1. Problem 10.18 Suppose that the continuous random variable X has the prob ability density function f (x) = (/)(/x)+1 for x >and f(x) =

1. Problem 10.18 Suppose that the continuous random variable X has the prob ability density function f (x) = (α/β)(β/x)α+1 for x >βand f(x) = 0 for x ≤β for given values of the parameters α>0 and β>0. This density is called the Pareto density, which provides a useful probability model for income distributions among others.

(a) Calculate the expected value, the variance and the median of X.

(b) Assume that the annual income of employed measured in thousands of dollars in a given country follows a Pareto distribution with α = 2.25 and

β =2.5. What percentage of the working population has an annual income of between 25 and 40 thousand dollars?

(c) Why do you think the Pareto distribution is a good model for income distributions? Hint: use the probabilistic interpretation of the density function f (x).