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Unformatted text preview: EE 178 Tuesday, March 11, 2008 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 ( n i =1 A i ) vs. 1 2 2 n if A 1 ,A 2 ,...,A n are independent and P( A i ) = 3 / 4 for all i . 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 8 ) vs. 1680, if X N (0 , 2). e. M Z ( s ) vs. M X ( s ) M Y ( s ) if Z = X + Y , and X and Y are uncorrelated. f. P { XY < 16 } vs. 3 / 4, if X 0 and Y 0 are independent and E( X ) = E( Y ) = 2. 2. Function of two random variables Let X 1 and X 2 be independent r.v.s, where X 1 is distributed exponentially with param eter 1 and X 2 is distributed exponentially with parameter 2 . Let B be a Bernoulli r.v., independent of X 1 and X 2 , with P { B = 0 } = P { B = 1 } = 1 2 . Define Z = braceleftbigg min { X 1 ,X 2 } if B = 0 , max { X 1 ,X 2 } if B = 1 . Find the pdf of Z . 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 = 1 with probabaility 1 / 2 and = 1 with probability 1 / 2 be the signal transmitted over an additive uniform noise channel with output Y = + Z , where the noise Z U[ 3 / 2 , 3 / 2] is independent of ....
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This note was uploaded on 09/29/2008 for the course EE 178 taught by Professor Hewlett during the Spring '08 term at Stanford.
 Spring '08
 Hewlett

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