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?

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Author:Amos Gilat
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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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