X has Gamma(r, λ) distribution. Prove that Y=λX has Gamma(r,1) distribution.
I thought that you might be able to merely substitute the pdf for a Gamma(r, λ) distribution and then multiply that by lambda to then compare that to the pdf of a Gamma(r, 1) distribution. However this did not work. Any hints are greatly appreciated!
You need to use the rule for change of variables, which for $Y=g(X)$, the density function of $Y$, denoted by $f(y)$ is $$f(y)=f(x)\left|\frac{dX}{dY}\right|=f(g^{-1}(y))\left|\frac{dX}{dY}\right|$$ where $g^{-1}(y)$ denotes the inverse function.
So for $$f(x)=\Gamma(r,\lambda)=\frac{\lambda^r}{\Gamma(r)}x^{r-1}e^{-\lambda x}$$ and $Y=\lambda X$ you have $$f(y)=\frac{\lambda^r}{\Gamma(r)}\left(\frac{y}{\lambda}\right)^{r-1}e^{-\lambda \frac{y}{\lambda}}\times\frac{1}{\lambda}=\frac{1}{\Gamma(r)}y^{r-1}e^{-y}=\Gamma(r,1)$$