📂Mathematical Statistics

Multivariate Probability Distributions in Mathematical Statistics

Definition ¹

1. A Random Vector is defined as $X = (X_{1} , \cdots , X_{n})$ for $n$ number of probability variables $X_{i}$ defined in sample space $\Omega$. The range $X(\Omega)$ of $X$ is also called a space.
2. A function that satisfies the following $F_{X} : \mathbb{R}^{n} \to [0,1]$ is called the Joint Cumulative Distribution Function of $X$. $$ F_{X}\left( x_{1}, \cdots , x_{n} \right) := P \left[ X_{1} \le x_{1} , \cdots , X_{n} \le x_{n} \right] $$
3. If there exists a function satisfying the following for some $h_{1} , \cdots , h_{n} >0$, it is known as the Moment Generating Function of $X$. $$ M_{X} (t_{1}, \cdots , t_{n}) := E \left[ e^{\sum_{k=1}^{n} t_{k} X_{k} } \right] = E \left[ \prod_{k=1}^{n} e^{t_{k} X_{k}} \right] \\ |t_{1}| < h_{1} , \cdots , |t_{n} | < h_{n} $$

Discrete

• D1: If the space of $X$ is a countable set, $X$ is said to be a Discrete Random Vector.
• D2: The following $p_{X} : \mathbb{R}^{n} \to [0,1]$ is called the Joint Probability Mass Function of the discrete random vector $X$. $$ p_{X} (x_{1} , \cdots , x_{n}) := P \left[ X_{1} = x_{1} , \cdots , X_{n} = x_{n} \right] $$
• D3: The following $P_{X_{k}} (x_{k})$ about $1 \le k \le n$ is called the Marginal Probability Mass Function. $$ P_{X_{k}} (x_{k}) := \sum_{x_{1}} \cdots \sum_{x_{k-1}}\sum_{x_{k+1}} \cdots \sum_{x_{n}} p_{X} (x_{1} , \cdots , x_{n}) $$
• D4: $S_{X}:= \left\{ \mathbb{x} \in \mathbb{R}^{n} : p_{X}(\mathbb{x}) > 0 \right\}$ is referred to as the Support of $X$.

Continuous

• C1: If the cumulative distribution function $F_{X} = F_{X_{1} , \cdots , X_{n}}$ of probability variable $X$ is continuous at all $\mathbb{x} \in \mathbb{R}^{n}$, then $X$ is considered a Continuous Random Vector.
• C2: The following $f_{X} : \mathbb{R}^{n} \to [0,\infty)$ is known as the Joint Probability Density Function of the continuous random vector $X$. $$ F_{X} (x_{1}, \cdots, x_{n}) = \int_{-\infty}^{x_{1}} \cdots \int_{-\infty}^{x_{n}} f_{\mathbb{x}} (t_{1} , \cdots , t_{n}) dt_{1} \cdots d t_{n} $$
• C3: The following $f_{X_{k}} (t_{k})$ about $1 \le k \le n$ is known as the Marginal Probability Density Function. $$ f_{X_{k}}(t_{k}) := \int_{\infty}^{x_{1}} \cdots \int_{\infty}^{x_{k-1}} \int_{\infty}^{x_{k+1}} \cdots \int_{\infty}^{x_{n}} f_{X}(t_{1} , \cdots , t_{n}) dt_{1} \cdots d_{k-1} d_{k+1} \cdots d_{n} $$
• C4: $S_{X} := \left\{ \mathbb{t} \in \mathbb{R}^{n} : f_{X} ( \mathbb{t} ) > 0 \right\}$ is referred to as the Support of $X$.

---

• Originally, Random Vector is translated as 확률 벡터(Random Vector), but to avoid confusion with terms like Stochastic or Probabilistic after graduating high school, it is kept in its original wording.
• Originally, Joint Cumulative Distribution Function is translated as 결합 확률 분포, but to avoid potential confusion with independence or dependence, it is kept in its original wording.
• Originally, Marginal Distribution is translated as 주변 분포, but similar to how Marginal in economics might not convey its meaning well, it is kept in its original wording.

Explanation

Multivariate probability distribution is a generalization of univariate probability distribution to multiple dimensions, and while it inherently differs due to having multiple variables, at least at the undergraduate level of mathematical statistics, it can sufficiently differ through calculus skills. Let’s take a look at how they differ:

• 1: What should not be confused is that the random vector $X : \Omega^{n} \to \mathbb{R}^{n}$ is still a function. Hence, its range can be thought of, and through this, they can be classified into discrete and continuous types regarding multivariate.
• C2: Continuous joint density functions are generally defined to meet the following, excluding $A \subset \mathbb{R}^{n}$ where the probability is $0$, according to the fundamental theorem of calculus. $$ {{ \partial^{n} } \over { \partial x_{1} \cdots \partial x_{n} }} F_{X} (\mathbb{x}) = f(\mathbb{x}) $$
• D3, C3: Although the equation is complex, simply put, it changes the joint probability distribution exclusively to the distribution concerning probability variable $X_{k}$. Contrary to how the word marginal in economics corresponds with the concept of differentiation, in mathematical statistics, it involves integrating or summing up to eliminate variables of no interest.