# The Weibull distribution

### 8.4.16 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

k
λ

(

x − θ
λ

)2 e−((x−θ)λ)2 (11)

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:

weibull(2,1,3)

or:

weibull(2,1,0,3)

Output:

6
e9

#### 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:

weibull_icdf(2.2,1.5,0.4,0.632)

Output:

1.89977657604