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In many instances we may be interested in how several random variables behave jointly. In such cases we use a joint probability mass function or probability density function.
 

For the discrete case:

The joint pmf = p(x,y) = P(X=x and Y=y)

For the continuous case:

f(x,y) is a joint pdf and is defined so that P[XÎ (a,b) and YÎ (c,d)] = 

Just as we defined independent events, we can define independent random variables: X and Y are independent if for any two values x,y

Discrete: p(x,y) = p_X(x)× p_Y(y) where p_X and p_Y are the marginal pmf’s of X and Y

Continuous: f(x,y) = f_X(x)×f_Y(y) where f_X and f_Y are the marginal pdf’s of X and Y

If the above do not hold, then X and Y are said to be dependent.
 

The expected value of a function h(X,Y) is given by

All of these concepts can be readily extended to more than 2 variables...

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