### 7.4.15 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}. The weibull (or
weibulld) command computes this, where it can take arguments
k,λ,θ and x, where the θ can be left out and
will default to 0. If you enter

weibull(2,1,3)

or

weibull(2,1,0,3)

you will get

6/exp(9)

#### The cumulative distribution function for the Weibull distribution: weibull_cdf weibulld_cdf

The command weibull_cdf computes
the cumulative distribution function for the Weibull distribution.
Like weibull, it takes parameters k, λ and
θ, where θ will default to 1 if it is omitted.
The Weibull cumulative distribution
function is given by the formula
weibull_cdf(k,λ,θ,x) = 1 −
e^{−((x−θ)/λ)2}.
If you enter

weibull_cdf(2,3,5)

or

weibull_cdf(2,3,0,5)

you will get

1-exp(-25/9)

and if you enter

weibull_cdf(2.2,1.5,0.4,1.9)

you will get

0.632120558829

If you give weibull_cdf an extra argument (which will
require that θ be explicitly included), you will get the probability that
the random variable lies between two values;
weibull_cdf(k,λ,θ,x,y) = Prob(x ≤ X ≤ y). If you
enter

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

for example you will get

0.410267239944

#### The inverse distribution function for the Weibull distribution: weibull_icdf weibulld_icdf

Given a value h, the inverse distribution function gives
the value of x with Prob(X ≤ x) = h.
The weibull_icdf command will compute the
inverse distribution for the Weibull distribution. This uses the
arguments k, λ and θ as well as h, although
θ can be omitted and will default to 0.
If you enter

weibull_icdf(2.2,1.5,0.4,0.632)

you will get

1.89977657604