Science:Math Exam Resources/Courses/MATH105/April 2018/Question 05 (b)

MATH105 April 2018

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Question 05 (b)

According to sociologists, the net income of the people can be modelled with a continuous random variable $X$, which has the following CDF: $$F(x)={\begin{cases}1-\left({\frac {A}{x}}\right)^{b},&{\text{if }}x\geq A,\\0,&{\text{if }}x<A,\end{cases}}$$ where $A$ and $b$ are positive constants and $b\neq 1$ ($A$ is the minimum income and $b$ is the tail index). (b) The Pareto principle (or 80-20 law) states that 80% of all the people receive 20% of all the income, and conversely, 20% of the people receive 80% of the income. Assume that the minimum income is $10, and that by earning more than $40 one can get in the top 20%, i.e. the chance that an arbitrary person gets $40 or less is 80%. Find the value of the tail index $b$.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Recall the definition of CDF, $F(x)=P(X\leq x)$.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Since the minimum income is $10, we have $A=10$. On the other hand, we assume that the chance that an arbitrary person gets $40 or less is 80%, which implies that $$\mathbb {P} (X\leq 40)=0.8={\frac {4}{5}}.$$ By the definition of CDF, we have $F(x)=P(X\leq x)$. Therefore, we get $F(40)=\mathbb {P} (X\leq 40)={\frac {4}{5}}$. Plugging $x=40$ into the given CDF $F$, $$F(40)=1-\left({\frac {10}{40}}\right)^{b}.$$ Finally, we solve $1-\left({\frac {10}{40}}\right)^{b}=F(40)={\frac {4}{5}}$ to get $$\left({\frac {1}{4}}\right)^{b}={\frac {1}{5}}\iff b(-\ln 4)=-\ln 5\iff b={\frac {\ln 5}{2\ln 2}}.$$ Answer: $\color {blue}{\frac {\ln 5}{2\ln 2}}$