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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.

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