9.4.16  The Weibull distribution

The probability density function for the Weibull distribution: weibull weibulld

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.
weibulld is a synonym for weibull.

• 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.
Input:

or:

Output:

The cumulative distribution function for the Weibull distribution: weibull_cdf weibulld_cdf

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)/λ)2}.

Examples.

• Input:
  weibull_cdf(2,3,5)
  or:
  weibull_cdf(2,3,0,5)
  Output:

  1−e

  − 25 9

• Input:
  weibull_cdf(2.2,1.5,0.4,1.9)
  Output:

  0.632120558829

• Input:
  weibull_cdf(2.2,1.5,0.4,1.2,1.9)
  Output:

  0.410267239944

The inverse distribution function for the Weibull distribution: weibull_icdf weibulld_icdf

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.
Input:

Output: