I've searched all Google and I did not find any good and clear explanation about this:
Let's suppose that we want use Pareto Distribution with X axis expressing ranges of some income proxy and Y axis, some population proxy.
$$\color{blue}{ PDF(x) = \alpha/ x^{\alpha+1}}$$
Where $\color{blue}{x >=1}$ and $\color{blue}{\alpha>1}$
I know the $\color{blue}{PDF(x)}$ shows for continuous and increasing income bands, something related to the proportion of the population that earn at least the corresponding to that range. So $\color{blue}{PDF(1)}$ is the bigger value because it's the lower income range.
It there somebody that can explain to me without excessive formalism why does $\color{blue}{CDF(x)}$ generates population size proxy within the given range of $\color{blue}{x}$ values?
$\color{blue}{\qquad\qquad\qquad\qquad\qquad CDF(x) = 1 - (1/x)^{\alpha}}$
So we have
$\qquad\qquad\qquad\qquad\qquad \color{blue}{P=1 - CDF(x)}\;$ (1)
Where $\color{blue}{P}$ is proportion from population with income greater than $\color{blue}{x}$
If I take for granted that $\color{blue}{CDF(x)}$ sums the population it's not so hard to understand why $\color{blue}{ \int x PDF(x) dx}$ returns a proxy for total income.
From $\color{blue}{1}$ to $\color{blue}{\infty}$ it gives $\color{blue}{\alpha/(\alpha-1)}$. From $\color{blue}{x}$ to $\color{blue}{\infty}$ it gives $\color{blue}{\alpha/(\alpha-1)(x^{1-\alpha})}$
So the rate $\color{blue}{I}$ of superior income on total income is $\color{blue}{I = x^{1-\alpha}\;}$ (2)
If I select values for income and populations proportions in (1) and (2) and combine them, we have 2 equations with 2 incognitos.( $\color{blue}{x}$ and $\color{blue}{\alpha}$):
$$\color{blue}{\alpha = 1/(1-\log_{P} I)}$$
For instance, let's suppose that we want to define a Pareto Distribution where $\color{blue}{P = 20\%}$ of richer population win $\color{blue}{I = 80\%}$ of total income (one of the classic assumptions), so:
$$\color{blue}{\alpha = 1/(1-\log_{20\%} 80\%) \sim 1.161}$$
The $\color{blue}{x}$ that reaches 80% of the population is:
$$\color{blue}{x = 80\%/20\% = 4}$$
And it's right, because $\color{blue}{CDF(4) = 80\%}$, i.e.,
$\color{blue}{1 - CDF(4) = 20\%}$ are the richer people.
Using (1) the rate of superior income on total income is $\color{blue}{x^{1-\alpha}=80\%}$
So everything works smoothly but I dont't understand, after all, why $\color{blue}{CDF(x)}$ is related to total population.
Why?
PS: In this case, I'm not interested about Lorenz Curve. It's related to a explanation about a $\color{blue}{CDF(x)}$ behaviour.
UPDATE
The below answer fully clarified my doubt.
In fact, like @StubbornAtom has showed, $\color{blue}{CDF(x)}$ is built-in associated to some $\color{blue}{x}$ (income, revenue, profit, counting) proportion in a universe (population, items, products)
Suppose that $\color{blue}{x}$ is income and $\color{blue}{CDF(x)}$ is proportion of incomers lesser than $\color{blue}{x}$.
So there is no useful $\color{blue}{PDF(x)}$ interpretation in Pareto Distribution for Pareto Principle application. It's obviously related to a slope (difference of proportion of population below income $\color{blue}{x}$ on income difference), for each income $\color{blue}{x}$:
$$\color{blue}{\color{blue}{PDF(X) =\Delta CDF(x) /\Delta x }}$$
And it's explain why $\color{blue}{PDF(X_M)}$ is the bigger value. The proportion variation with income starts high, then slows and after it fades slowly.
I don't know if you would find this helpful, but here is a possible explanation of the genesis of the Pareto distribution:
Suppose $N_x$ is the number of individuals in a community having income at least $x$.
Then, $$N_x\propto x^{-a}\quad,\,a>0$$
Consequently, the proportion of individuals having income at least $x$ in the community is also proportional to $x^{-a}$ for $a>0$.
Let $X$ be the random variable denoting income with distribution function $F$. Then based on the above observation, one can assume that $$1-F(x)\propto x^{-a}\quad,\,a>0$$
That is, $$1-F(x)=Ax^{-a}\quad,\text{ for some }A$$
Or, $$F(x)=1-Ax^{-a}$$
Let $k$ be the least income earned by an individual in the given community.
Then, $$P(X\geqslant k)=1-F(k)=1$$
So, $$A=k^a$$
Finally the distribution function looks like
$$F(x)=\left[1-\left(\frac{k}{x}\right)^a\right]\mathbf1_{x\geqslant k}$$
, from which the density of a Pareto distribution with shape $a$ and scale $k$ is derived as
$$f(x)=\frac{ak^a}{x^{a+1}}\mathbf1_{x\geqslant k}\quad,\,a>0,k>0$$
The above description is a version of the celebrated Pareto law, and $a$ here is called the Pareto constant.