TITLE: Intuitive meaning of the probability density function at a point
QUESTION [6 upvotes]: I understand how to integrate probability density functions to find probability within a certain range.
However, what I don't understand is what it would mean to set the variable (say $x$ or $y$) to a certain value and evaluate the pdf as you would evaluate any other function.  
I know this happens when dealing with conditional density functions in a multivariable situation. But I'm a little confused as to what this actually signifies.

REPLY [6 votes]: Here's an example that may convey some intuition.
Suppose we consider how much it rains in a given year in Los Angeles.  This random variable, which we shall call $R$, has a distribution, which is a PDF that we may denote $f_R(r)$.
Suppose, for the sake of argument, that $f_R(15) = 0.1$.  Now what does that mean?  Obviously, it doesn't mean that there's a $0.1$ probability that it will rain exactly $15$ inches.  That probability is zero, since the rainfall will differ from exactly $15$ inches* almost surely.
Instead, what it means is that the probability of it raining about $15$ inches is $0.1$ "per inch."  That is, if we ask how likely it is for the rainfall to be in a given range around $15$ inches, the answer will be approximately $0.1$ times the width of the interval in inches.  Obviously, the answer won't be exactly $0.1$ times the width of the interval, in general, but for intervals that are "small enough," the answer will be "close enough" (in a delta-epsilon kind of way for continuous PDFs).
*Ignoring the atomic nature of rain for the moment.