How do I find the transformation between the following random variables?

Question

Let $U$ $\sim$ $U(0,1)$ .

The Probability Integral Transform theorem states that for any continuous random variable $Y$ with cumulative distribution function $F(y)$ and inverse cumulative distribution function $F^{-1}$:

1. $F(Y)$ is a $U(0,1)$ random variable.
2. $F^{-1}(U)$ is a random variable with distribution function $F$.

I am being asked to apply this theorem to find the following transformation between the random variable $U$ $\sim$ $U(0,1)$ and $V$ which has the following cumulative distribution function:

$$F_{V}(v) = e^{({\frac{v-\alpha}{\beta})}^{-\alpha}}$$ for $\alpha$ $\lt$ $v$ $\lt$ $\infty$,

and $$F_{V}(v) = 0$$ otherwise.

So far, I have begun by:

Let $U$ $\sim$ $U(0,1)$.

$$u = F(v)$$ $$u = e^{({\frac{v-\alpha}{\beta})}^{-\alpha}}$$

but I am finding it hard to rearrange for $v$ to find the transformation between $u$ and $v$

Answer 1

Hint:

Taking natural logarithms gives you $$ \ln u=\left(\frac{\beta}{v-\alpha}\right)^\alpha\ . $$ Can you see how to proceed from here?