Editing Conditional independence (section)

==Conditional independence of random vectors==
Two [[random vector]]s <math>\mathbf{X}=(X_1,\ldots,X_l)^{\mathrm T}</math> and <math>\mathbf{Y}=(Y_1,\ldots,Y_m)^{\mathrm T}</math> are conditionally independent given a third random vector <math>\mathbf{Z}=(Z_1,\ldots,Z_n)^{\mathrm T}</math> if and only if they are independent in their conditional cumulative distribution given <math>\mathbf{Z}</math>. Formally:

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|equation = {{NumBlk||<math>(\mathbf{X} \perp\!\!\!\perp \mathbf{Y}) \mid \mathbf{Z} \quad \iff \quad F_{\mathbf{X},\mathbf{Y}|\mathbf{Z}=\mathbf{z}}(\mathbf{x},\mathbf{y}) = F_{\mathbf{X}\,\mid\,\mathbf{Z}\,=\,\mathbf{z}}(\mathbf{x}) \cdot F_{\mathbf{Y}\,\mid\,\mathbf{Z}\,=\,\mathbf{z}}(\mathbf{y}) \quad \text{for all } \mathbf{x},\mathbf{y},\mathbf{z}</math>|{{EquationRef|Eq.3}}}}
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where <math>\mathbf{x}=(x_1,\ldots,x_l)^{\mathrm T}</math>, <math>\mathbf{y}=(y_1,\ldots,y_m)^{\mathrm T}</math> and <math>\mathbf{z}=(z_1,\ldots,z_n)^{\mathrm T}</math> and the conditional cumulative distributions are defined as follows.

: <math>\begin{align}
F_{\mathbf{X},\mathbf{Y}\,\mid\,\mathbf{Z}\,=\,\mathbf{z}}(\mathbf{x},\mathbf{y}) &= \Pr(X_1 \leq x_1,\ldots,X_l \leq x_l, Y_1 \leq y_1,\ldots,Y_m \leq y_m \mid Z_1=z_1,\ldots,Z_n=z_n) \\[6pt]
F_{\mathbf{X}\,\mid\,\mathbf{Z}\,=\,\mathbf{z}}(\mathbf{x}) &= \Pr(X_1 \leq x_1,\ldots,X_l \leq x_l \mid Z_1=z_1,\ldots,Z_n=z_n) \\[6pt]
F_{\mathbf{Y}\,\mid\,\mathbf{Z}\,=\,\mathbf{z}}(\mathbf{y}) &= \Pr(Y_1 \leq y_1,\ldots,Y_m \leq y_m \mid Z_1=z_1,\ldots,Z_n=z_n)
\end{align}</math>