The significance of the pdf 
is that 
is the probability that
the r.v. is in
the interval 
, written as:
This is an operational definition of .
Since is unitless (it is a probability), then has units of
inverse r.v. units, e.g., 1/cm or 1/s or 1/cm
,
depending on the units of 
.
Figure 4
shows a typical pdf and illustrates
the interpretation of the probability of finding the r.v. in with
the area
under the curve from to 
.
We can also determine the probability of finding the r.v. somewhere in the
finite interval 
:
which, of course, is the area under the curve from 
to 
.
As with the definition of discrete probability distributions, there are some
restrictions on the pdf. Since is a probability density, it must be
positive
for all values of the r.v. .
Furthermore, the probability of finding the r.v. somewhere on the
real axis must be unity.
As it turns out, these two conditions are the only necessary conditions for
to be a legitimate pdf, and are
summarized below.
Note that these restrictions are not very stringent, and in fact allow one to apply Monte Carlo methods to solve problems that have no apparent stochasticity or randomness. By posing a particular application in terms of functions that obey these relatively mild conditions, one can treat them as pdf's and perhaps employ the powerful techniques of Monte Carlo simulation to solve the original application. We now define an important quantity, intimately related to the pdf, that is known as the cumulative distribution function, or cdf.
