This problem set is due on Friday 2/20. Before working on it, you need to read Section 1.2 of Bertsekas and Tsitsiklis’s textbook. Please further read to Section 1.5 by Tuesday 2/17. Feel very free to discuss it over email!

You may consult whoever and whatever you like, but be sure to list your collaborators, write up your solutions in your own words without referring to any notes you might have jotted down from the consultation, and be ready to solve similar problems from scratch.

In addition to the following exercises, please do problems 5, 6, and 7 from the textbook. In problem 7, one way to specify the sample space is to describe the type of a sample as a data structure (that is, write something like “A sample is either … or … and …”).

1. Write a program using a random number generator to perform the experiment described in problem 7 from the textbook. The program should return a representation of the sample obtained; describe the format of this representation in English. (It is preferred that you do this programming in groups of 2 or 3.)

2. Consider the following probability distribution: the sample space is the set of ordered pairs (x,y) where x and y is each a real number between −1 and 1; the probability of an event is proportional to its area.

   1. What is the probability that y > 0, and why?
   2. What is the probability that x > 0 or y > 0, and why?
   3. What is the probability that 2x > y and 2y > x, and why?
   4. What is the probability that x² + y² < 1, and why?

3. Prove from the probability axioms that P(A) ≤ P(A ∪ B) for any two events A and B.