Question

Use the inverse transformation method to present an approach for generating a random variable from the Weibull distribution

$F\left(t\right)=1-e^{-a{t}^{\beta }}t\ge 0$

Short answer

The required method is used for the solution and it is explained below.

Step-by-step solution

Given Information

We have given Weibull distribution

$F\left(t\right)=1-e^{-a{t}^{\beta }}t\ge 0$

Simplify

Finding$F^{-1}$. forlocalid="1648201218153" $y\in (0,1)$we have

$F\left(t\right)=1-{{e^{-}}^{at}}^{\beta }=y$

$1-y={e^{-at}}^{\beta }$

$\log (1-y)=-a{t}^{\beta }$

$-\frac{1}{a}\log (1-y)=t^{\beta }$

localid="1648201140915" $t={\left(-\frac{1}{a}\log (1-y)\right)}^{\frac{1}{\beta }}$

So, the method is as follows. choose a random number (call it y) from the interval $(0,1)$and declare that localid="1651486180967" $t={\left(-\frac{1}{a}\log (1-y)\right)}^{\frac{1}{\beta }}$. from the universality of the uniform, we have that $x$follows the distribution of X

$\log (1-y)=-a{t}^{\beta }$