Math 480, Fall 2016, Assignment 3

No doubt many people feel that the inclusion of mathematics among the arts is unwarranted. The strongest objection is that mathematics has no emotional import. Of course this argument discounts the feelings of dislike and revulsion that mathematics induces....

- Morris Kline, Mathematics in Western Culture

Carefully define the following terms, then give one example and one non-example of each:

1. Information (emitted by an event $E$).
2. Conditional information.
3. Mutual information (of two events).
4. Essential containment (of events).
5. Essential equality (of events).
6. System of events.
7. Induced probability assignment (on a system of events).
8. Amalgamation (of a system of events).
9. Refinement (of a system of events).
10. Join (of two systems of events).
11. Mutual information (of two systems of events).

Carefully state the following theorems (you do not need to prove them):

1. Theorem relating exhaustive events to essential equality.
2. Theorem relating mutually exclusive events to essential equality.

Solve the following problems:

1. Section 2.1, problems 2, and 3.
2. Section 2.2, problems 1, 3, and 7. (In problem 1, you are really considering four different systems of events for a common finite probability space. Each of these becomes a probability space in its own right, using the induced probability assignment.)
3. Consider the systems of events $(\mathcal{S}_1,\dots,\mathcal{S}_4)$ from section 2.2, problem 1. Compute the mutual information for several pairs of these systems.

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