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
or
you will get
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
or
you will get
and if you enter
you will get
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
for example you will get
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
you will get