Weibull distribution

Weibull(shape, scale)

The Weibull distribution is often used to represent failure time in reliability models. It is similar in shape to the gamma distribution, but tends to be less skewed and tail-heavy. It is a continuous distribution over the positive real numbers.

The Weibull distribution has a cumulative density on [math]\displaystyle{ x\ge 0 }[/math] given by:

[math]\displaystyle{ F(x) = 1 - exp\left({-\left({x\over{scale}}\right)^{shape}}\right) }[/math]

and F(x) = 0 for x < 0. The density on [math]\displaystyle{ x\ge 0 }[/math] is given by:

[math]\displaystyle{ f(x) = {{shape}\over{scale}} \left({x\over{scale}}\right)^{shape-1} exp\left(-(x/{scale})^{shape}\right) }[/math]

Library

Distribution

Example

Weibull(10, 4) →

Parameter Estimation

Suppose you have sampled historic data in Data, indexed by I, and you want to find the parameters for the best-fit Weibull distribution. The parameters can be estimated using a linear regression as follows:

Index bm := ['b', 'm'];

Var Fx := (Rank(Data, I) - 0.5)/Size(I);

Var Z := Ln(-Ln(1 - Fx));

Var fit := Regression(Z, Array(bm, [1, Ln(Data)]), I, bm);

Var shape := fit[bm = 'm'];

Var b := fit[bm = 'b'];

Var scale := Exp(-b/shape);

[shape, scale]

See Also

• Dens_Weibull
• CumWeibull
• Gamma
• Normal
• Regression
• Exp
• Ln