PRproblems by dheerajkushwaha

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									Q1. Consider a two-category classification problem with two-dimensional
feature vector X = ( x1, x 2 ). The two categories are ω1 and ω 2 , with equal

prior and 0-1 loss function.


                ⎡0⎤
p( X ω1 ) ~ N ( ⎢ ⎥, Σ1 )
                ⎣1⎦
                 ⎡2⎤
p( X ω 2 ) ~ N ( ⎢ ⎥, Σ 2 )
                 ⎣0⎦
                   1
P(ω1 ) = P(ω 2 ) =
                   2
     ⎡1 0⎤        ⎡1 1 ⎤
Σ1 = ⎢   ⎥, Σ 2 = ⎢1 2⎥,
     ⎣0 1⎦        ⎣    ⎦

Generate 100 bi-variate random training samples from each of the two
densities.

(a) Plot these samples in the two-dimensional feature space. Sketch the Bayes
    decision boundary when the true parameters are known.
(b) Now we assume that µ1, µ2 , Σ1, Σ 2 are not known. Find the

    maximum-likelihood estimates of µ1, µ2 , Σ1, Σ 2 using these training

    samples.
(c) Find the decision boundary using the estimated parameters. And draw the
    decision boundary in the same figure as part (a). Whether the two decision
    boundary are the same, why?
(d) Repeat parts (b) and (c) by drawing another set of 100 random samples
    from each class. Whether the decision boundary using different training
    sets are the same, why?

								
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