CS6800 15F: Homework 06

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General Instructions

1. Feel free to work with others on this assignment. However, you must acknowledge with whom you worked, and you must write up your own solutions.

2. All of the problems below are from the text, Cover and Thomas, 2nd Edition.

Problems

1. You have a pot of money that you wish to invest, and two opportunities have arisen: (1) Someone has offered you a 5% return on your money for a short-term loan. (2) Over that same time period, there is a horse race with three horses offering uniform fair odds, but you happen to know that the odds of the horses winning is not uniform but rather (1/2, 1/4, 1/4).

   You will invest in either (1) or (2) repeatedly.

   • What is the optimal betting strategy for (2), and what is the corresponding doubling rate?
   • Which opportunity would cause your wealth to increase at the greatest rate?

2. Problem 6.8

3. Consider a binary channel with one-sided errors: When transmitting a 0, there is a 50/50 chance that a 0 or 1 is received; however, when transmitting a 1, a 1 is always received.
   • What is the capacity of this channel? (An analytic expression would be ideal, but a numeric answer is also acceptable.)
   • What input distribution achieves this capacity? (The same parenthetic comment applies.)

4. Problem 7.7

5. Problem 7.19