The joint probability distribution function of (X,Y) is denote by f(xi,yi).
The concept of independent events extends to a similar definition for independent random variables. Two random variables X and Y are said to be independent if:$$P(X=x,Y=y) = P(X=x).P(Y=y)$$
In simple terms, X and Y are independent if knowing the value of one does not change the distribution of the other. Thus, if X and Y are independent, then:$$E(XY) = E(X)E(Y)$$
It follows that if X and Y are independent, then:$$Cov(X ,Y)= 0,\,or \,Corr(X,Y) = 0 $$
Dependent variables may also be uncorrelated, if the relationship is non-linear, a u-curve for instance.
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