Obtaining marginal densities from joint probability density function

Question

I'm given a joint probability density function of X and Y.

$$f(x, y)=c \mathrm{e}^{-y}|y-x|, \quad y>0 \text { and }-1 \leq x \leq 1$$

After normalizing I obtain $c = \frac{1}{\frac{-2}{e}+3}$. Now I am interested in the marginal densities of X and Y. How do I obtain these? I know to integrate with respect to each of the variables.

Answer 1

Guide:

If we want to compute $f_X(x)$, we are fixing the $x$ and we want to integrate over the $y$. Hence we should integrate from $0$ to $\infty$.

If we want to compute $f_Y(y)$, we are fixing the $y$ and we want to integrate over the $x$. Hence we should integrate from $-1$ to $1$.

Answer 2

You have found $c$ by using: $$\begin{align}1&=\iint_{y>0, -1\leq x\leq 1} f_{X,Y}(x,y)~\mathrm d (x,y)\\[1ex]&= \int_{-1}^1\int_0^\infty c\, \mathrm e^{−y} \lvert y−x\rvert~\mathrm d y~\mathrm d x\\[1ex]&=\int_{-1}^1\left(\int_0^x c\mathrm e^{-y}(x-y)\mathrm d y+\int_x^\infty c\mathrm e^{-y}(y-x)\mathrm d y\right)~\mathrm d x\end{align}$$

Surely you can then see that:

$$f_X(x)=\int_{y>0} f_{X,Y}(x,y)~\mathrm d y\\f_Y(y)=\int_{-1\leq x\leq 1} f_{X,Y}(x,y)~\mathrm d x$$

Brought to you by the fact that the supports are seperable.