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MTHE/STAT 353 — Winter 2017 Homework Assignment 4 due Friday, March 3, in class Section and problem numbers refer to the 3rd edition of the Ghahramani textbook. 1. Section 10.1, #9. 2. Section 10.1, #15. 3. Section 10.1, #18. There is a typo in the problem statement: the sequence X1 , X2 , . . . should be an infinite sequence, not a finite one. Hint: Theorem 10.2 in the text may be useful. 4. Let G = (V, E) be a finite graph (the number of vertices and edges, |V | and |E| respectively, are both finite), where V is the set of vertices and E is the set of edges. Suppose that |E| ≥ 1 (i.e., there is at least one edge) and there are no self-loops (i.e., every edge joins two distinct vertices). We wish to show that there is at least one subset W ⊂ V of vertices with the property that the number of edges with one endpoint going to a vertex in W and the other endpoint going to an endpoint in W c (the elements of V not in W ) is at least |E|/2. (a) Let each vertex in G be colored red or blue, each with probability 1/2, independently from vertex to vertex. Let X denote the number of edges that have one endpoint going to a red vertex and the other endpoint going to a blue vertex. Find E[X]. (b) From the answer in part(a), complete the argument that that there is at least one subset W with the property described above.