Weibull Distribution - Related Distributions

Related Distributions

• The translated Weibull distribution contains an additional parameter. It has the probability density function

for and f(x; k, λ, θ) = 0 for x < θ, where is the shape parameter, is the scale parameter and is the location parameter of the distribution. When θ=0, this reduces to the 2-parameter distribution.

• The Weibull distribution can be characterized as the distribution of a random variable X such that the random variable

is the standard exponential distribution with intensity 1.

• The Weibull distribution interpolates between the exponential distribution with intensity 1/λ when k = 1 and a Rayleigh distribution of mode when k = 2.

• The Weibull distribution can also be characterized in terms of a uniform distribution: if X is uniformly distributed on (0,1), then the random variable is Weibull distributed with parameters k and λ. This leads to an easily implemented numerical scheme for simulating a Weibull distribution.

• The Weibull distribution (usually sufficient in reliability engineering) is a special case of the three parameter Exponentiated Weibull distribution where the additional exponent equals 1. The Exponentiated Weibull distribution accommodates unimodal, bathtub shaped* and monotone failure rates.

• The Weibull distribution is a special case of the generalized extreme value distribution. It was in this connection that the distribution was first identified by Maurice Fréchet in 1927. The closely related Fréchet distribution, named for this work, has the probability density function

• The distribution of a random variable that is defined as the minimum of several random variables, each having a different Weibull distribution, is a poly-Weibull distribution.