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 $0<x<y<1$.
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 : $F(x,y) = \int_{-\infty}^{x}\int_{-\infty}^{y} f(u,v)dvdu$
Thanks you in advance.
Best Answer
There are some issues here:
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:
$$\begin{array} \,F(x,y) &=\int\limits_{x'=-\infty}^{x}\int\limits_{y'=-\infty}^{y} 2 \, 1_{x<y}\, 1_{0<x<1}\, 1_{0<y<1}\, dy'\, dx' \\&= \int\limits_{x'=0}^{y}\int\limits_{y'=x'}^{y} 2\, dy'\, dx' \\ &= \int\limits_{x'=0}^{y}2(y-x')\, dx' \\ &= y^2 \end{array}$$