# Joint and Marginal distribution

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 $$f_y(y) = \int_C f_{x, y}(x, y) dx.$$

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.

• 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 $S$. Nov 3 '20 at 7:25
• @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 $\mathbb{R}^2$. Nov 3 '20 at 7:32
• 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 $S$? Nov 3 '20 at 8:00
• @BoltzBooz I've updated my answer Nov 3 '20 at 17:04