Math 480, Fall 2016, Assignment 1

By one of those caprices of the mind, which we are perhaps most subject to in early youth, I at once gave up my former occupations; set down natural history and all its progeny as a deformed and abortive creation; and entertained the greatest disdain for a would-be science, which could never even step within the threshold of real knowledge. In this mood of mind I betook myself to the mathematics, and the branches of study appertaining to that science, as being built upon secure foundations, and so, worthy of my consideration.

- Mary Shelley, Frankenstein

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

1. Finite probability space.
2. Outcome.
3. Probability distribution (a.k.a. probability assignment).
4. Event.
5. Probability (of an event, as opposed to an outcome).
6. Mutually exclusive events.
7. Exhaustive events.
8. Random variable (with values in a set $A$).
9. Pushforward distribution (associated with a random variable).
10. Product distribution (on a Cartesian product of two finite probability spaces).
11. Frequentist interpretation of probability theory.
12. Bayesian interpretation of probability theory.

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

1. Useful observations and results (facts 1.2.1 - 1.2.6).

Solve the following problems:

1. Section 1.1, problems 1 and 2.
2. Section 1.2, problem 1.
3. Read about the D'Alembert-Laplace controversy in Section 1.1. From a frequentist perspective, how could you resolve the controversy? What about from a Bayesian perspective?

--------------------End of assignment--------------------