Support on Probability Functions I would like to ask a somewhat general question: How do I determine the correct support for a given function? I'm thinking in terms of piecewise functions, and with a focus on probability. I will give some examples to illustrate what I mean.
Consider the joint probability function:
$$f(x,y) = \begin{array}{lr}
xy, & -1 < x < 1, & 0 < y < a \\
0, & else
\end{array}$$
where "a" is chosen to make the function a valid pdf. If I want to find the joint CDF, I would evaluate the integral:
$$F(x,y) = \int_{-\infty}^{y} \int_{-\infty}^{x} uvdudv$$
Where u and v are dummy variables. However, as I understand it, since f(x,y) is zero outside the box proscribed by the bounds given, it follows that the correct integral would have the bounds:
$$F(x,y) = \int_{0}^{y} \int_{-1}^{x} uvdudv \\ \text{y<a & x<1}$$
Since any other part would be zero. Then suppose I want to find $$F_X(x)$$ in which case I would want to find F(x,y) where x is a constant and the limit of y goes to infinity. But since 0<y<a, does this mean that I actually would need to evaluate:
$$F_X(x) = F(x,a)$$
I've become considerably confused since my professor's notes are... rather jumbled and messy. I'm looking for resources that would be able to explain support and provide explicit examples of how to evaluate various integrals in the context of probability.

Given the posted density:
$$f_{XY}(x,y)=xy\mathbb{1}_{(-1;1)}(x)\mathbb{1}_{(0;a)}(y)$$
there's not any value of $a$ such that it is a valid joint density.
But worst....if you try to calculate $f_X$ you get
$$f_X(x)=\int_0^a xy dy=\frac{a^2}{2}x$$
Which is NEGATIVE in $x \in(-1;0)$

I'm looking for resources that would be able to explain support and provide explicit examples of how to evaluate various integrals in the context of probability.

The best book I have ever read on this matter is the Papoulis
