Let $$ f = \left\{ \begin{array} 2e^{-x-y}& \text{if } y > x > 0\\ 0 &\text{otherwise}, \end{array} \right. $$ why can't this function be factorized as a function of $x$ times a function of $y$ for all pairs $(x,y)$ in the domain?
Im struggling to see why it can't just be factorised as $2e^{-x}e^{-y}$?
Because this is not a correct description of $f(x,y)$ at, for example, the point $(3,1)$. The formula $2\exp(-x)\exp(-y)$ predicts a non-zero result, while the actual density $f(3,1)$ is $0$.
Remark: Somewhat informally, if the joint density of $X$ and $Y$ truly factors, then $X$ and $Y$ are independent. But in this example, $X$ and $Y$ are not independent. If we know that $X$ is bigger than $6$, then we know for sure that $Y$ cannot be less than $2$.
