Normally distributed and uncorrelated does not imply independent

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.

The Pearson correlation coefficient indicates the strength of a linear relationship between two variables, but its value generally does not completely characterize their relationship.<p>While independence refers to every relationship between two variables, when we use correlation we&#x27;re usually only referring to one type of relationship, a linear relationship.

Uncorrelated and <i>jointly</i> 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:<p>Let X ~ N(0, 1), and Y = X^2.<p>Cov(X,Y) = 0, though they&#x27;re obviously not independent.

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

Not a normally distributed example, but:<p>(1,1), (0,0), (1,-1)<p>X and Y are uncorrelated but not independent.