What is exactly probability density? I am quite confused:
$$ f_x(x) = 4x^2 \text{ for } 0<x<1 \text{ else } f_x(x) = 0$$
Now for the equivalent probability mass function it's easy to see. On the X-axis we have the value of the random variable $X$ and on the Y-axis we have the corresponding probability of that exact value $p_{X_i}$
But this can't be the same for continuous random variables, because for example the function that I defined has for the value $x=1$ the value on the y-axis is $y=4*1^2 = 4 $; which can't be a probability because $4>1$. Now I understand that singletons have probability of zero for a continuous random variable, but still  I can't properly understand what is going on on the Y-axis in simple words in contrast with discrete random variables.
So how is the proper way of understanding what exactly is? I mean the probability has to do with the areas and the borders of those areas.
 A: By your question I assume that you are in a introductory course to probability. So I will keep it as simple as possible
A valid probability density function is just a non-negative Real valued Integrable(take Riemann integrable for simplicity) function such that $\int_{-\infty}^{\infty}f(x)\,dx=1$ .
Such a function $f$ is said to be the pdf of a continuous rv $X$ if $\int_{a}^{b}f(x)\,dx=P(X\in [a,b])=P(a\leq X\leq b)$. Note that the interval can be $(a,b),[a,b),(a,b]$ also .
Now you can easily see that $f(x)=\begin{cases} 4x^{3}\,,0<x<1\\ 0,\text{otherwise}\end{cases}$ . Satisfies that it is non-negative and $\int_{-\infty}^{\infty}f(x)\,dx=\int_{-\infty}^{0} 0\,dx+\int_{0}^{1}4x^{2}\,dx+\int_{1}^{\infty} 0\,dx = 1$ . So this is a valid PDF.
If you want to think in terms of probability. Let $X$ be a uniform$[0,1]$ random variable.
So if $Y=\sqrt[4]{X}$ . Then $Y$ is also a random variable.
So this $Y$ will have the pdf $4y^{3}\,,0<y<1 $ and $0$ elsewhere.
That is for this particular pdf $f$. $\int_{a}^{b}f(x)\,dx$ represents the probability of picking a number
$x$ uniformly and randomly from $[0,1]$ such that $a\leq\sqrt[4]{x}\leq b$ .
In words , given $a$ and $b$. $\int_{a}^{b}f(x)\,dx$ is the probability that the fourth root of a uniformly and randomly chosen number from $[0,1]$ is greater than equal to $a$ and less than equal to $b$.
A: The formal definition of probability density is through Radon–Nikodym theorem. But I think that's not what you are looking for based on the description of this question.
To give a quick insight of probability density, let's focus on the word "density".
Say if you have a rock in hand, how would you describe "how large the stone is?"
Maybe you would say

"the size is $2\mathrm{m}^{3}$"

or

"the weight is bout $2\mathrm{t}$"

which corresponds to volume and mass in physics. These are two measures to describe how large the stone is. In physics, $\textit{density}=\textit{mass}/\textit{volume}$, so what density does is telling us how to link these two measures.
Now return to mathematics world, suppose you are asked how large a interval
$[a,b]$ is, you can tell him

the length of this interval is $b-a$

or

the probability of random variable $X$ falling into this interval is $p$

which corresponds to Lebesgue measure and (induced) probability measure,
what probability density does is nothing but building a bridge between these two measures. That's why we call this $f$ a density, the intuition behind it is exactly the same as the original concept in physics.
