How to choose the bounds of integration for the density function.

Question

$X $and $Y$ are two random variables with joint density $f_{X,Y}(x,y) = c(x+y)$ where $0 < x < y < 1$.

To find $c$, I know that I must integrate the density and equate that to one to find c i.e

$\iint c(x+y)$ = 1 but I am struggling on working out the bounds in which to integrate over. Can anyone clarify a general idea of how I choose the bounds?

Much appreciated.

Answer 1

The collection of points $R=\{(x,y)\in\mathbb{R}^2\mid 0<x<y<1\}$ is shown in green in the picture below. Note that for each fixed value of $x$, the values of $y$ ranges from $x$ to $1$. Meanwhile $x$ ranges from $0$ to $1$. Hence $$ 1=\int_{R} f\, dA=\int_0^1\int_{y=x}^1 c(x+y)\,dy\,dx $$ which you can compute and find the value of $c$.

Answer 2

Hint: $$0 < x < y < 1 \implies 0 < y < 1 \text{ and } 0 < x < y.$$