Chapters IID and II a : SUN new and sun proofs of WUN Last - PowerPoint PPT Presentation

Chapters IID and II a : SUN new

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Distributed ) ( Identically Defy variables { Xilie ± random of collection A distributed it for and all seek identically is have i. je I we any - IPCX ; ex ) IP ( Xi ex ) . different definite 'm than slightly one Note : text . the in

have = RCB ) MA ) A. B . Then EI suppose get we = { play it " Alka Ex ) a. * , if x > I 1 = { far , it " a. * , if x > I 1 IPC IBEX ) =

let f- EAD lebesgue , and let under I measure 7 } of is : nth decimal - { w Elo , D w An - [ Ho , %) So A - - - u [ 9%0,9%0 ) = ' shoo ) v , - [ Hoo , shoo ) v C' Huo , Az - . 99% . . ) - - - u [ 94% . . ' %ooJu - [ Hmo , % . o ) u ( ' 7ham , Az - last ' to By the BC An ) has An = Each . x distributed . { An ) new identically the example , are { Ian } NEIN

theorem fer ay identically distributed iff { Xilie # is - Iefffxj ) ) CECHXI ) ) have f : Rt IR measurable - we c- I all i. j for . be given We want FI ( ⇐ ) let seek . C- I all for i IP ( Xj Ex ) . IPC Xis x ) = have Efffxi ) ) - Hea for f Note - , we = ( Xi Ex ) , .

- IPCXJ Ex ) ( ⇒ ) fer all lP( Xi ex ) have - we Using The uniquest all i. j C- I and KEIR . is enough This Theorem , the extension result them PIXIE B) =lP( Xj EB ) for ay that to prove B EIR . Borel Eff ( Xj ) ) for IEC flxi ) ) show to proceed We through working function f by Borel measurable ay - negative fxn , ① indicate - fxns ④ simpletons ③ non cases : fins general ④

f- = HB Borel B Tun where is Suppose . = { Xlw ) EB # Xi if 1 - ' CB ) f- ( Xi ) i " Xlw ) # B if O i - IEHIX ; ) ) - RCXJEB ) - IPCXIEB ) ' IB ) ) - lP( Xi - ( Hxi ) ) - - limff ) Ica ) then simple lie , Now if f is ' ( Ci ) = f- use . Now Bi where Ci IB ; f- = E - IECHXJ ) ) ( Efflxi ) ) to expectation show of - linearity .

- negative function , can tried If f is we non a - negation fn sequence of u .li simple , : IR → R non a . f. ( x ) { - " Lion Hx ) ) if ftxkn { f n ) Tf ( recall co : if flx ) > n n to MCT argue Use - Eff ( Xj ) ) " I Elfnlxj ) ) IE ( fnl Xi ) ) = IECHXI ) ) = " T - by decomposing general f Finally , handle - f- : - f ft Dan .

Cer ( identical distribution implies equal moments ) distributed , then all identically { Xilie * If are moments agree : and central moments " ) - Mk ) = Efcxj - m ) IEC ( Xi . apply " II and - y ) ( x f- ( x ) choose - - result . previous ④

⇐ Define Iiit ) " if " did called { Xi lien are A collection identically distributed . and independent they are 7 } ath decimal = { we lo , D : is An let . { Han ) iid Then are .

version 2) therm ( SLLN , iid , and suppose that Xi , Xz , Suppose are - - for The . Then mean the common is MEIR - - t Xu ) = I ( X , t variables Sn convey random almost surely : to m m ) - 1 pl hnmsn - - - . dense . and PI Technical p ④ ,