20.4.16  Weibull distribution

The probability density function for the Weibull distribution.

The Weibull distribution depends on three parameters; k>0, λ > 0 and a real number θ. The probability density at x is given by

The weibull or weibulld command compute this density function.

• weibull takes three mandatory and one optional argument:
  • k, a positive integer.
  • λ, a positive real number.
  • Optionally θ, a real number (by default 0).
  • x, a real number.
• weibull(k,λ ⟨,θ ⟩, x) returns the value of the Weibull density function, given in (11).

Example

or:

The cumulative distribution function for the Weibull distribution.

The weibull_cdf (or weibulld_cdf) command computes the cumulative distribution function for the Weibull distribution.

• weibull_cdf (or weibulld_cdf) takes three mandatory arguments and two optional arguments:
  • k, a positive integer.
  • λ, a positive real number.
  • Optionally θ, a real number (by default 0).
  • x, a real number.
  • Optionally, y, a real number. If this optional argument is included, then θ must also be included.
• weibull_cdf(k,λ ⟨,θ ⟩,x) returns Prob(X ≤ x) for the Weibull distribution with parameters k, λ and θ.
• weibull_cdf(k,λ ⟨,θ ⟩,x,y) returns Prob(x ≤ X ≤ y).

In this case, the Weibull cumulative distribution function is given by the formula weibull_cdf(k,λ,θ,x)= 1−e^{−(x−θ)2λ2}.

Examples

or:

The inverse distribution function for the Weibull distribution.

The weibull_icdf (or weibulld_icdf) command computes the inverse distribution for the Weibull distribution.

• weibull_icdf (or weibulld_icdf) takes three mandatory arguments and one optional argument:
  • k, a positive integer.
  • λ, a positive real number.
  • Optionally θ, a real number (by default 0).
  • h, a real number between 0 and 1.
• weibull_icdf(k,λ ⟨,θ ⟩,h) returns the inverse distribution for the Weibull distribution with parameters k, λ and θ; namely, the value of x for which Prob(X ≤ x)=h.

Example