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I need to clarify something I came across in my calculations. I will appreciate any clarification or being pointed to the right literature to understand this concept.

Given the joint distribution of two random variables, $X$ and $Y$, each defined on $\mathbb{R}$, $$f_{x, y}(x, y)$$ which holds for only when $g(x,y) \le 0$, and zero otherwise. This condition $g(x, y)$ forces $X$ and $Y$ to jointly draw from $A$ and $B$, respectively.

Questions:

  1. Is it possible that the marginal distribution of, say $x$, $f_x(x)$ can draw from $C$ such that $A \subset C$, OR even $C \subset A$?
  2. In calculating the marginal distribution for $y$, which is correct: $$f_y(y) = \int_A f_{x, y}(x, y) dx$$ or $$f_y(y) = \int_C f_{x, y}(x, y) dx.$$
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You already have defined $f_{X,Y}$ to be zero outside of a particular region $S:=\{(x,y) : g(x,y) \le 0\} \subseteq \mathbb{R}^2$. The marginal is just $\int_{-\infty}^\infty f_{X,Y}(x,y) \, dx$. Of course, the integrand is zero for a lot of values, but that doesn't make this integral invalid.

What you seem to be interested in is the smaller set $S_1 := \{x : \text{there exists $y$ such that $(x,y) \in S$}\}$. Then $\int_{-\infty}^\infty f_{X,Y}(x,y) \, dx = \int_{S_1} f_{X,Y}(x,y) \, dx$. You can think of $S_1$ as what you would get if you looked at the $x$-coordinates of points in $S$.

You can define $S_2$ analogously, and you would have $S \subseteq S_1 \times S_2$; the left-hand side is your original set in $\mathbb{R}^2$, and the right-hand side is the smallest rectangle that contains it.


Update: I think you are asking for the relationship between the sets $\{x : f_{X,Y}(x,y) > 0 \text{ for some $y$}\}$ and $\{x : f_X(x) > 0\}$; the first set is the plausible set of possible $X$ values you could draw from the joint density, and the second set is the same but for the marginal density.

Since $f_X(x) = \int_{-\infty}^\infty f_{X,Y}(x,y) \, dy$, we have $(f_X(x) > 0) \implies (f_{X,Y}(x,y) > 0 \text{ for some $y$})$." So $$\{x : f_{X,Y}(x,y) > 0 \text{ for some $y$}\} \supseteq \{x : f_X(x) > 0\}.$$ However, the reverse inclusion need not hold. If for a specific value of $x$ there are only finitely many or countably many $y$ such that $f_{X,Y}(x,y) > 0$, then the integral $\int_{-\infty}^\infty f_{X,Y}(x,y) \, dy$ is zero.

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  • $\begingroup$ Thank you for quick response. I understand the first two paragraphs of your answer. My question is actually this: is it possible that $f_X(x)$ is not zero for region $S_3 := \{x: \text{there exists } y \text{ such that } S \subset (x, y)\}$? That is to say, the $x$ that are in $S_3$ can be larger than the $x$ that are in $S$. $\endgroup$
    – BoltzBooz
    Nov 3 '20 at 7:25
  • $\begingroup$ @BoltzBooz Can you clarify the definition of $S_3$? I don't understand the notation $S \subset (x,y)$; the left-hand side is a subset of $\mathbb{R}^2$, while the right-hand side is a point in $\mathbb{R}^2$. $\endgroup$
    – angryavian
    Nov 3 '20 at 7:32
  • $\begingroup$ Sorry for that. I am not used to mathematical notations. I meant to ask, is it possible that after deriving $f_X(x)$, that $f_X(x)$ is not zero for region $S_3 := \{x: x \in \mathbb{R} \}$ and that the $x$ that are in $S_3$ can be outside the bounding region of the valid $x$ in $S$? $\endgroup$
    – BoltzBooz
    Nov 3 '20 at 8:00
  • $\begingroup$ @BoltzBooz I've updated my answer $\endgroup$
    – angryavian
    Nov 3 '20 at 17:04

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