Krdytkyu

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?

'Relative likelihood' is indeed misleading. Look at it as a limit instead:
$$
f(x)=lim_{h to 0}frac{F(x+h)-F(x)}{h}
$$
where $F(x) = P(X leq x)$

In general, if $X$ is a random variable with values of a measure space $(A,mathcal A,mu)$ and with pdf $f:Ato [0,1]$, then for all measurable set $Sinmathcal A$,
$$P(Xin S) = int_S fdmu $$
So, if $A=Bbb R$ (and $mu=lambda$), then
$$P(a<X<b)=int_a^b f(x)dx$$
So, $f(x) = displaystylelim_{tto 0} frac1{2t}int_{x-t}^{x+t} f =lim_{tto 0} frac1{2t} P(|X-x|<t) $ for example.. We can call it 'relative likelihood'..

Intro statistics focuses on the PDF as the description of the population, but in fact it is the CDF (cumulative density function) that gives you a functional understanding of the population, as points on the CDF denote probabilities over a relevant range of measures. If you look at all stats from this perspective, then the PDF is just the description of probability change with respect to a change around a point along the measure at hand. The values on the PDF therefore only give you a look at the spread. For example, given two normal distributions $N(mu_1, sigma_1^2)$ and $N(mu_2, sigma_2^2)$, if you choose any value of $x$ to get point $p_n=mu_n+xcdotsigma_n$ for the respective distributions and get $X_1[p_1 ] > X_2[p_2 ]$, then this just means $sigma_1 < sigma_2$. Similar relationships exist for other distributions.

I am not sure if Jester is still interested, as it's been 5 years, but I think I found a less confusing anwer than in Wikipedia.

In contrast to discrete random variables, if X is continuous, f(X) is a function whose value at any given sample is not the probability but rather it indicates the likelihood that X will be in that sample/interval. For example if the value of the PDF around a point (can be generalized for a sample) x is large, that means the random variable X is more likely to take values close to x. If, on the other hand, f(x)=0 in some interval, then X won't be in that interval

Of course a more practical way of thinking it is that the probability of X being in an interval is given by the integral of the PDF.

You might want to look at the link below for more details: http://mathinsight.org/probability_density_function_idea

The ratio of the pdf $f(x)$ at two points, $r_x = f(x_0)/f(x_1)$, is not a measure of relative probability (or "relative likelihood") for the two outcomes for the random variable $X$. The ratio depends on the metric. That is, with a variable transformation, $z=z(x)$, with the pdf for $Z$ given by $h(z)$, the ratio $r_z=h(z_0)/h(z_1)neq r_x$, in general, even though the two ratios refer to the same two outcomes. For monotonic transformations, $f(x),dx = h(z),dz$.

Numerical values of the pdf have no value on their own. The metric, $dx$, is required for probability interpretations (ie. $f(x),dx$). Wikipedia got this wrong, so I have corrected it.