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# hw6 - EE 178 Probabilistic Systems Analysis Handout#14...

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EE 178 Handout #14 Probabilistic Systems Analysis Thursday, February 19, 2009 Homework #6 Due Thursday, February 26 1. Uncorrelation vs. Independence. Let X and Y be random variables with joint pdf f X,Y ( x, y ) = braceleftbigg c if 0 ≤ | x | ≤ | y | , 0 ≤ | y | ≤ 1 0 otherwise, where c is a constant. a. Find c . b. Are X and Y independent? Justify your answer. c. Are X and Y uncorrelated? Justify your answer. 2. Correlation Coefficient. Let X and Y have correlation coefficient ρ X,Y . What is the correlation coefficient between X and aY + b , where a and b are constants and a negationslash = 0 ? 3. Modified additive noise channel. Consider the additive noise channel shown in the figure below. a b X Z Y = b ( aX + Z ) The signal X and the noise Z have zero mean, average powers P and N, respectively, and are uncorrelated, and a and b are constants. Find the best linear MSE estimate of X given Y and its MSE in terms only of P , N , a , and b . 4. Repetition code. We wish to transmit 10 3 data bits U 1 , U 2 , . . . , U 10 3 , where U i Bern(0 . 5), for 1 i 10 3 , over a binary symmetric channel.

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hw6 - EE 178 Probabilistic Systems Analysis Handout#14...

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