2019 November 14,
Carleton College, Joshua R. Davis
Today in class we studied Markov's and Chebyshev's inequalities. The following exercises use some of the other inequalities in Section 10.1.
A. Section 10.7 Exercise 5. (You will need Jensen's inequality.)
B. Section 10.7 Exercise 6. (You will need to use the Cauchy-Schwarz inequality in a "creative way".)
Recall from class that X is Pareto-distributed if fX(x) = α x-α - 1 on support [1, ∞). The Pareto distribution has been used to model many social and natural phenomena, such as income disparities.
C. Find a simple expression for the kth moment of X. (It is sufficient to work through the first few cases and guess the pattern.) Your expression will be valid only for certain values of the parameter α; clearly state this restriction.
D. Working from the definition of the MGF, show that X does not have an MGF.
I was going to have you investigate how Chebyshev's inequality applies to this X, but let's not do that on homework. Consider studying it as you prepare for our final exam.