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 is a random variable uniformly distributed on and , then it’s straightforward to show that and , so that
and hence and 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.”