I have a basic question. Say I have two random variables, $X$ and $Y$. I can standardize them by subtracting the mean and dividing by the standard deviation, i.e., $X_{standardized} = \frac{(X – E(X))}{(SD(X))}$.
Is the correlation of $X$ and $Y$, $Cor(X, Y)$, the same as the covariance of the standardized versions of $X$ and $Y$? That is, is $Cor(X, Y) = Cov(X_{standardized}, Y_{standardized})$?
Best Answer
$$\begin{align} \operatorname{corr}(X,Y)&=\frac{E\Big((X-E(X))\times(Y-E(Y))\Big)}{SD(X)\times SD(Y)}\\ \operatorname{Cov}(X_{\text{standardized}}, Y_{\text{standardized}}) &=E\Bigg[\Bigg(\frac{(X - E(X))}{(SD(X))}-0\Bigg)\times\Bigg(\frac{(Y - E(Y))}{(SD(Y))}-0\Bigg)\Bigg]\\ &= \frac{E\Big((X-E(X))\times(Y-E(Y))\Big)}{SD(X)\times SD(Y)} \end{align}$$So, Yes!