Calculating with Joint density

Question

Given two random variables $X$ and $Y$ with the joint density

$$f(x,y) = axy \mathbb{1}_{\{X>0, Y>0, X+Y \leq 1\}},$$

how can one calculate the parameter $a$?

Answer 1

The integral over the region where $1_{\{x>0,\, y>0,\, x+y\le 1 \}}=0$ is $0$, so you just need to integrate over the region where $x>0,$ $y>0,$ and $x+y\le1.$ You can write $$ \int_0^1 \cdots\cdots \,dx $$ and then inside that you got an integral with respect to $y$.

For any fixed value of $x$ between $0$ and $1$, you have $y$ going from $0$ to $1-x,$ so that $x+y$ remains $\le 1.$ Thus you have $$ \int_0^1 \left( \int_0^{1-x} \cdots\cdots \, dy \right) dx. $$ (And where the dots are you need $axy.$)

(You can also do it with the roles of $x$ and $y$ interchanged, getting $\displaystyle \int_0^1 \left( \int_0^{1-y}\cdots\cdots \, dx \right) dy$.)