If I have a continuous joint distribution function $f(x,y) = 12y^{2}$ for $0 \leq x \leq y \leq 1$ and $0$ everywhere else. How do I determine the limits of the integration, while evaluating expectation $E[X,Y]$? In short how do I handle inequalities of the type $0 \leq x \leq y \leq 1$?
The books that I refer to directly write the integrals, but I want to know what the inequality means. Thanks
It is a standard double integration. Your support is given by $0\le x\le y \le 1$, i.e., a triangle with vertices $(0,0)$, $(1,1)$ and $(0,1)$. Your density function is simple so it doesn't matter the order of integration. But generally, you should take into consideration what order will be more convenient. Lets say we start from $x$, so if you go from the $y$ axis in the $x$ "direction", you get that $x$ goes from $0$ to $x=y$, i.e., $\int \int_{x\in[0,y]}f_{X,Y}(x,y)dxdy$ and $y$ is in $[0,1]$, thus $$ \int_{y\in[0,1]} \int_{x\in[0,y]}f_{X,Y}(x,y)dxdy\, . $$