mtsample

# mtsample - Z = | X-Y | 3(20 points Multiplicative noise...

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EE 278 Saturday, July 18, 2009 Statistical Signal Processing Handout #8 Sample Midterm This is a sample midterm. 1. (20 points) Inequalities. Label each of the following statements with =, , , or None . Label a statement with = if equality always holds. Label a statement with or if the corresponding inequality holds in general and strict inequality holds sometimes. If no such equality or inequality holds in general, label the statement as None . You have to justify your answers. a. (2 points) P( A | B ) vs. P( B | A ) b. (2 points) p X ( x ) vs. p X,Y ( x,y ) for X and Y discrete random variables. c. (2 points) f X ( x ) vs. f X,Y ( x,y ) for X and Y continuous random variables. d. (3 points) E p X 1 X 1 + X 2 P + E p X 2 X 1 + X 2 P vs. 1. e. (3 points) E( X log 2 X ) vs. 2 if X > 0 and E ( X ) = 2. f. (4 points) P { ( XY ) 2 > 16 } vs. 1 / 8 if E( X 4 ) = E( Y 4 ) = 2. g. (4 points) E(Var( X | Y 2 )) vs. E(Var( X | Y )). 2. (15 points) Function of exponential random variables. Let X and Y be independent exponen- tially distributed random variables with parameter λ . Find the pdf of the random variable

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Unformatted text preview: Z = | X-Y | . 3. (20 points) Multiplicative noise channel. The signal X and the noise Z are independent random variables and each is uniform in the interval (0 , 1). We observe the output Y = XZ . a. (10 points) Find the joint pdf of the random variables X and Y . b. (10 points) Find the MMSE estimate of X given Y . 4. (20 points) Fading channel. Let the signal X = b 1 with probability 1 / 2 2 with probability 1 / 2 and the channel gain A with pdf f A ( a ) = b λe-λa a ≥ otherwise be independent random variables. The channel output Y = AX is observed. a. (10 points) Find the optimal decoding rule D ( y ) for deciding whether X = 1 or 2, i.e., the rule that minimizes the probability of decoding error. b. (10 points) Find the probability of error for the decoding rule in part (a). Page 2 of 2 EE 278, Spring 2009...
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## This note was uploaded on 05/20/2011 for the course EE 278 taught by Professor Balajiprabhakar during the Winter '09 term at Stanford.

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mtsample - Z = | X-Y | 3(20 points Multiplicative noise...

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