20061ee131A_1_131a7sol

20061ee131A_1_131a7sol - EE 131A Homework#7 Solution Winter...

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Homework #7 Winter 2006 Solution K. Yao 1. (3.74) Y = A cos( Wt ) + c, E [ Y ] = E [ A ]cos( Wt ) + c = m cos( Wt ) + c V ar [ Y ] = cos 2 ( Wt ) V ar [ A ] = σ 2 cos 2 ( Wt ) . 2. (3.78) a. E [ Y ] = Z -∞ g ( x ) f X ( x ) dx = Z - a -∞ ( x + a ) f X ( x ) dx + Z a - a 0 f X ( x ) dx + Z a ( x - a ) f X ( x ) dx = Z - a -∞ xf X ( x ) dx + Z a xf X ( x ) dx + aF X ( - a ) - a (1 - F X ( a )) E [ Y 2 ] = Z - a -∞ ( x + a ) 2 f X ( x ) dx + Z a ( x - a ) 2 f X ( x ) dx = Z - a -∞ x 2 f X ( x ) dx + Z a x 2 f X ( x ) dx + 2 a Z - a -∞ xf X ( x ) dx - 2 a Z a xf X ( x ) dx + a 2 F X ( - a ) + a 2 (1 - F X ( a )) V AR ( Y ) = E [ Y 2 ] - ( E [ Y ]) 2 b. Since pdf of a Laplacian RV is an even function and Y=g(X) is an odd function, then E[Y]=0. E [ Y 2 ] = 2 Z a x 2 β 2 e - βx dx + 2 a 2 F X ( - a ) = βe - βx β 2 x 2 + 2 βx + 2 ( - β ) 3 x = a + 2 a 2 ( 1 2 e - βa ) = 2 a 2 e - βa + 2 ae - βa β + 2 e - βa β 2 3. (3.79) Y = K + LX, E [ Y ] = K
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This note was uploaded on 06/09/2010 for the course EE 131A taught by Professor Lorenzelli during the Spring '08 term at UCLA.

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20061ee131A_1_131a7sol - EE 131A Homework#7 Solution Winter...

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