Download Assignment 4

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.