TITLE: Joint to marginal distribution does not make sense
QUESTION [5 upvotes]: Let the joint distribution of $X$ and $Y$ be
$$
f_{X,Y}(x,y) = 
\begin{cases}
cxe^{-2y} &0\leq x\leq 1, y> 0 \\
c(2-x)e^{-2y} &1\leq x\leq 2, y> 0 \\
0 &\mathrm{else}
\end{cases}
$$

Find the value of $c$.
$$\underbrace{\int _1^2\int_0^\infty c(2-x) e^{-2y} \, dy \, dx}_{c/4} +  \underbrace{\int _0^1\int_0^\infty cxe^{-2y} \, dy \, dx}_{c/4} = 1 \implies c=2 $$
Are $X$ and $Y$ independent, are $X$ and $Y$ uncorrelated?

Hence, I calculated $f_X(x)=c=2$, but this does not make any sense because now $1 = \int_{-\infty}^{+\infty} f_X(x) \, dx= \int_0^2 f_X(x) \, dx= \Big[2x\Big]^2_0 = 4$.

REPLY [4 votes]: $X,Y$ are independent because the joint density is the product of a function of $x$ and a function of $y.$ A fortiori they are uncorrelated.