability functions.

Cumulative hazard function

The Weibull distribution has the cumulative hazard function (for positive parameters b, c, and θ):

H(t) = (x-θ) / b^c