### 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

k
λ

⎛
⎜
⎜
⎝

x−θ
λ

⎞
⎟
⎟
⎠

2

e−(x−θ)2λ2. (11)

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

weibull(2,1,3)

or:

weibull(2,1,0,3)

6
e9

##### 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

weibull_cdf(2,3,5)

or:

weibull_cdf(2,3,0,5)

1−e

−

25
9

weibull_cdf(2.2,1.5,0.4,1.9)

0.632120558829 |  |  |  |  |  |  |  |  |  |

weibull_cdf(2.2,1.5,0.4,1.2,1.9)

0.410267239944 |  |  |  |  |  |  |  |  |  |

##### 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

weibull_icdf(2.2,1.5,0.4,0.632)

1.89977657604 |  |  |  |  |  |  |  |  |  |