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

According to sociologists, the net income of the people can be modelled with a continuous random variable , which has the following CDF:
where and are positive constants and ( is the minimum income and is the tail index).
(b) The Pareto principle (or 8020 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 . 
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, . 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Since the minimum income is $10, we have . On the other hand, we assume that the chance that an arbitrary person gets $40 or less is 80%, which implies that By the definition of CDF, we have . Therefore, we get . Plugging into the given CDF , Finally, we solve to get Answer: 