Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 34 / 2 / b / Solution

For the scale family, substitute $u=y/\sigma$ in the expectation:This depends on $f,g$ but not on $\sigma$. It holds whenever the expectation exists, including an extended expectation for nonnegative $g$; an undefined difference of two infinite integrals is not an expectation.