OOOOH, thisiscoolthisiscoolthisiscool.

So today in my multivariate class we learned that two sets of variables are independent if and only if they are uncorrelated—but only if the variables come from a bivariate (or higher multivariate dimension) normal distribution. In general, correlation is not enough for independence.

ELABORATION!

Let X be defined as:

meaning that it is a matrix containing two sets of variables, X₁ and X₂, and has a bivariate normal distribution. In this case, X₁ and X₂ are independent, since they are uncorrelated (cov(X₁, X₂)=cor(X₁,X₂) = 0, as seen in the off-diagonals of the covariance matrix).

But what happens if X does not have a bivariate distribution? Now let Z ~ N(0,1) (that is, Z follows a standard normal distribution) and define X as:

So before we even do anything, it’s clear that X₁ and X₂ are dependent, since we can perfectly predict X² by knowing X¹ or vice versa. However, they are uncorrelated:

(The expected value of Z³ is zero, since Z is normal and the third moment, Z³, is skew.)

So why does zero correlation not imply independence, as in the first example? Because X² is not normally distributed (a squared standard normal variable actually follows the chi-square distribution), and thus X is not bivariate normal!

Sorry, I thought that was cool.