Lecture12

# Lecture12 - Characterization of Sums Central Limit Theorem...

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1 ± Central Limit Theorem The CLT is truly the “ubiquitous” tool in probability and statistics. It describes the statistical behavior of a sum of I.I.D RVs as their number grows large ( ). Denoting the sum by ( i.e. of n IID RV’s) we define where n n S n n X X X S ... 2 1 + + = (*) n nm S Z n n σ = ) var( ) ( 2 i i X X E m = = Characterization of Sums Theorem: Consider a sequence of IID RV’s with finite mean m and finite variance Let be defined as (*) , then One may also think of an infinite convolution of PDF’s which gives rise to a gaussian. +∞ = 1 } { n n X 2 n Z dx e z Z P z x n n = < 2 2 2 1 ) ( lim π ) ( i x f PDFs Convolution Example: Consider a sequence of independent RV’s such that Does the RV converge in the M.S. sense? Define ,apply Cauchy’s criterion: (m>n) } { n X 0 } { = n X E < = 1 2 } { n n X E i S = = n i i n X S 1 0 ) ( } ) {( } ) {( 1 2 1 2 2 = = + + = m n i m n i i m n X E X E S S E

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2 as since the series is known to converge for n=0 m n ⎯→ i s m n X S . .
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## This note was uploaded on 09/24/2009 for the course ECE 514 taught by Professor Krim during the Fall '08 term at N.C. State.

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Lecture12 - Characterization of Sums Central Limit Theorem...

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