User:IssaRice/Pareto distribution

I still don't know how this is derived, but here's how to make sense of the 80/20 rule:

What does it mean to say that the top 20% own 80% of the wealth? It means that ${\frac {\int _{c}^{\infty }xf(x)\,dx}{\int _{x_{m}}^{\infty }xf(x)\,dx}}=0.80$ where we choose the cutoff c according to $\int _{c}^{\infty }f(x)\,dx=0.20$.

Why? $\int _{c}^{\infty }f(x)\,dx$ is the fraction of people with wealth above the cutoff c; we're just integrating the pdf of the distribution.

$\int _{c}^{\infty }xf(x)\,dx$ is the amount of wealth owned by the people with wealth above c. This is because at each wealth level x, such people are a fraction f(x)dx of people, and they each own wealth x, so the wealth owned by them is xf(x)dx. Now just sum over all of them from c to infinity.

The above can also be expressed as ${\frac {\mathbb {E} [X\mid X>c]}{\mathbb {E} [X]}}=0.80$ where the cutoff c is chosen by $\mathbb {P} (X>c)=0.20$.

Using $\alpha =(\log 5)/(\log 4)$ in the Pareto distribution, we get $c={\frac {1}{0.20^{1/\alpha }}}={\frac {1}{0.20^{(\log 4)/(\log 5)}}}$. The mean is ${\frac {\alpha }{\alpha -1}}\approx 7.213$ and $\int _{c}^{\infty }xf(x)\,dx\approx 5.77$. Sure enough, 5.77/7.213 is very close to 0.80.