This problem is from the textbook by John Rice, "Mathematical Statistics and Data Analysis":
Let $X$ and $Y$ have the joint density function
$f(x,y)=k(x-y),$ for $0\le y\le x\le 1$ and $0$ elsewhere
Sketch the region over which the density is positive.
How would you do this question? How do you account for $k$? I should note that $k$ is some constant.
Best Answer
The region where $f(x,y)$ is positive is a triangle in the $(x,y)$ plane bounded by the lines $y=x$, and the $x$ axis, both between $x=0$ and $x=1$, and the line $x=1$ between $y=0$ and $y=1$.
To calculate $k$ use the fact that $k\int_0^1\int_0^x (x-y)dydx=1$.