Perfectly Correlated Random Variables Math, CS, Data
Suppose $X$ and $Y$ have finite nonzero variances $\sigma_X^2$ and $\sigma_Y^2$. Define $C$ to be the correlation between $X$ and $Y$. If $\vert C \vert=1$, then $Y$ must be an affine linear function of $X$ (and vice versa) with probability one. In particular, $Y = mX + b$ where
Consider the variance of $Y-mX$. By a simple variance formula, we can expand this into
The variance of the random variable $Y-aX$ is zero. A random variable with zero variance has to be a constant with probability one, so $Y-mX=b$ with probability one for some constant $b$. Taking expectations of both sides shows that $b$ must be $\E Y - m \E X$.
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