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# oldmidterm1 - 2. Two independent uniform random variables....

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UCSD ECE153 Handout #19 Prof. Young-Han Kim Tuesday, May 3, 2011 Midterm Examination (Fall 2008) (Total: 120 points) There are 3 problems, each problem with 4 parts, each part worth 10 points. Your answer should be as clear and readable as possible. In particular, if the answer involves a pmf or pdf, make sure to identify the values or intervals for which the pmf or pdf is nonzero. 1. Coin with random bias. A random variable P is drawn uniformly from the interval [0 , 1]. Then a coin with bias P is Fipped 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 Fips are heads? (b) ±ind the probability that the second Fip is heads given that the ²rst Fip is heads. (c) Is the second Fip independent of the ²rst Fip? (d) What is the conditional pdf of the bias P given the ²rst Fip is heads?

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Unformatted text preview: 2. Two independent uniform random variables. Let X and Y be independently and uniformly drawn from the inverval [0 , 1]. (a) ±ind the pdf of U = max( X,Y ) . (b) ±ind the pdf of V = min( X,Y ). (c) ±ind the pdf of W = U-V . (Hint: In case you are stuck, you can start working on part (d) ²rst.) (d) ±ind the probability P {| X-Y | ≥ 1 / 2 } . 3. One-bit quantization of Gaussian sources. Let X ∼ N (0 , 1) and let Y = b 1 , if X ≥ ,-1 , otherwise. Thus Y encodes the sign of X . (a) ±ind the pmf of Y . (b) What is the conditional pdf of X given the observation that X is nonnegative? In other words, ²nd f X | Y ( x | 1). 1 (c) Find the minimum MSE (mean squared error) estimator of X given Y . That is, ±nd the estimator g ( y ) that minimizes the MSE E b ( X-g ( Y )) 2 B . (d) What is the associated MSE? 2...
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## This note was uploaded on 07/07/2011 for the course EECS 153 taught by Professor Kim during the Spring '11 term at UCSD.

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oldmidterm1 - 2. Two independent uniform random variables....

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