# How does “$F_Z(z) = \iint_{D_z}f_{XY}(x,y)\,dx\,dy$” read in plain English?

See: this book. Page-123.

How does the following integral notation read in plain English?

$$F_Z(z) = \iint_{D_z}f_{XY}(x,y)\,dx\,dy$$ original image

As far as I can tell,

The value of $$F_Z(z)$$ is equal to the value of integration of the function $$f(x, y)$$ over the rectangular region $$Dz$$, where $$Dz$$ is the region bounded by $$(X, Y)$$

1. Am I correct?

2. Is the region $$(X, Y)$$ rectangular?

• I certainly see no reason to assume the region is rectangular. Also it makes no sense to have a region bounded by a pair of random variables. – spaceisdarkgreen Oct 1 '18 at 3:32

Parsing the language down to your special case, on line 4.10 the text defines $$D_z$$ as the set of points $$(x,y)$$ with $$f_{XY} (x,y) \leq z$$, $$f_{XY}$$ being the joint density. This need not be rectangular at all.
• Is the region bounded by $(x, y)$ or $(X,Y)$? Why? – user366312 Oct 1 '18 at 3:35