Problem Set 11 Solutions Due: 5PM on Friday, May 6 This is a mini-problem set. The first problem reviews basic facts about expectation. The second and third are typical final exam questions. Problem 1. Answer the following questions about expectation. (a)There are several equivalent definitions of the expectation of a random variable

Problem Set 9 Solutions Due: Monday, April 25 at 9 PM Problem 1. There are three coins: a penny, nickel, and a quarter. When these coins are flipped: The penny comes up heads with probability 1/3 and tails with probability 2/3 The nickel comes up heads with probability 3/4 and tails with probability 1/4. The quarter comes up heads with

Due: Monday, April 11 at 9 PM Problem 1. An electronic toy displays a 4x4 grid of colored squares. At all times, four are red, four are green, four are blue, and four are yellow. For example, here is one possible configuration:

Problem 1. An undirected graph G has width w if the vertices can be arranged in a se- quence V1,2,3,…,Vn such that each vertex v; is joined by an edge to at most w preceding vertices. (Vertex vj precedes if i.) Use induction to prove that every graph with width at most w is (w+1)-colorable

Problem set 3 Solutions Due: Tuesday, February 22 at 9 PM Problem 1. An urn contains 75 white balls and 150 black balls. while there are at least 2 balls remaining in the urn, you repeat the following operation. You remove 2 balls elected arbitrarily and then: If at least one of the two balls is black, then you discard one black ball and put the other ball back in the urn

Problem set 1 Solutions Due: Monday February 7 at 9 PM Problem 1. The connectives A(and), V(or), and =(implies)come often not only in com uter programs, but also everyday speech. But devices that compute the nand operation

1 Conditional Expectation and Total Expectation There are conditional expectations, just as there are conditional probabilities. If R is a random variable and e is an event, then the conditional expectation Ex(r e)is defined

Problem 1. The following two parts are not related. Try them, to make sure you un- derstand the jargon of random variables distributions, probability density functions, etc. Ask your TA if you don't understand/remember what some phrase means. (a)Suppose X1, X2, and X3 are three mutually independent random variables, each having the uniform distribution

The Law of Total Probability is handy tool for breaking down the computation of a prob- ability into distinct cases. More precisely, suppose we are interested in the probability of an event E: Pr(). Suppose also that the random experiment can evolve in two different ways; that is, two different cases X and X are possible. Suppose also that it is easy to find the probability of each

Problem 1. Find closed-form generating functions for the following sequences. Do not concern yourself with issues of convergence