If $X$ and $Y$ are two independent random variables: Then why is it true that
$$\rho (x,y)=\rho_x(x)\times\rho_y(y)$$
I just don't understand if the variables are independent; what justifies their probability densities to be written as a product?
I can see from the vote counts that people want to close this question as it appears unclear; and on reflection I too agree that it is a little unclear.
So in response to the comments below, I acknowledge that it can't be proved, all I am looking for here is a little motivation for the formula. I have looked on-line and they just state the formula without logic or reasoning (which is not helpful).
Distribution theory is not my strong point so I am somewhat unfamiliar with this formula; If someone could simply post a link that explains it a bit more intuitively (or give an intuitive answer) I would be most grateful.
The definition I'm familiar with is: random variables $X$ and $Y$ are independent if for all real numbers $a, b$, $$ \mathbb P(X \le a, Y \le b) = \mathbb P(X \le a) \mathbb P(Y \le b)$$
EDIT:
In case $X$ and $Y$ have densities, $$ \mathbb P(X \le a) \mathbb P(Y \le b) = \int_{-\infty}^a dx\; f_X(x) \int_{-\infty}^b dy\; f_Y(y) = \int_{-\infty}^a dx \int_{-\infty}^b dy\; f_X(x) f_Y(y) $$ which says that $f_X(x) f_Y(y)$ is a joint density for $X$ and $Y$.