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@jemoka / Jemoka Knowledge Base / raw/concept/kbhcovariance.md
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--- title: "covariance (probability)" source: https://www.jemoka.com/posts/kbhcovariance/ --- \begin{equation} cov(x,y) = E[(X-E[X])(Y-E[Y])] = E[XY]-E[X]E[Y] \end{equation} (the derivation comes from FOIling the two terms and applying properties of expectation. we want to consider: if a point goes way beyond its expectation, does the corresponding point change for another? \begin{equation} (x-E[x])(y-E[y]) \end{equation} if both points are varying . Instead of using this unbounded value, we sometimes use a normalized value named correlation: \begin{equation} \rho(X,Y) = \frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}} \end{equation} we can express this in terms of a covariance matrix