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4. Given a set of two states of nature = {₁,w₂} with prior probabilities P(₁) = 0.4 and P(w₂) = 0.6. Furthermore, each pattern is represented by a two-dimensional feature vector x with Gaussian pdf p(x | wi) ~ N(Hi, Σ;) (1≤ i ≤2). Specifically, we have mean vectors ₁ = (0,0)*, 2 = (−1, 2)ª and covariance matrices Σ1 = 22 - (1₂2) ²2 - (²3) = 0 24 Please answer the following questions: a) Prove that Ep¹ -(105) and E5¹ = (-0.5 0.5); 1 = b) Prove that both Σ₁ and ₂ are positive definite; c) Specify the discriminant function for each state of nature; d) Given a pattern x = (0, 2)', which state of nature should we decide on x?

4. Given a set of two states of nature = {₁,w₂} with prior probabilities P(₁) = 0.4 and P(w₂) = 0.6. Furthermore, each pattern is represented by a two-dimensional feature vector x with Gaussian pdf p(x | wi) ~ N(Hi, Σ;) (1≤ i ≤2). Specifically, we have mean vectors ₁ = (0,0)*, 2 = (−1, 2)ª and covariance matrices Σ1 = 22 - (1₂2) ²2 - (²3) = 0 24 Please answer the following questions: a) Prove that Ep¹ -(105) and E5¹ = (-0.5 0.5); 1 = b) Prove that both Σ₁ and ₂ are positive definite; c) Specify the discriminant function for each state of nature; d) Given a pattern x = (0, 2)', which state of nature should we decide on x?

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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= 4. Given a set of two states of nature = {w₁,w₂} with prior probabilities P(₁) = 0.4 and P(w₂) = 0.6. Furthermore, each pattern is represented by a two-dimensional feature vector x with Gaussian pdf p(x|w₁) ~ N(i, Σ;) (1≤ i ≤2). Specifically, we have mean vectors ₁ = (0,0)*, ₂ = (−1,2)¹ and (12) (²2). 02 Please answer the following questions: a) Prove that E¹ = (1 (0.5) and Ezt Σ₂² = (-0.5 b) Prove that both Σ₁ and ₂ are positive definite; c) Specify the discriminant function for each state of nature; d) Given a pattern x = (0, 2)', which state of nature should we decide on x? covariance matrices Στ = Σ2 = - (-0.5 -0.5);
Transcribed Image Text:= 4. Given a set of two states of nature = {w₁,w₂} with prior probabilities P(₁) = 0.4 and P(w₂) = 0.6. Furthermore, each pattern is represented by a two-dimensional feature vector x with Gaussian pdf p(x|w₁) ~ N(i, Σ;) (1≤ i ≤2). Specifically, we have mean vectors ₁ = (0,0)*, ₂ = (−1,2)¹ and (12) (²2). 02 Please answer the following questions: a) Prove that E¹ = (1 (0.5) and Ezt Σ₂² = (-0.5 b) Prove that both Σ₁ and ₂ are positive definite; c) Specify the discriminant function for each state of nature; d) Given a pattern x = (0, 2)', which state of nature should we decide on x? covariance matrices Στ = Σ2 = - (-0.5 -0.5);
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