User:IssaRice/Pareto distribution: Difference between revisions

Revision as of 20:16, 5 February 2020

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.

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 What does it mean to say that the top 20% own 80% of the wealth? It means that <math>\frac{\int_c^\infty xf(x)\, dx}{\int_{x_m}^\infty xf(x)\, dx} = 0.80</math> where we choose the cutoff c according to <math>\int_c^\infty f(x)\, dx = 0.20</math>.
+Why? <math>\int_c^\infty f(x)\, dx</math> is the fraction of people with wealth above the cutoff c; we're just integrating the pdf of the distribution.
+<math>\int_c^\infty xf(x)\, dx</math> 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.