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Probability simply refers to the likelihood of something occurring. We may talk about the probabilities of particular outcomes鈥攈ow likely they are鈥攚hen we're unclear about the result of an event. Statistics is the study of occurrences guided by probability.

Proposition:

The following holds

\({\rm{Cov(X,Y) = E(XY) - E(X) \times E(Y)}}\)

The

correlation coefficient

of \({\rm{X}}\) and \({\rm{Y}}\) is

\({{\rm{\rho }}_{{\rm{X,Y}}}}{\rm{ = }}\frac{{{\rm{Cov(X,Y)}}}}{{{{\rm{\sigma }}_{\rm{X}}}{\rm{ \times }}{{\rm{\sigma }}_{\rm{Y}}}}}\)

Assume that \({\rm{X}}\) and \({\rm{Y}}\) are independent, the following holds from the proposition

\(\begin{aligned}{\rm{Cov(X,Y) = E(XY) - E(X) \times E(Y)}}\\{\rm{ = E(X)E(Y) - E(X)E(Y)}}\\{\rm{ = 0}}{\rm{.}}\end{aligned}\)

When\({\rm{Cov(X,Y) = 0}}\), then\({\rm{\rho = 0}}\), which is obvious.