Regarding independence of random variables I am just wondering if there is something trivial that I have missed regarding this.
If I've understood it right, two random variables are called independent if and only if:  $f(x_1,x_2)=f_{x_1}(x_1)*f_{x_2}(x_2)$ 
Where the left hand side denotes the joint density function, and the right hand side denotes the product of the marginal densities.
But when I am asked to determine if two random variables X and Y are independent given the joint density function, I use the definition above. Yet it seems that every exercise regarding this, is followed up by another question where they ask for the marginal densities. 
If Im using the definition above I've already calculated the marginal densities, and hence solved the follow-up exercises. So am I doing something wrong, or is the book just confusing me?
 A: Actually the condition that you mention is sufficient but is not necessary. This because densities are in not determined by the distribution.
If $f$ is a density for the distribution and $A$ is a Borel-measurable subset of $\mathbb R^2$ with Lebesgue measure $0$ then also $f1_{A^{\complement}}$ will work as density. That gives you the freedom to find a density $f(x_1,x_2)$ and next to that marginal densities $f_{1},f_{2}$ with: $$\{(x_1,x_2)\in\mathbb R^2\mid f(x_1,x_2)\neq f_1(x_1)f(x_2)\}\neq\varnothing$$
in spite of independence.
So you can use the formula for proving independence but you better not use it when it comes to defining independence.
A: Your definition of independent random variables is missing the important requirement that (for jointly continuous random variables),
$$f_{X,Y}(x,y) = f_X(x)f_Y(y) ~ \textbf{for all real numbers}~x~\text{and}~y.\tag{1}$$
In some cases, it is possible to determine that $X$ and $Y$ are dependent random variables without determining the marginal densities and verifying 
whether $(1)$ is satisfied or not.  For example, if the support of $f_{X,Y}$ is not a rectangular region with sides parallel to the axes (e.g. the support is a rotated square with vertices $(1,0), (0,1),(-1,0), (0,-1)$ or the triangle with vertices $(0,0), (1,1), (0,1)$), then $X$ and $Y$ are dependent random variables.  This is essentially noting (visually after sketching the support on a piece of paper or in one's mind) that there are points $(x,y)$ for which $f_{X,Y}(x,y) = 0$ while both $f(x)$ and $f_Y(y)$ are clearly nonzero, and so $(1)$ obviously does not hold. In an answer to a different question, I have called this the eyeball test for dependence.
Try it for $(x,y) = \left(\frac 34, \frac 34\right)$ in the example cited above.
In short, there can be good reason for asking for the actual marginal densities from those who used their imagination as described above.
