📂Probability Theory

Joint and Marginal Distributions Defined by Measure Theory

Definition ¹

Let’s assume that a probability space $( \Omega , \mathcal{F} , P)$ is given.

1. Joint Distribution: If there are two random variables $X$ and $Y$ defined in $( \Omega , \mathcal{F} , P)$, the distribution of the random vector $(X,Y) : \Omega \to \mathbb{R}^2$ for a Borel set $B \subset \mathcal{B} \left( \mathbb{R}^2 \right)$ is defined as $$ \begin{align*} P_{(X,Y)} (B) :=& P \left( (X,Y) \in B \right) \\ =& \int_{B} f_{(X,Y)} (x,y) d m_{2} (x,y) \end{align*} $$ and if there exists $f_{(X,Y)}$ that satisfies this, $X$ and $Y$ are said to have a joint density.
2. Marginal Distribution: For a Borel set $A \subset \mathbb{R}$, the following is referred to as the marginal distribution: $$ P_{X} (A) := P_{(X,Y)} ( A \times \mathbb{R} ) \\ P_{Y} (A) := P_{(X,Y)} ( \mathbb{R} \times A ) $$

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• If you haven’t encountered measure theory yet, you can ignore the term “probability space.”

Formulas

Introducing the formula for the sum of two random variables $X+Y$. Since the sum of random variables directly leads to the concept of the average, its importance is considered to be significant.

• For $X$ and $Y$ that have a joint density, marginal density is found as follows. $$ f_{X} (x) = \int_{\mathbb{R}} f (X,Y) (x,y) dy \\ f_{Y} (y) = \int_{\mathbb{R}} f (X,Y) (x,y) dx $$

• If $X$ and $Y$ have a joint density $f_{X,Y}$, $$ f_{X+Y} (z) = \int_{\mathbb{R}} f_{X,Y} (x , z - x) dx $$

Derivation

If we define $y ' = x + y$, according to the Fubini’s theorem, $$ \begin{align*} f_{X+Y} (z) =& P ( X+Y \le z ) \\ =& P_{X,Y} \left( \left\{ (x,y) : x + y \le z \right\} \right) \\ =& \iint_{ \left\{ (x,y) : x + y \le z \right\} } f_{X,Y} (x,y) dx dy \\ =& \int_{\mathbb{R}} \int_{- \infty}^{z-x} f_{X,Y} (x,y) dy dx \\ =& \int_{- \infty}^{z} \int_{\mathbb{R}} f_{X,Y} (x,y ' - x) dx dy ' \end{align*} $$

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