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Weibull Distribution

Definition

The Weibull pdf is positive only for positive values of x, and is zero otherwise. For strictly positive values of the shape parameter b and scale parameter a, the density is

Background

Waloddi Weibull offered the distribution that bears his name as an appropriate analytical tool for modeling the breaking strength of materials. Current usage also includes reliability and lifetime modeling. The Weibull distribution is more flexible than the exponential for these purposes.

To see why, consider the hazard rate function (instantaneous failure rate). If f(t) and F(t) are the pdf and cdf of a distribution, then the hazard rate is

Substituting the pdf and cdf of the exponential distribution for f(t) and F(t) above yields a constant. The example below shows that the hazard rate for the Weibull distribution can vary.

Parameters

Suppose you want to model the tensile strength of a thin filament using the Weibull distribution. The function wblfit gives maximum likelihood estimates and confidence intervals for the Weibull parameters.

rng('default');  % For reproducibility
strength = wblrnd(0.5,2,100,1);  % Simulated strengths
[p,ci] = wblfit(strength)
p =

    0.4768    1.9622


ci =

    0.4291    1.6821
    0.5298    2.2890

The default 95% confidence interval for each parameter contains the true value.

Example

The exponential distribution has a constant hazard function, which is not generally the case for the Weibull distribution. The plot shows the hazard function for exponential (dashed line) and Weibull (solid line) distributions having the same mean life. The Weibull hazard rate here increases with age (a reasonable assumption).

t = 0:0.1:4.5;
h1 = exppdf(t,0.6267) ./ (1-expcdf(t,0.6267));
h2 = wblpdf(t,2,2) ./ (1-wblcdf(t,2,2));
plot(t,h1,'--',t,h2,'-')

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