Positive random variables $X$ and $Y$ satisfy a scale model with parameters $\delta > 0$, if $\mathbb P(Y\le t) = \mathbb P(\delta X\le t), \forall t > 0$, or equivalently, $G(t) = F\left(\dfrac t\delta\right), \delta > 0, t > 0$.
To Prove:
b) Show that if $X$ and $Y$ satisfy a shift model with the parameter $\Delta$, then $e^X$ and $e^Y$ satisfy a scale model with parameter $e^\Delta$.
c) Suppose a scale model holds for $X$, $Y$. Let $c > 0$ be a constant. Does $X'= X^c, Y' = Y^c$ satisfy a scale model? Does $\log X', \log Y'$satisfy a shift model?
My work:
b) $\mathbb P(e^Y \leq t) = \mathbb P(e^X + e^\Delta \leq t)$
$\mathbb P(Y \leq\log(t)) = \mathbb P(X \leq \log(t-e^\Delta))$
Is $\Delta$ the same as $\delta$ or is it totally different?
c)
No idea about how to start.
b. $$P(e^Y \le t) = P(e^{X+\Delta} \le t) = P(e^\Delta e^X \le t)$$
c. $$P(Y^c \le t) = P(Y \le t^{1/c}) = P(\delta X \le t^{1/c}) = P(\delta^c X^c \le t)$$ $$P(\log(Y^c) \le t) = P(Y \le e^{t/c}) = P(\delta X \le e^{t/c}) = P(\log((\delta X)^c) \le t) = P(\log(\delta^c) + \log(X^c) \le t)$$