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

Question

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.

(c) Based on the reasoning for part (b) state and prove a tighter version of Markov’s inequality of the following form: New inequality: For any numbers a, c > 0 and a random variable X such that X c > 0 always, P(X a) ________ . By tighter we want that the right-hand side is smaller than E(X)/a which we already know by Markov’s. (Hint: Consider the new random variable Y = X c and apply Markov’s to Y . This will find the statement of the right-hand side and also give a proof immediately. It’s not particularly pretty so don’t expect much simplification.)

Explanation / Answer

a) Given that 1/ x fraction of country can make x*$50,000 of income

If the income is below -$25,000 , so 1/x = -1/2

                                                                                X= -2

Then the fraction of population to make $75,000 or more = 1/3 – (-1/2)

                                                                                                                =1/3+1/2

                                                                                                                =5/6

b) if the income at least $35,000 ,   1/x = 7/10

Then the fraction of population to make $75,000 or more =1/3+7/10

                                                                                                                    =17/30

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