Calculating joint cdf from joint pdf

Question

I have the following joint distribution function

$f_{xy} = (12/5)(x+y^2)$ on $0<y<x<1$ (zero everywhere else)

I know I must integrate along both $y$ and $x$ to find this:

$$F(x,y) = \int \int f(x,y) \, dx\, dy$$

But I'm unsure about what those bounds are. I know the triangular area defined by the area is bounded by $y=x,$ $x=1,$ and $y=0,$ but unsure how that helps me out here.

I'm aware about how the cdf is defined for every pair $(x,y)$ of real numbers. So that means there will be two cases here.

Case 1 $(y>x)$: Here $Y∈[0,x]$ and $X∈[Y,x].$

Case 2 $(y≤x)$: I'm having difficulty figuring out what the bounds here would be.

The integration for case 1 feels a little straightforward but I'm a little lost on approaching Case 2

It's been a while since I've done this so I'm a bit rusty but any tips or advice would be great! Thank you

Answer 1

In order to find $F(x,y)$ you have to discern the following cases:

• $x\geq1\wedge y\geq1$ then $F(x,y)=1$
• $x<0\vee y<0$ then $F(x,y)=0$
• $x\geq 1\wedge 0<y<1$
• $0<x<1\wedge y\geq x$
• $0<x<1\wedge y<x$

Make a picture to get some grip on this information.