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 .

Figure 4 Typical Probability Distribution Function (pdf) View figure

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*.