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

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MATH105 April 2018

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Question 05 (c)
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). (c) For which values of $b$ does the random variable $X$ has a finite expected value?

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 that the expected value of $X$ is $\mathbb {E} (x)=\int _{0}^{\infty }xf(x)dx$ , where $f$ is PDF.

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Solution
By the Hint, it is enough to find $b$ which make the integral $\mathbb {E} (x)=\int _{0}^{\infty }xf(x)dx$ converges. Recall the PDF found in part (a): $f(x)={\begin{cases}{\frac {A^{b}\cdot b}{x^{b+1}}},&{\text{if }}x>A,\\0,&{\text{if }}x<A.\end{cases}}$ Plugging this into the integral, we get ${\begin{aligned}\int _{0}^{\infty }xf(x)dx&=\int _{10}^{\infty }x\cdot {\frac {A^{b}\cdot b}{x^{b+1}}}dx=\lim _{\tau \to \infty }\int _{10}^{\tau }x\cdot {\frac {A^{b}\cdot b}{x^{b+1}}}dx\\&=A^{b}\cdot b\lim _{\tau \to \infty }\int _{10}^{\tau }{\frac {1}{x^{b}}}dx=A^{b}\cdot b\cdot {\frac {1}{1-b}}\lim _{\tau \to \infty }\left[x^{-b+1}\right]_{10}^{\tau }\\&={\frac {A^{b}\cdot b}{b-1}}\lim _{\tau \to \infty }\left(10^{-b+1}-\tau ^{-b+1}\right)<+\infty \end{aligned}}$ if $-b+1<0$ . i.e., $b>1$ . Therefore, the random variable $X$ has a finite expected value for $b>1$ . Answer: $\color {blue}b>1$