Consider the three-dimensional normal distribution p(x|ω) with mean μ and covariance matrix Σ where
μ = (123) , Σ = (3 2 42 0 24 2 3 )
Compute the matrices representing the eigenvectors and eigenvalues Φ and Λ to answer the following:
a. Find the probability density at the point x = (.75 0 1)?
b. Construct an orthonormal transformation y =ΦT x. Show that for orthonormal transformations, Euclidean distances are preserved (i.e., ‖?‖= ‖?‖)
In a particular binary hypothesis testing application, the conditional density for a scalar feature y given class w1 is ?((?|?1)=?1 ?−?210
Given class w2 the conditional density is ?((?|?2)=?2 ?−(?−2)22
a. Find k1 and k2, and plot the two densities on a single graph using Mat-lab/Octave
b. Assume that the prior probabilities of the two classes are equal, and that the cost for choosing correctly is zero. If the costs for choosing incorrectly are C12 = 1 and C21 = √5, what is the expression for the Bayes risk?
c. Find the decision regions which minimize the Bayes risk, and indicate them on the plot you made in part (a)
d. For the decision regions in part (c), what is the numerical value of the Bayes risk?
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