Why is the following true for the covariance $cov(X, Y)$ and variances $Var(X)$, $Var(Y)$ of two rvs $X$ and $Y$: $$ \frac{Cov(c X, Y)}{\sqrt{Var(c X) Var(Y)}} = \frac{c Cov(X, Y)}{\sqrt{c^2 Var(X) Var(Y)}} = \frac{Cov(X, Y)}{\sqrt{Var(X) Var(Y)}} $$ I cannot see how the $c^2$ within the square root can disappear just because we have $c$ as a factor in the numerator.
2026-04-02 20:38:41.1775162321
Correlation of random variables after multiplication by a constant
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Recall that $\sqrt{ab} = \sqrt{a}\sqrt{b}$. Then,
$$\frac{c Cov(X, Y)}{\sqrt{c^2 Var(X) Var(Y)}} = \frac{c Cov(X, Y)}{\sqrt{c^2} \sqrt{ Var(X) Var(Y)}}= \frac{c Cov(X, Y)}{|c|\sqrt{Var(X) Var(Y)}} = \mathrm{sgn}(c)\frac{Cov(X, Y)}{\sqrt{ Var(X) Var(Y)}}$$