EECS 564________________________PROBLEM SET #2________________________Winter 1999

ASSIGNED: January 15, 1999
DUE DATE: January 22, 1999

READ: Sections 3.4-3.5 of Srinath, and Chapter 5 through p. 156 on estimation.
We will skip 3.6 until after we study estimation, and put off 3.9 until later.
We will skip Chapter 4 until after we have studied the Karhunen-Loeve expansion.
THIS WEEK: Binary and m-ary hypothesis testing with more complex decision regions.

1. Under H0, random variable r has a unit (normalized) Gaussian distribution: r~N(0,1).
Under H1, r has pdf pr(R)=0.5e-|R|. Simplify the LRT as much as possible.
There are three different possible ranges of threshold n, leading to 1, 3 or 5 decision regions.
2. Under H0, random variable r has a unit (normalized) Gaussian distribution: r~N(0,1).
Under H1, r is Gaussian: mean m and variance s2. Simplify the LRT as much as possible.
Consider these special cases: (1) m=0; (2) m=infinity; (3) s=0; (4) s=1; (5) s=infinity.
3. Denote the function L(R)=likelihood ratio=pr|H1(R|H1)/ pr|H0(R|H0).
Consider L(R) to be a function of random variable r. Prove the following:
(1) E[Ln|H1]=E[Ln+1|H0]; (2) E[L|H0]=1; (3) E[L|H1]-E[L|H0]=Var[L|H0].
The point of this problem is to get you used to the idea of L(R) as a random variable.
4. We know random variable r is Gaussian with variance s2. Its mean could be any of:
H1: -2m; H2: -m; H3: 0; H4: m; H5: 2m where m is known
The criterion is MEP (min Pr[error]) and the 5 hypotheses are equally likely a priori.
(1) Draw the decision regions on the R-axis. (2) Compute the attained Pr[error].
5. We wish to estimate a=Pr[heads] of a coin from observation r=#heads in n independent flips.
1. Compute aMLE(R). Show it is unbiased and compute the mean square error.
2. Compute the Cramer-Rao bound. Use it to show that aMLE(R) is efficient.

"An atheist is someone who watches Notre Dame and Southern Methodist University play football
and doesn't care who wins"--Dwight D. Eisenhower.