University of Illinois
Spring 2015
ECE 561: Problem Set 2
Neyman-Pearson HT, M-ary HT, Composite HT
Due:
Reading:
Tuesday, February 17 in class
Lecture Notes Chapters 2, 3, 4; Poor, Chapter 2; Levy, Chapter 2.
1. [Neyman-Pearson Hypothesis Testing Continu
ammuniahoh A ng
H 'I y '1 mo + Z.
[4, .' y s [4, + Z
k (y) 5 I (;_j-'/4J)z
% r73 r C zrz =7 [ (21)
>
My Lid) =1 r. E C/o'COD
cfw_ O < [7/ 7T! CarC _ Z,
1 7 2.
W : L 9 + .+,
03 31 < WM. L LEM.
) :1
KB ('5) 7; WVHQ
0 Aw; F660 Minimh H ~11:st TeJhn
Km)! (7
Complete Metric Spaces
Definition: Let (X,d) be a metric space. A sequence
(x_n) is called Cauchy if
A sequence (x_n) is called convergent if there
exists an x in X such that
Definition: A metric space is caled complete if every Cauchy is
convergent.
Defi
Separation version of Hahn-Banach's theorem
Theorem B: Let C be a convex not-empty set and x not in C
a) There exists a linear map f:X-> R such that
f(y) f(x) for all yin C
b) If in addition X is tvs and C is open, then f is continuous, non
trivial and
f(
Banach Spaces
1) Basic definitions and Hahn Banach
Definition: 1) A vector space X equipped with a seminorm
is a called a semi-normed space.
2) A semi-norm is not degenerated if |x|=0 iff x=0.
3) A vector space equipped with a norm is called a normed
spac
The Hahn Banach Theorem
Definition: Let X be a vector space. A map q:X-> R_+
is subadditive if
q(x+y) q(x) + q(y)
A subadditive q is sublinear if in addition q(sx)=sq(x)
holds for s>0.
Theorem: Let X be a vector space, q sublinear,
Y be a subspace, and f:
Locally convex topological vector spaces
Proposition: A map T:X->Y between topological spaces is
continuous if and only if for every x, every open neighborhood
W of T(x) there exists an open neigborhoud V of x such that
V is contained in T^cfw_-1(W).
Defi
University of Illinois
Spring 2015
ECE 561: Problem Set 1
Statistical Decision-Making Framework, Binary Bayesian and Minimax
Hypothesis Testing
Due:
Reading:
Tuesday, February 3 in class
Lecture Notes Chapters 1 and 2; Poor, Chapter 2.
1. [Statistical Dec