131A_1_Lecture13-2_Winter_2012

131A_1_Lecture13-2_Winter_2012 - EE 131A Probability...

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UCLA EE131A (KY) 1 EE 131A Probability Professor Kung Yao Electrical Engineering Department University of California, Los Angeles Lecture 13-2
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UCLA EE131A (KY) 2 Central Limit Theorem (1) In probability, statistics, and engineering applications, Gaussian rv is often used. Why? The Central Limit Theorem provides one strong reason. Let X 1 , X 2 , …, be an infinite sequence of independent rv’s all with the same mean and variance 2 . Denote the partial sum Y(n) = X 1 + X 2 + … X n . The mean of Y(n), Y(n) , is given by 1n n1 Y(n) 1 2 n X ...X 1 n 1 n xx μ E{Y(n)} = ... (x x ... x )f (x ,. ..,x )dx . ..dx    
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UCLA EE131A (KY) 3 Central Limit Theorem (2) The mean of the sum of n independent rv’s is the sum of the means of the rv’s. 12 n n1 n X 1 1 X 2X n2 n xx = ... (x x ... x )f (x )dx f (x ). ..f (x )dx . ..dx        23 n 2n X 2 2 X 3 X n 3 n = ... ( μ x ... x )f (x )dx f (x ). ..f (x )dx . ..dx n n nX n n x = ( μμ . . . x) f (x)dx = ( ... μ ) = n μ .
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UCLA EE131A (KY) 4 Central Limit Theorem (3) The variance 2 Y(n) is given by The variance of the sum of n independent rv’s is the
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This note was uploaded on 02/29/2012 for the course ELECTRICAL 131a taught by Professor Yao during the Spring '12 term at UCLA.

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131A_1_Lecture13-2_Winter_2012 - EE 131A Probability...

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