I understand the basic principles of bivariate distributions and their marginal counterparts. I am stuck on a slightly more sophisticated question however.

Given the following bivariate distribution:

$$ f(x,y) = \begin{cases} q &\quad 0 \leq x \leq 1, \ 0 \leq y \leq x\\ 0 &\quad\text{otherwise} \\ \end{cases} $$

determine the parameter $q$ and the marginal distributions $f_X(x), f_Y(y)$.

I am aware that the marginal distributions in this case would be: \begin{align} f_Y(y) &= \int^{1}_{y} f(x,y)dx \\[8pt] f_X(x) &= \int^{x}_{0} f(x,y)dy \end{align} But I am not sure how I can go about calculating $q$ to get to that point.

  • 1
    $\begingroup$ Hint: the total volume trapped between the $f(x,y)$ surface and the $x$-$y$ plane is $1$. So, $q$ equals the multiplicative inverse of the area of the support of $f(x.y)$. $\endgroup$ – Dilip Sarwate May 22 '18 at 21:39