Problem Set 4 Solutions Due: Monday, February 28 at 9 PM Problem 1. Prove all of the following statements except for the two that are false; for those, provide counterexamples. Assumen 1. When proving each statement, you may assume all its predecessors (a)a =(mod n) Solution. Every number divides zero, so n (a-a), which means a a (mod n). (b)a≡b(modn) impliesa(modn)
1 The Number-Picking Game Here is a game that you and I could play that reveals a strange property of expectation. 3, First, you think of a probability density function on the natural numbers. Your distri- bution can be absolutely anything you like. For example, you might choose a uniform distribution on 1, 2, ... 6, like the outcome of a fair die roll. Or you might choose a bi- probability, provided that,...,n. You can even give every natural number a non-zero nomial distribution on 0, 1 he sum of all probabilities is 1
The expectation or expected value of a random variable is a single number that tells you a lot about the behavior of the variable. Roughly, the expectation is the average value, where each value is weighted according to the probability that it comes up. Formally, the expected value of a random variable r defined on a sample space s is: (B)=∑R()Pr(o) To appreciate its signficance, suppose S is the set of students in a class, and we select a student uniformly at random. Let r be the selected student's exam score. Then
