Positive random variables X and Y satisfy a scale model with parameters gt; 0 if P(Y t) = P(X t) for all t gt; 0, or equivalently, G(t) = F(t=),...

Positive random variables X and Y satisfy a scale model with parameters δ > 0 if P(Y ≤

t) = P(δX ≤ t) for all t > 0, or equivalently, G(t) = F(t=δ), δ > 0, t > 0.

(a) Show that in this case, log X and log Y satisfy a shift model with parameter log δ.

(b) Show that if X and Y satisfy a shift model with parameter ∆, then eX and eY satisfy a

scale model with parameter e∆.

(c) Suppose a scale model holds for X, Y . Let c > 0 be a constant. Does X' = Xc, Y' = Yc

satisfy a scale model? Does log X', log Y' satisfy a shift model?