Index: The Book of Statistical ProofsGeneral Theorems ▷ Probability theory ▷ Correlation ▷ Definition

Definition: The correlation of two random variables $X$ and $Y$, also called Pearson product-moment correlation coefficient (PPMCC), is defined as the ratio of the covariance of $X$ and $Y$ relative to the product of their standard deviations:

\[\label{eq:corr} \mathrm{Corr}(X,Y) = \frac{\sigma_{XY}}{\sigma_X \sigma_Y} = \frac{\mathrm{Cov}(X,Y)}{\sqrt{\mathrm{Var}(X)} \sqrt{\mathrm{Var}(Y)}} = \frac{\mathrm{E}\left[ (X-\mathrm{E}[X]) (Y-\mathrm{E}[Y]) \right]}{\sqrt{\mathrm{E}\left[ (X-\mathrm{E}[X])^2 \right]} \sqrt{\mathrm{E}\left[ (Y-\mathrm{E}[Y])^2 \right]}} \; .\]

Metadata: ID: D71 | shortcut: corr | author: JoramSoch | date: 2020-06-02, 20:34.