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CAF123
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Homework Statement
The joint probability density function of ##X## and ##Y## is given by $$f(x,y) = \frac{1}{8}(y^2 - x^2)e^{-y},\,\,\,\, x \in\,[-y,y]\,\,,y \in\,(0, \infty)$$
Compute the marginal densities of ##X## and ##Y##.
The Attempt at a Solution
I know the defintions are $$ F_X(x) = \int_{- \infty}^{\infty}\,f(x,y)\,dy\, \text{and}\, F_Y(y) = \int_{- \infty}^{\infty}\,f(x,y)\,dx.$$
Am I correct in saying that if the domains of x and y are just numbers then the limits on the integrals are the endpoints of these domains? I.e if in the example above ##x \in\, [2,4], \,\,\text{and}\,\, y\,\in\,[1,3] F_Y(y) = \int_2^4 f(x,y)\,dx,\,\,\text{while}\,\, F_X(x) = \int_1^3 f(x,y)\,dy##, right?
However, in this case, the domain of x depends on y. So x is less than y=x and greater than y = -x. Drawing this, I see it is the portion enclosed by x = |y|, but above the x-axis (since y is greater than 0). Hence, to get the limits when computing ##F_X(x)##, I should say y is from x to infinity. When I check the answers, they have y from |x| to infinity. Can't I replace |x| = x since y is above 0? Or did I miss something else? Many thanks