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:
weibull(2,1,3)
or:
weibull(2,1,0,3)
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:
- Input:
weibull_cdf(2.2,1.5,0.4,1.9)
Output:
- Input:
weibull_cdf(2.2,1.5,0.4,1.2,1.9)
Output:
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: