Q. 10.5 - A First Course in Probability

Question

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

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

Step-by-Step Solution

Answer

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

Step 1: Given Information

We have given Weibull distribution 

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

Step 2: Simplify

Finding$F^{-1}$. for$y\in (0,1)$ we have

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

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

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

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

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

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

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