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.


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 $f_Y$.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.