In the sense that it is used in statistics, i.e., as an assumption in

applying a statistical test, a random sample from a population means a set

of random variables X1,...., Xn that are identically distributed and and

mutually independent (mutually independent is stronger than pairwise

independence). The random variables are mutually independent if their

joint distribution is equal to the product of their marginal distributions.

In the case of joint normality, independence is equivalent to zero

correlation but not in general. Independence will imply zero correlation

(if the random variables have second moments) but not conversely. Not that

not all random variables have a first moment let alone a second moment and

hence there may not be a correlation coefficient.

However if the correlation coefficient of two random variables

(theoretical) is not zero then the random variables are not independent.

Zero correlation is sort of the next best thing to independence but it is

not equivalent to it.

For some purposes zero correlation is sufficient but not for most

statistical tests.

Donald E. Myers

Department of Mathematics

University of Arizona

Tucson, AZ 85721

myers@...
http://www.u.arizona.edu/~donaldm
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