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I'm trying to find $f_x$ and $f_y$ given a joint probability distribution $$f(x,y) = \frac18 (y^2 -x^2)e^{-y}$$ defined on the interval $0 \leq y \leq \infty$, $-y \leq x \leq y$

Naturally I've tried integrating on the intervals and found:

$$ f_x(x) = \int_0^\infty \mathrm{d}y\ f(x,y)= \frac18 (2 -x^2)$$ $$ f_y(y) = \int_{-y}^y \mathrm{d}x\ f(x,y)= \frac16 e^{-y}y^3$$

But $f_x$ isn't normallized to one.

I believe the error is on the integration interval but I don't know what exactly is wrong. Can anybody point the correct direction?

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  • $\begingroup$ Hint: Start from the true joint density, namely, $$f(x,y)=\tfrac18(y^2-x^2)e^{-y}\mathbf 1_{|x|<y},$$ and apply the fully general formula $$f_X(x)=\int_\mathbb Rf(x,y)\mathrm dy.$$ In the present case, $$f_X(x)=\int_\mathbb R\tfrac18(y^2-x^2)e^{-y}\mathbf 1_{|x|<y}\mathrm dy=\int_{|x|}^\infty\tfrac18(y^2-x^2)e^{-y}\mathrm dy=\cdots$$ $\endgroup$ – Did May 25 '15 at 22:47
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It always helps to sketch the support of the joint distribution; i.e., the region of $(X,Y)$ such that the density is positive. In your case, you can see that this region must be in the upper half plane ($y \ge 0$), and for a given value of $y$, $x$ must be between $-y$ and $y$; so this region is triangular and is bounded to the right by $y = x$ and to the left by $y = -x$.

Thus, the marginal density of $X$ is given by the integral $$f_X(x) = \int_{y=|x|}^\infty f_{X,Y}(x,y) \, dy$$ since we require $y$ to be at least as large as $|x|$ (otherwise the joint density is zero). This corresponds to picking an $x$-value, drawing a vertical line, and integrating over a $y$-interval that corresponds to the intersection of this vertical line and the support of the joint distribution; in a sense, it is like "collapsing" the support onto the $x$-axis, where the marginal density at each $x$-value is the integral of the joint density for all corresponding $y$-values.

The marginal density of $Y$ is easier to see: $$f_Y(y) = \int_{x=-y}^y f_{X,Y}(x,y) \, dx$$ as you wrote. This is analogous to "collapsing" the support onto the $y$-axis, "summing" up the joint density in horizontal rather than vertical lines.

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