Gamma Distribution deviance function unchanged, why?

Question

Consider the expressions for the deviance function of the gamma distribution:

Show that if each datum y_i replaced by 100y_i (say a change of measurement unitsfrom meters to centimeters) that the numerical value of the gamma deviance function does not change.

Answer 1

If $X\sim Gamma(a;b)$ the mean is $\mu_X=\frac{a}{b}$

If you multiply by 100 the rv you obtain a new gamma

$$Y\sim Gamma(a;\frac{b}{100})$$ with a new mean that is $\mu_Y=100\frac{a}{b}=100 \mu_X$

Now simply substituting the data in your expression you will realize that in this case the deviance is a "Scale Invariant Function"