Gamma distribution and pdf Let $X \sim \mathsf{Gamma}(2,3)$ and $Y = 2X.$
Find the pdf of $Y$ at $y=13.5.$
Attempt: $f_X(x)= 2*[1/9*\Gamma(2)]*x*e^{-x/3}.$
Do I have to integrate now?

Another option:
Suppose $X $~ Gamma(2,3).  Then the moment generating function for $X$ is:
$$f(t) = E(e^{tX}) = (1-\theta t)^{-k} = (1-3t)^{-2}$$
Also, the moment generating function for $2X$ is: $$E(e^{t2X}) = E(e^{(2t)X}) = f(2t) = (1-3(2t))^{-2}$$
So we see that $2X$ has moment generating function $(1 - 6t)^{-2}$, so that $2X$ ~ Gamma(2,6).
