Normally distributed and uncorrelated does not imply independent | Hacker News Reader

Normally distributed and uncorrelated does not imply independent

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24 points666_howitzer9 comments

True. Normal distribution is a way a set of data is distributed. That can never imply independance. Think about a scenario: there is no correlation between growth in sales revenue and growth in website traffic. But that does not mean that the two datasets are independant.

Could you clarify this statement? Independence is defined in terms of distributions (the joint distribution can be split up into a product of marginals), so I'm not sure how "the way a set of data is distributed" and "can never imply independence" jive.

The Pearson correlation coefficient indicates the strength of a linear relationship between two variables, but its value generally does not completely characterize their relationship.

While independence refers to every relationship between two variables, when we use correlation we're usually only referring to one type of relationship, a linear relationship.

Uncorrelated and jointly normally Gaussian distributed implies independent. As I recall, there is a careful proof in one of Feller I or II.

The classic standard normal + chi-squared example is also one worth remembering:

Let X ~ N(0, 1), and Y = X^2.

Cov(X,Y) = 0, though they're obviously not independent.

But then the requirement that the rv's be jointly normal is violated. The "jointly normal + uncorrelated" combination is special. There aren't too many other named distributions that have the property that uncorrelated implies independence.

Can someone provide a real-life example of a data set that this warning applies to?

Not a normally distributed example, but:

(1,1), (0,0), (1,-1)

X and Y are uncorrelated but not independent.

It's kind of hard to say "independent" about any discrete population, because once you know you are sampling from a discrete population, you can look up the values of one variable given the other. So you'd only find that the variables are "independent" if you're looking at a cartesian product; in literally any other situation the variables are dependent.

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