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oldfinal3 - UCSD ECE153 Prof Young-Han Kim Handout#37...

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UCSD ECE153 Handout #37 Prof. Young-Han Kim Thursday, May 26, 2011 Final Examination (Spring 2010) (Total: 180 points) Your answer should be as clear, readable (and short) as possible. 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 n -th 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.

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