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.