oldmidterm1sol - UCSD ECE153 Handout #20 Prof. Young-Han...

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Unformatted text preview: UCSD ECE153 Handout #20 Prof. Young-Han Kim Tuesday, May 4, 2010 Solutions to Midterm (Fall 2008) 1. Coin with random bias. Let P be a random variable distributed uniformly over [0 , 1]. A coin with (random) bias P is flipped three times. Assume that the value of the bias does not change during the sequence of tosses. (a) What is the probability that all three flips are heads? (b) Find the probability that the second flip is heads given that the first flip is heads. (c) Is the second flip independent of the first flip? (d) What is the conditional pdf of the random bias P given the first flip is heads? Solution: Let X i , i = 1 , 2 , 3 , denote the outcome of the i-th coin flip. (a) By the law of total probability P { X 1 = H, X 2 = H, X 3 = H } = integraldisplay 1 P { X 1 = H,X 2 = H,X 3 = H | P = p } f P ( p ) dp = integraldisplay 1 p 3 f P ( p ) dp = integraldisplay 1 p 3 dp = 1 4 . (b) By the definition of conditional probability, P { X 2 = H | X 1 = H } = P { X 2 = H,X 1 = H } P { X 1 = H } . Again by the law of total probability P { X 1 = H } = integraldisplay 1 P { X 1 = H | P = p } f P ( p ) dp = integraldisplay 1 pdp = 1 / 2 , 1 and...
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oldmidterm1sol - UCSD ECE153 Handout #20 Prof. Young-Han...

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