# Joint Probability Distribution

Joint probability is the probability of two events happening together, and their joint probability distribution is the corresponding probability distribution on all possible outcomes of those events.

The joint probability distribution function of (X,Y) is denoted 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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