Joint Probability Distribution

Denote by f(xi,yi), f is called the joint probability distribution function of (X,Y).

The concept of independent events leads quite naturally to a similar definition for independent random variables.

Two random variables X and Y are said to be independent if

$$P(X=x,Y=y) = P(X=x).P(Y=y)$$

Roughly speaking, 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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