Description

Department of Electrical Engineering and Computer Sciences
EECS 126: Probability and Random Processes
Discussion 11
1. BSC: MLE & MAP
You are testing a digital link that corresponds to a BSC with some error probability
(a) Assume you observe the input and the output of the link. How do you find the MLE of
?
(b) You are told that the inputs are i.i.d. bits that are equal to 1 with probability 0.6 and to 0 with probability 0.4. You observe n outputs (n is a positive integer). How do you calculate the MLE of ?
2. Hypothesis Testing for Uniform Distribution
Assume that
• If X = 0, then Y ∼ Uniform[−1,1].
• If X = 1, then Y ∼ Uniform[0,2].
Using the Neyman-Pearson formulation of hypothesis testing, find the optimal randomized decision rule r : [−1,2] →{0,1} with respect to the criterion
randomized
s.t.
where β ∈ [0,1] is a given upper bound on the false positive probability.
3. Bayesian Hypothesis Testing for Gaussian Distribution
Assume that X has prior probabilities P(X = 0) = P(X = 1) = 1/2. Further
• If X = 0, then
• If X = 1, then
Assume µ0 < µ1 and σ0 < σ1.
Using the Bayesian formulation of hypothesis testing, find the optimal decision rule r : R→ {0,1} with respect to the minimum expected cost criterion
.
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