Multivariate Distribution

Definition

Joint CDF

The Distribution Function $F_{\mathbf{X}}:{\mathbb{R}}^n\to [0,1]$ of a Random Vector $\mathbf{X}=(X_1,\dots ,X_n{)}^{⊺}$ is defined as $F_{\mathbf{X}}(\mathbf{x})=P(\mathbf{X}\le \mathbf{x})=P({\mathbf{X}}_1\le x_1,\dots ,{\mathbf{X}}_n\le x_n)$ where $\mathbf{x}=(x_1,\dots ,x_n{)}^{⊺}$

Joint PDF

$p_{X_1,\dots ,X_n}(x_1,\dots ,x_n)=P(X_1=x_1,\dots ,X_n=x_n)$

The joint probability Distribution Function of $n$ discrete Random Vector $X_1,X_2,\dots ,X_n$

$f_{X_1,\dots ,X_n}(x_1,\dots ,x_n)=\frac{{\partial }^n{F}_{X_1,\dots ,X_n}(x_1,\dots ,x_n)}{\partial x_1\dots \partial x_n}$

The joint probability Distribution Function of $n$ continuous Random Vector $X_1,X_2,\dots ,X_n$

Expected Value of a Multivariate Function

Continuous

E[g(\mathbf{X})] = \idotsint\limits_{x_{n} \dots x_{1}} g(\mathbf{x})f(\mathbf{x})dx_{1} \dots \dots dx_{n}

Discrete

$E[g(\mathbf{X})]=\sum _{x_n}\cdots \sum _{x_1}g(\mathbf{x})f(\mathbf{x})$

Marginal Distribution of a Multivariate Function

Marginal CDF

$F_{X_1}(x_1)=\underset{x_2,\dots ,x_n\to \infty }{\lim }{F}_{\mathbf{X}}(\mathbf{x})$

Marginal PDF

f_{X_{1}}(x_{1}) = E[g(\mathbf{X})] = \idotsint\limits_{x_{n} \dots x_{1}} f(x_{2}, \dots, x_n)dx_{2} \dots \dots dx_{n}

Conditional Distribution of a Multivariate Function

$f_{2,\dots ,n|1}(x_2,\dots ,x_n|x_1)=\frac{f_{\mathbf{X}}(\mathbf{x})}{f_{X_1}(x_1)}$

$f_{1|2,\dots ,n}(x_1|x_2,\dots ,x_n)=\frac{f_{\mathbf{X}}(\mathbf{x})}{f_{X_2,\dots ,X_n}(x_2,\dots ,x_n)}$

Properties

Linearity

If $\exists E[g_1(X_1,X_2)],E[g_2(X_1,X_2)]$, then $E[k_1{g}_1(X_1,X_2)+k_2{g}_2(X_1,X_2)]=k_{1}E[g_1(X_1,X_2)]+k_{2}E[g_2(X_1,X_2)]$