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# hw3sol - UCSD ECE 153 Prof Young-Han Kim Handout#10...

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UCSD ECE 153 Handout #10 Prof. Young-Han Kim Thursday, October 23, 2008 Solutions to Homework Set #3 (Prepared by TA Halyun Jeong) 1. Read Sections 6.5.1, 8.6.1–8.6.2 in the text. Try to work on all examples. 2. Coin with random bias. You are given a coin but are not told what its bias (probability of heads) is. You are told instead that the bias is the outcome of a random variable P Unif[0 , 1]. To get more information about the coin bias, you ﬂip it independently 10 times. Let X be the number of heads you get. Thus X B(10 , P ). Assuming that X = 9, ﬁnd and sketch the a posteriori probability of P , i.e., f P | X ( p | 9). Solution: In order to ﬁnd the conditional pdf of P, apply Bayes’ rule for mixed random variables to get f P | X ( p | x ) = p X | P ( x | p ) R 1 0 p X | P ( x | p ) f P ( p ) dp f P ( p ) . Now it is given that X = 9, thus for 0 p 1 f P | X ( p | 9) = p 9 (1 - p ) R 1 0 p 9 (1 - p ) dp = p 9 (1 - p ) 1 110 = 110 p 9 (1 - p ) . Figure 1 compares the unconditional and the conditional pdfs for P . It may be seen that given the information that 10 independent tosses resulted in 9 heads, the pdf is shifted towards the value 9 10 . 3. Signal or no signal (from Spring 2008 midterm). Consider a communication system that is operated only from time to time. When the communication system is in the “normal” mode (denoted by M = 1), it transmits a random signal S = X with X = ( +1 , with probability 1 / 2 , - 1 , with probability 1 / 2 . 1

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 f P (p) f P|X (p|9) Figure 1: Comparison of a priori and a posteriori pdfs of P When the system is in the “idle” mode (denoted by M = 0), it does not transmit any signal ( S = 0). Both normal and idle modes occur with equal probability. Thus S = ( X, with probability 1 / 2 , 0 , with probability 1 / 2 . The receiver observes Y = S + Z, where the ambient noise Z Unif[ - 1 , 1] is indepen- dent of S . (a) Find and sketch the conditional pdf
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hw3sol - UCSD ECE 153 Prof Young-Han Kim Handout#10...

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