TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Levajkovic)/Übungen 2023W/HW03.4

The Pareto distribution (the 80-20 rule).

The Pareto distribution is a power-law probability distribution, i.e. for a fixed baseline m its probability density function depends on the parameter $\alpha$ having the form

$f(x)={\begin{cases}{\frac {\alpha m^{\alpha }}{x^{\alpha +1}}},\ x\leq m,\\0,\ x<m\end{cases}}$

Assume $X$ is a random variable that follows such a distribution.

(a) Find the cumulative distribution function of $X$.

(b) The Pareto principle is also known as the 80-20 rule. This principle describes a variety of phenomena. For example, it means that 80% of the wealth is owned by 20% of the people. It may be applied to fundraising (20% of the donors contributing towards 80% of the total), in business management (80% of sales come from 20% of clients), in computer

science (80% of a piece of software can be written in 20% of the total allocated time), etc. The 80-20 rule is only exact for the Pareto distribution with $\alpha ={\frac {\ln 5}{\ln 4}}=1.16$. Assuming $\alpha =m=1$, compute the 0.80-quantile of $X$.

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Lösungsvorschlag von Lessi

--Lessi 2024-02-07T13:04:11Z

Let $f_{X}(x)={\begin{cases}{\frac {\alpha m^{\alpha }}{x^{\alpha +1}}},&x\geq m,\\0,&x<m\end{cases}}$ be the pdf of a Pareto distribution.

a)

We compute the cdf of that pdf by simply integrating $\int _{m}^{x}{\frac {\alpha m^{\alpha }}{t^{\alpha +1}}}dt=1-({\frac {m}{x}})^{\alpha }$, which gives us $F_{X}(x)={\begin{cases}1-({\frac {m}{x}})^{\alpha },&x\geq m,\\0,&x<m\end{cases}}$

b)

Given that $\alpha =m=1$ we can define the cdf of $X$ as $F_{X}(x)={\begin{cases}1-{\frac {1}{x}},&x\geq 1,\\0,&x<1\end{cases}}$. Now that the quantile $p=0.8$ is obviously $>0$ we can use the former case of our cdf and invert it, which gives us $x_{p}=F^{-1}(p)=5$.