University of California, Berkeley - EECS 126dis12-sol

Document Content and Description Below

UC Berkeley Department of Electrical Engineering and Computer Sciences EECS 126: Probability and Random Processes Discussion 12 Spring 2018 1. Hypothesis Testing for Gaussian Distribution Assume... that X has prior probabilities P(X = 0) = P(X = 1) = 1/2. Further • If X = 0, then Y ∼ N (µ0, σ2 0 ). • If X = 1, then Y ∼ N (µ1, σ2 1 ). 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 min r:R→{0,1} E[I{r(Y ) 6= X}]. Solution: According to the theory the optimal decision rule is given by r(y) = ( 0, if f(y | X = 0) > f(y | X = 1) 1, if f(y | X = 0) < f(y | X = 1). The condition f(y | X = 0) < f(y | X = 1) can be written as  1 σ 2 0 − 1 σ 2 1  x 2 − 2 µ0 σ 2 0 − µ1 σ 2 1  x + µ 2 0 σ 2 0 − µ 2 1 σ 2 1 − 2 ln σ 2 1 σ 2 0  > 0, and if we let a < b be the two roots of this quadratic, then the optimal decision rule can be written as r(y) = ( 0, if y ∈ (a, b) 1, if y ∈ (−∞, a) ∪ (b, ∞). 2. Hypothesis Testing for Uniform Distribution Assume that • If X = 0, then Y ∼ Uniform[−1, 1]. • If X = 1, then Y ∼ Uniform[0, 2]. [Show More]

Last updated: 1 year ago

Preview 1 out of 3 pages