Example 4 Consider the random variables and with the joint probability mass function as tabulated in Table 1. The marginal probabilities and are as shown in the last column and the last row respectively.
Table 1
Joint Probability Density Function
If and are two continuous random variables and their joint distribution function is continuous in both and , then we can define joint probability density function
by provided it exists.
and 
with the joint probability mass function as tabulated in Table 1. The marginal probabilities 
and 
are as shown in the last column and the last row respectively. 
and 
, then we can define joint probability density function 
by
provided it exists.