Writing out cdf for joint pdf

Question

Given over the region $0<x<2, 0<y<2, F(x,y)=\frac{1}{44}x^3y+\frac{3}{176}x^2y^2+\frac{1}{44}xy^3$

And we extend the its definition over the entire xy-plane:

$F(x,y)=0, x\leq0 , or, y\leq0$

$F(x,y)=1, x\ge2 , or, y\ge2$

$x\ge2, 0<y<2, F(x,y)=F(2,y)=\frac{2}{11}y+\frac{3}{44}y^2+\frac{1}{22}y^3$

$y\ge2, 0<x<2, F(x,y)=F(x,2)=\frac{2}{11}x+\frac{3}{44}x^2+\frac{1}{22}y^3$

then $F(x,y)$ is written as:

$ F(x,y)=\begin{cases} 0 & x\leq0 , or, y\leq0\\ \frac{1}{44}x^3y+\frac{3}{176}x^2y^2+\frac{1}{44}xy^3 & 0<x<2, 0<y<2 \\ \frac{2}{11}y+\frac{3}{44}y^2+\frac{1}{22}y^3 &x\ge2, 0<y<2 \\ \frac{2}{11}x+\frac{3}{44}x^2+\frac{1}{22}y^3 & y\ge2, 0<x<2 \\ 1 & x\ge2 , or, y\ge2 \end{cases} $

What I don't get is what order to write those functions in the cdf. Because for non joint cdf I just write them up in order of the intervals. Like in this case, obviously the 0 and 1 conditions are written at the top and bottom respectively but in this case I don't know whether I should be writing $F(x,2)$ before $F(2,y)$ as such. Cause the region is a square and I don't believe thinking in terms of what region covers the most area makes sense.

For example I don't know why it couldn't be written like this:

$ F(x,y)=\begin{cases} 0 & x\leq0 , or, y\leq0\\ \frac{2}{11}y+\frac{3}{44}y^2+\frac{1}{22}y^3 &x\ge2, 0<y<2 \\ \frac{1}{44}x^3y+\frac{3}{176}x^2y^2+\frac{1}{44}xy^3 & 0<x<2, 0<y<2 \\ \frac{2}{11}x+\frac{3}{44}x^2+\frac{1}{22}y^3 & y\ge2, 0<x<2 \\ 1 & x\ge2 , or, y\ge2 \end{cases} $

Answer 1

• You probably mean $x\ge2$ and $y \ge 2$, not or, otherwise you have intersecting regions.
• The order in which you define the branches does not matter whatsoever, as long as the regions of definition do no intersect.
• You computed $F(x,2)$ incorrectly since this is not supposed to depend on $y$ at all and your expression does.