TITLE: What does the value of a probability density function (PDF) at some x indicate?
QUESTION [20 upvotes]: I understand that the probability mass function of a discrete random-variable X is $y=g(x)$. This means $P(X=x_0) = g(x_0)$.
Now, a probability density function of of a continuous random variable X is $y=f(x)$. Wikipedia defines this function $y$ to mean 

In probability theory, a probability density function (pdf), or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value.

I am confused about the meaning of 'relative likelihood' because it certainly does not mean probability! The probability $P(X<x_0)$ is given by some integral of the pdf.
So what does $f(x_0)$ indicate? It gives a real number, but isn't the relative likelihood of a specific value for a CRV always zero?

REPLY [2 votes]: For continuous probability distributions, it is not useful to talk about $P(X=x_0)$, since this probability is zero. It is only useful to talk about the probabilities of $X$ being in sets with a positive length, like a nontrivial interval, or a union of several intervals. So, instead of asking the probability of $X=x_0$, we ask the probability that $X$ is near $x_0$. To be precise, we talk about the probability that $X$ is in an interval of length $\delta$ centered at $x_0$, for some $\delta>0$. That is,
$$
P(x_0-\delta/2<X<x_0+\delta/2)
$$
As $\delta\to 0$, the above probability will approach zero. How quickly will it approach zero? It turns out that there will exist a number $f(x_0)$ for which
$$
P(x_0-\delta/2<X<x_0+\delta/2)\approx \delta f(x_0)
$$
This means that in $P(x_0-\delta/2<X<x_0+\delta/2)$, the probability that $X$ in an interval around $x_0$ is approximated by $f(x_0)$ times the length of that interval. Furthermore, the relative error of this approximation approaches zero as the length of the interval approaches zero.
This qnuanty $f(x_0)$ is also the value of the pdf of $X$, and the preceding discussion is a way of giving it precise meaning.