# Correlation and Independence

## If X and Y are correlated to a third random variable Z, does it mean that X and Y are also correlated? Answer is no.

Correlation plots.

$X \sim \mathcal{N}(0,1), Y \sim \mathcal{N}(0,1), Z = X + Y + \epsilon \text{, where } \epsilon \sim \mathcal{N}(0,1)$

## Can dependent variables be zero correlated? Answer is yes.

\begin{align} U & \sim \mathcal{N}(0,1) \\ cov(U, U^2) & = E[U^3] = 0\end{align}

by definition of the odd central moments of the Normal Distribution. Therefore, $U$ and $U^2$ are strictly uncorrelated but they remain dependent through the function $f(X) = X^2$.

Written on March 15, 2016