Let $X_n$ be a sequence of random variables with density $\gamma^{n,1}(x) = \dfrac{x^{n-1}}{\Gamma(n)}e^{-{x}}$ for each $n \ge 1$ (Gamma law).
I am trying to calculate the density of $Z_n=\dfrac{X_{n+1}-n-1}{\sqrt{n+1}}$
What I did:
The variable change we have here is monotonic function so $f_{Z_n}(z)dz=f_{X_n}(x)dx=\gamma^{n,1}(x)dx$.
Is it correct?
In general if random variable $X$ has density $f_X(x)$ then for $a>0$ and $b\in\mathbb R$ random variable $aX+b$ has density: $$f_{aX+b}(x)=\frac1a f_X\left(\frac{x-b}a\right)$$
Note that: $$F_{aX+b}(x):=\mathsf P(aX+b\leq x)=\mathsf P\left(X\leq\frac{x-b}a\right)=F_X\left(\frac{x-b}a\right)$$
If $F_X$ is differentiable then $f_X$ can be found as derivative of $F_X$.