2. The reasoning behind Markov’s inequality says that if the average income in a

Question

2. The reasoning behind Markov’s inequality says that if the average income in a country is $50,000, then at most half the country makes $100,000, and at most one third makes $150,000, and so on. In general, at most a 1/x-fraction of the country can make x × $50, 000. The reasoning was that this must be true because otherwise the high-earners would pull up the average all by themselves, even if the rest of the population made $0. Here we used the assumption that the income was never below zero. This problem is about what we can say when we change this assumption.

(a)Suppose that it is in fact possible for someone to have negative income, but that no one will have an income below -$25,000 (but the average income is still $50,000). What is the largest fraction of the population that can make $75,000 or more? (Hint: Using the fact that the average must be $50,000, you can set up an equation for the unknown and then solve. We didn’t have to do this for the original version.)

(b)Now suppose that not only is everyone’s income positive, but that an especially generous country guarantees that everyone makes at least $35,000 per year. Assuming the average income is $50,000 as before, now find the largest fraction of the population that can make $75,000 or more.

Explanation / Answer

it is given that 1/x fraction of countrry can make x*50000 of income

a) suppose if the income is below -$25,000 here the fraction will be

1/x = - 50000/ 25000 = -2

fraction of the population that can make $75,000 or more

(50000/75000 ) - (-2) = ( 2/3) + 2= 8/3

b) suppose that income is atleast $ 35,000 herethe fraction will be 1/x = 50000/35000=10/7

fraction of the population that can make $75,000 or more

(50000/75000 ) +(10/7) = 2/3 + 10/7 = 18/7

Related Questions

Navigate

• Browse (All)
• Browse 2
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.