Vector random variable

In probability theory, a vector random variable is a column vector of $n$ random variables:

$$\vec{X}=\left[\begin{array}{c}{X}_1\\ X_2\\ ⋮\\ X_n\end{array}\right]$$

This occurs when several quantities are of interest. Each event involving $\vec{X}$ has a region in ${\mathbb{R}}^n$. For a continuous vector random variable, the joint CDF is given by:

$$F_X(\vec{X})=F_{X_1{X}_2\cdots X_n}(x_1,x_2,\cdots ,x_n)=P(X_1\le x_1\wedge X_2\le x_2\cdots X_n\le x_n)$$

The PDF is given by:

$$f_X(\vec{x})=\frac{{\partial }^n{F}_X(\vec{X})}{\partial x_1\partial x_2\cdots \partial x_n}$$ $$1={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\cdots {\int }_{-\infty }^{\infty }{f}_X(\vec{x})d{x}_{1}d{x}_2\cdots d{x}_n$$

In the discrete case, the joint PMF is given by:

$$P_X(\vec{X})=P(X_1=x_1\wedge X_2=x_2\wedge \cdots \wedge X_n=x_n)$$

The conditional PMF is given by:

$$P_{X_n}(x_n|x_1{x}_2\cdots x_{n-1})=\frac{P_X(\vec{X})}{P_{X_1{X}_2\cdots X_{n-1}}(x_1,x_2,\cdots ,x_{n-1})}$$

If the RV is pairwise independent, then:

$$F_X(\vec{X})=F_{X_1}(x_1)F_{X_2}(x_2)\cdots F_{X_n}(x_n)$$ $$f_X(\vec{x})=f_{X_1}(x_1)\cdots f_{X_n}(x_n)$$