# finding CDF bounds of joint distribution

i'm doing exercise (i'm a student) for probability and there is something that I don't understand, How do we manage to find bound of $$CDF$$ given a $$PDF$$.

the PDF : $$f(x,y) = 2$$ , is obviously uniformly distributed have the constraint of $$0.

When I want to find the CDF I do this :

$$\int_{0}^{x}\int_{0}^{y} 2dvdu = \int_{0}^{x} 2y du = 2xy$$

I know that i'm wrong because $$x$$ and $$y$$ are dependant each other and here i'm counting also the part where where $$x>y$$ but I dont know to bound the integrals to respect the dependance. I followed the formula of a CDF given PDF : $$F(x,y) = \int_{-\infty}^{x}\int_{-\infty}^{y} f(u,v)dvdu$$

There are some issues here:

• The density is only positive when $$0 so you could use indicator variables, for example something like $$f(x,y)=2 \, 1_{x
• There can be confusion between the limits of integration and the variables being integrated over, so perhaps use something like $$F(x,y)=\int\limits_{x'=-\infty}^{x} \int\limits_{y'=-\infty}^{y} f(x',y')\, dy'\, dx'$$
• The CDF is going to come in pieces, depending on the values being sought, and you probably need to work them all out, such as
• $$0 \le y$$
• $$x \le 0 < y$$
• $$0 < x < y < 1$$
• $$0 < x < 1 \le y$$
• $$0 < y \le x < 1$$
• $$0 < y \le 1 \le x$$
• $$1 \le x < y$$
• $$1 < y \le x$$

though the first two are easy ($$0$$) as are the last two ($$1$$). Here is an example for $$0 < y < x < 1$$ (concentrate on the limits in the second line and see why they take these values), and remember some of the others will be different:

$$\begin{array} \,F(x,y) &=\int\limits_{x'=-\infty}^{x}\int\limits_{y'=-\infty}^{y} 2 \, 1_{x

• Thanks you for you answer Henry :), but I didn't how did you replaced -inf to x and -inf to 0 in your last step. Commented Dec 2, 2021 at 17:50
• @BilalBrarou The indicator variables are $0$ for $x \le 0$ or $y \le 0$ or $y \le x$ so the pdf is $0$ too and you can drop those parts of the integration Commented Dec 2, 2021 at 17:55
• Oh ok I understand now thx. Commented Dec 2, 2021 at 17:59