The Pareto distribution is often used to model the distribution of personal income in a society. It has a lower limit m, which could for example denote the minimum wage, and a long right tail reflecting very high incomes at the upper end. A Pareto distribution with a minimum income of $20 per hour and a shape parameter of 2 has the CDF FX(x) = (
1 −
20 x , 20 ≤ x < ∞, 0, otherwise.

a. Find the PDF, fX(x), and sketch it.

b. Under this model, if we ranked everyone in order of their income, what income would correspond to the person at the lower 5% point of the distribution? What about the person at the upper 5% point?

c. What is the median income?

d. What is the probability of an income over $100 per hour?
What about an income over $1000 per hour? Don’t worry – it’s just a model.