HW3_ES250

# HW3_ES250 - Harvard SEAS ES250 Information Theory Homework...

This preview shows pages 1–2. Sign up to view the full content.

Harvard SEAS ES250 – Information Theory Homework 3 (Due Date: Oct. 25 2007) 1. Let X p ( x ), x = 1 , 2 , · · · , m , denote the winner of a horse race. Suppose the odds o ( x ) are fair with respect to p ( x ), i.e., o ( x ) = 1 p ( x ) . Let b ( x ) be the amount bet on horse x , b ( x ) 0, m 1 b ( x ) = 1. Then the resulting wealth factor is S ( x ) = b ( x ) o ( x ), with probability p ( x ). (a) Find the expected wealth ES ( X ). (b) Find W * , the optimal growth rate of wealth. (c) Suppose Y = 1 , X = 1 or 2 0 , otherwise If this side information is available before the bet, how much does it increase the growth rate W * ? (d) Find I ( X ; Y ) 2. Many years ago in ancient St. Petersburg the following gambling proposition caused great conster- nation. For an entry fee of c units, a gambler receives a payoff of 2 k units with probability 2 - k , k = 1 , 2 , · · · . (a) Show that the expected payoff for this game is infinite. For this reason, it was argued that c = was a ”fair” price to pay to play this game. Most people find this answer absurd. (b) Suppose that the gambler can buy a share of the game. For example, if he invests c/ 2 units in the game, he receives 1 2 a share and a return X/ 2, where Pr( X = 2 k ) = 2 - k , k = 1 , 2 , · · · .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern