distribution of a transformed gamma random variable

Question

I have $X=Gamma(a,b)$ and $Y=cX$ where c is a positive constant; I need to find the distribution of Y using the moment generating function method.

I know
$m_Y(t) = E e^{tY} = E e^{tcX}$
If i consider $E e^{tcX}$ as the m.g.f. of X evaluated at tc $m_x(tc)$ then I get that that such m.g.f. is equal to $\frac{b^a}{(b-tc)^a}$ for $tc<b$, which to me looks like the m.g.f. of a Gamma(a,b), which means that X and Y have the same distribution.
Where am I wrong? Is the gamma distribution really scale invariant?

Answer 1

You have started with $E[e^{tX}]=\frac {b^a}{\left(b-t\right)^a}$ at least for $t <b$

If $Y=cX$, then $$E\left[e^{tY}\right] =E\left[e^{ tcX}\right] = \frac {b^a}{\left(b-tc\right)^a} = \frac {\left(\frac{b}c\right)^a}{\left(\frac bc-t\right)^a}$$ which, for $t<\frac bc$, is the moment generating function of a $\mathrm{Gamma}\left(a,\frac bc\right)$ distribution

and this makes sense when $b$ is a rate parameter.