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.

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

N <- 10000
X <- rnorm(N)
Y <- rnorm(N)
Z <- X + Y + rnorm(N)


cor(X,Z) # 0.57
cor(Y,Z) # 0.58
cor(X,Y) # 0.00

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 \(f(X) = X^2\).

U <- rnorm(10000)
U2 <- U^2
cor(U2,U) # 0 because the third (even) moment of the normal distribution is 0.
#However U and U^2 are dependent.