# My Favorite One-Liners: Part 122

Once in my probability class, a student asked a reasonable question — could I intuitively explain the difference between “uncorrelated” and “independent”? This is a very subtle question, as there are non-intuitive examples of random variables that are uncorrelated but are nevertheless dependent. For example, if $X$ is a random variable uniformly distributed on $\{-1,0,1\}$ and $Y= X^2$, then it’s straightforward to show that $E(X) = 0$ and $E(XY) = E(X^3) = E(X) = 0$, so that

$\hbox{Cov}(X,Y) = E(XY) - E(X) E(Y) = 0$

and hence $X$ and $Y$ are uncorrelated.

However, in most practical examples that come up in real life, “uncorrelated” and “independent” are synonymous, including the important special case of a bivariate normal distribution.

This was my expert answer to my student: it’s like the difference between “mostly dead” and “all dead.”

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