Independence (probability theory)
For real valued random vectors

Two random vectors X = (X 1, …, X m) T {\displaystyle \mathbf {X} =(X_{1},\ldots,X_{m})^{\mathrm {T} }} and Y = (Y 1, …, Y n) T {\displaystyle \mathbf {Y} =(Y_{1},\ldots,Y_{n})^{\mathrm {T} }} are called independent if

where F X (x) {\displaystyle F_{\mathbf {X} }(\mathbf {x})} and F Y (y) {\displaystyle F_{\mathbf {Y} }(\mathbf {y})} denote the cumulative distribution functions of X {\displaystyle \mathbf {X} } and Y {\displaystyle \mathbf {Y} } and F X, Y (x, y) {\displaystyle F_{\mathbf {X,Y} }(\mathbf {x,y})} denotes their joint cumulative distribution function. Independence of X {\displaystyle \mathbf {X} } and Y {\displaystyle \mathbf {Y} } is often denoted by X ⊥ ⊥ Y {\displaystyle \mathbf {X} \perp \!\!\!\perp \mathbf {Y} }.Written component-wise, X {\displaystyle \mathbf {X} } and Y {\displaystyle \mathbf {Y} } are called independent if
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