
Normally distributed and uncorrelated does not imply independent - 666_howitzer
http://en.wikipedia.org/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent
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ishadua
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.

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christopheraden
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.

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crimsonalucard
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.

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

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thearn4
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.

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christopheraden
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.

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

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qmalxp
Not a normally distributed example, but:

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

X and Y are uncorrelated but not independent.

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dietrichepp
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.

