10. A family of pdf’s that has been used to approximate the distribution of income, city population size, and size of firms is the Pareto family. The family has two parameters, k and , both 0, and the pdf is f(x; k, )


k x

k

1 k

 x 

0 x 

a. Sketch the graph of f(x; k, ).

b. Verify that the total area under the graph equals 1.

c. If the rv X has pdf f(x; k, ), for any fixed b , obtain an expression for P(X  b).

d. For  a 

b, obtain an expression for the probability P(a  X  b).