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.
