Let $f(x,y):=c \chi_{\{0<x<y<1\}}$ be the PDF of random variable $(X,Y)$
Find the PDF of random variable $X$.
I was able to find out that $c=2$, by using the normalization property of PDFs.
Question $1:$
In order to find the PDF $f_{X}(x)$, I would argue that
Since $P_{X}(A)=\int_{A}f_{X}(x)dx$ and note that $P_{(X,Y)}(A\times B)=\int_{A}\int_{B}f_{(X,Y)}(x,y)dydx$
Can I therefore state $f_{X}=\int f_{(X,Y)}(x,y)dy$?
Question $2:$
Lets say I have random variable $X$ that has any distribution (e.g. take the standard normal distribution (i.e. $\mathcal{N}(0,1)$))
and then I take constants $a > 0, b < 0$
Why will the distributions of $bX, a+X,a+bX$ still be identical to $X$. The mean should move, as well as the std in the above situations, like in the case of $\mathcal{N}(0,1)$
Any explanations for both questions are greatly appreciated.