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:
