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Unformatted text preview: UCSD ECE153 Handout #40 Prof. YoungHan Kim Thursday, June 2, 2011 Solutions to Final (Spring 2010) 1. Polya’s urn revisited (40 points). Suppose we have an urn containing one red ball and one blue ball. We draw a ball at random from the urn. If it is red, we put the drawn ball plus another red ball into the urn. If it is blue, we put the drawn ball plus another blue ball into the urn. We then repeat this process. At the nth stage, we draw a ball at random from the urn with n +1 balls, note its color, and put the drawn ball plus another ball of the same color into the urn. Let X be the number of red balls in the first three draws. (a) Find the pmf of X by specifying P { X = k } for k = 0 , 1 , 2 , 3. (b) Find the conditional pmf of X given the first ball is red by specifying P { X = k  the first ball is red } for k = 0 , 1 , 2 , 3. (c) Find the optimal decision rule D ( x ) ∈ { red , blue } for deciding the color of the first ball given X that minimizes the probability of decision error. (d) Find the corresponding probability of decision error. Solution: (a) From the midterm, we already know that p X ( k ) = 1 / 4 for k = 0 , 1 , 2 , 3. (b) Let R be the event that the first ball is red. Then, by simple calculation, p X  R ( k  R ) = k = 0 , 1 / 6 k = 1 , 1 / 3 k = 2 , 1 / 2 k = 3 ....
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 Spring '11
 YoungHanKim

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