oldfinal3sol

oldfinal3sol - UCSD ECE153 Handout#40 Prof Young-Han Kim...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: UCSD ECE153 Handout #40 Prof. Young-Han 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 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. (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 ....
View Full Document

Page1 / 4

oldfinal3sol - UCSD ECE153 Handout#40 Prof Young-Han Kim...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online