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Unformatted text preview: EE 178 Tuesday, March 10, 2009 Probabilistic Systems Analysis Handout #22 Sample Final Problems The following are old exam problems. To prepare for the final you will also need to review the lecture notes, the homework, midterm review, and midterm problems. 1. Inequalities Label each of the following statements with =, ≤ , ≥ or NONE. Label a statement with = if equality always holds. Label a statement with ≥ or ≤ if strict inequality is possible. If no such equality or inequality holds in general, label the statement as NONE. Justify your answers. a. P( ∪ 100 i =1 A i ) vs. 0 . 01, if max 1 ≤ i ≤ 100 P( A i ) ≤ 10 4 . b. E [Var( X + Y  Z )] vs. Var( X ) + Var( Y ) if X and Y are conditionally independent given Z . c. E( X 4  Y ) vs. (E( X 2  Y )) 2 . d. E( X 2 Y 6 ) vs. 30, if X ∼ N (1 , 1) and Y ∼ N (0 , 1) are independent. e. M Z ( s ) vs. M X ( s ) M Y ( s ) if Z = X + Y , and X and Y are uncorrelated. f. P { ( X + Y ) 2 < 2 } vs. 0 . 5, if X and Y are random variables with zero mean, E( X 2 ) = E( Y 2 ) = 1, and ρ X,Y = . 5. 2. Two Exponential Random Variables. Let X and Y be two independent random variables exponentially distributed with mean 1. Find the pdf of Z =  X Y  . (Hint: You may wish to sketch the event { Z ≤ z } on the x – y plane.) 3. Function of two random variables Let U and V be two random variables uniformly distributed over the set ( u,v ) such that ≤ u ≤ 1 and 0 ≤ v ≤ 1 and let X = UV . a. Find the joint cdf of X and U , F X,U ( x,u ). b. Find and sketch the pdf of X , f X ( x ). c. Find the best MSE estimate of U given X . 4. Additive uniform noise channel Let the random variable Θ = 1 with probabaility 1 / 2 and Θ = 1 with probability 1 / 2 be the signal transmitted over an additive uniform noise channel with output Y = Θ+ Z , where the noise Z ∼ U[ 3 / 2 , 3 / 2] is independent of Θ. a. Find the a posteriori probability of Θ = 1 given y . Plot this probability as a function of y ....
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This note was uploaded on 10/26/2011 for the course MATH 180C taught by Professor Eggers during the Spring '09 term at Aarhus Universitet.
 Spring '09
 Eggers
 Systems Analysis

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