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I have a joint probability density function, $f(x,y)$. However, I have a constraint associated:

$0< x < y < +\infty$.

So, when I calculate the marginal probability densities, how do I factor in the constraints to the integrands for both $F_x$ and $F_y$? Should I draw out the shaded regions in $\mathbb R^2$?

The function in question is $3e^{-2x-y}$. I can integrate - that's child's play.

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1 Answer 1

In general the marginal density of, say, $X$ is given by integrating the joint density over the whole real line, i.e. $$ f_X(x)=\int_{-\infty}^\infty f_{X,Y}(x,y)\,\mathrm dy. $$ Now, if $x\leq 0$ is fixed, then $f_{X,Y}(x,y)=0$ for all $y\in\mathbb{R}$ and hence $$ f_X(x)=0,\quad \text{if }\;x\leq 0. $$ If $x>0$, then $f_{X,Y}(x,y)$ is zero for $-\infty<y\leq x$ and non-zero for $x<y<+\infty$. Thus $$ f_X(x)=\int_{-\infty}^\infty f_{X,Y}(x,y)\,\mathrm dy=\int_x^\infty f_{X,Y}(x,y)\,\mathrm dy. $$

A similar argument applies for finding $f_Y$.

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