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?

  • $\begingroup$ 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. $\endgroup$ – 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.

The way it is written in that section is, admittedly, very confusing.

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  • $\begingroup$ Is the region bounded by $(x, y)$ or $(X,Y)$? Why? $\endgroup$ – user366312 Oct 1 '18 at 3:35
  • $\begingroup$ One is a pair of points, one is a pair of random variables. $\endgroup$ – Randall Oct 1 '18 at 3:36

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