We now consider two r.v.'s and , where and . We ask what is the probability that the first r.v. falls within and the second r.v. falls within , which defines the bivariate pdf :

Using this operational definition of , let us multiply and divide by the quantity , where we assume ,

It is readily shown that satisfies the properties for a legitimate pdf given in Eq. (24) and Eq. (25), and we can interpret as follows:

The quantity is known as the *marginal* probability distribution
function.
Now define the quantity ,

As with , it can be shown that is a legitimate pdf and can be interpreted as follows:

The quantity is called the *conditional* pdf.
The constraint that simply means that the r.v.'s and
are *not mutually exclusive*, meaning there
is some probability that both and will occur together.
Note that if and are independent r.v.'s, then and
reduce to the univariate pdf's for and :

and therefore for independent pdf's we find that the bivariate pdf is simply the product of the two univariate pdf's: