week3 - #$/0/12/3E.061/1(787(9 F.001.5G%H...

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!"#$ &’() *+,-.-/0/12/3 45.061/1 &(787(9 :;;< = & !"#$ &’() *+,-.-/0/12/3 45.061/1 :;;< = >;32?+;1 [email protected] *+,BC D,11 "E.3E2;+ :;;< = !"#$ &’() *+,-.-/0/12/3 45.061/1 & F.001 .5G H+51 *,11/-/0/2/;1 .+; IJK $L?.006 >/<;06 "?MM,1; N; E.O; 2N, G/12/5P?/1E.-0; ?+51 .5G 5 /G;52/3.0 -.001) ;.3E ;L?.006 0/<;06 2, -; /5 ;/2E;+ ?+5 KE;+; .+; Q5R&S M,11/-/0/2/;1T3,++;1M,5G/5P 2, < -.001 /5 ?+5 &) < U () &) C C C ) 5 V,N;O;+) 2E,1; M,11/-/0/2/;1 .+; 5,2 ;L?.006 0/<;06 B,+ .56 5 W & *X < -.001 /5 ?+5 & Y ) < U () &) C C C ) 5 ZB 5 U ’) 2E;5 *X ( /5 ?+5 & Y U *X ’ Y U &7[) *X & S U &7’ :;;< & !"#$ &’() *+,-.-/0/12/3 45.061/1 = n k " # $ % & 2 n !"#$ &’() *+,-.-/0/12/3 45.061/1 >;32?+; 8 Z5G;M;5G;52 $O;521 :;;< = !"#$ &’() *+,-.-/0/12/3 45.061/1 = Z5G;M;5G;52 $O;521 K,G.6 \?;12/,51 Z5G;M;5G;53; ];B/5/2/,5) $^._M0;1 K.<; V,_; & G?; !,5G.6) J32 ’() .2 12.+2 ,B 30.11 *,12;G -6 ‘M_ ,5 a+/G.6) J32 &8 V.5G,?2b KE;1; I,2;1 :;;< = !"#$ &’() *+,-.-/0/12/3 45.061/1 [ Z5G;M;5G;52 $O;521 *X $ a c " Y U *X $ c " Y *X a c $ " Y U *X $ c a " Y *X a c " Y $ .5G a .+; !"#$%$"#$"& ()$"&* d/O;5 " /B *X $ a c " Y U *X $ c " Y *X a c "Y ZB *X $ a c " Y W ( 2E;5 $ .5G a .+; /5G;M;5G;52 P/O;5 " /B .5G ,506 /B *X a c $ " Y U *X a c " Y /B .5G ,506 /B *X $ c a " Y U *X $ c " Y e;55 ]/.P+._ 3.5 -; G+.N5 ,+2E,P,5.006 :;;< = !"#$ &’() *+,-.-/0/12/3 45.061/1 " a 3 a $ 3 $ $ 3 $ a a 3 $ $ 3 " *X$c"Y *X$ 3 c"Y *X$c"Y *X$ 3 c"Y Z5G;M;5G;53; Z5G;M;5G;52 $O;521 Q3,52/5?;GS d/O;5 2E.2 $ .5G " .+; /5G;M;5G;52 P/O;5 " *X $ c " Y U *X $ c a " Y U *X $ c a 3 " Y *X a c " Y U *X a c $ " Y U *X a c $ 3 " Y *+,,Bb *X $ a 3 c "Y U *X $ c " Y @ *X $ c "Y *X a c "Y U *X $ c " Y Q & @ *X a c " Y S U *X $ c " Y *X a 3 c " Y >/<;N/1;) *X $ 3 a c " Y U *X $ 3 c " Y *X a c " Y KE;+;B,+;) $ .5G a .+; /5G;M;5G;52 P/O;5 " /B .5G ,506 /B a 2;001 ?1 5,2E/5P .-,?2 $ 2E.2 N; G/G5f2 .0+;.G6 <5,N B+,_ " /B .5G ,506 /B $ 2;001 ?1 5,2E/5P .-,?2 a 2E.2 N; G/G5f2 .0+;.G6 <5,N B+,_ " :;;< = !"#$ &’() *+,-.-/0/12/3 45.061/1 " a 3 a $ 3 $ $ 3 $ *Xac"Y *Xa 3 c"Y *X$c"Y *X$ 3 c"Y *X$c"Y *X$ 3 c"Y g
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!"#$ &’() *+,-.-/0/12/3 45.061/1 &(787(9 :;;< = Z5G;M;5G;52 $O;521 Q3,52/5?;GS !?02/M0; $O;521 $&) C C C ) $5 .+; /5G;M;5G;52 P/O;5 " /B .5G ,506 /B a,+ ;^._M0;) $O;521 $) a) .5G d .+; /5G;M;5G;52 P/O;5 " /B .5G ,506 /B *X $ a d c " Y U *X $ c " Y *X a c " Y *X d c " Y .5G *X $ a c " Y U *X $ c " Y *X a c " Y .5G *X $ d c " Y U *X $ c " Y *X d c " Y .5G *X a d c " Y U *X a c " Y *X d c " Y :;;< = !"#$ &’() *+,-.-/0/12/3 45.061/1 P { Ei | S } = P { Ei | S } for all K " {1, , n } i # K $ i # K ! 8 $^._M0; =C[.) *0.6/5P h.+G1 D.5<1 .5G "?/21 >;2 ;O;52 $ 3,++;1M,5G 2, . 1?-1;2 ,B 2E; 3.+G1 /5 G;3< ZB $ /1 /5G;M;5G;52 ,B 2E; 1?/2 NE.2 _?12 -; 2+?; .-,?2 $i Z2 _?12 3,52./5 ;L?.0 5?_-;+1 ,B ;.3E 1?/2j :;;< = !"#$ &’() *+,-.-/0/12/3 45.061/1 9 " V ] h 4 C C C k " KN, /5G;M;5G;52 G/; 2,11;1 *X Q/) lS Y U Q&7gSQ&7gS U &7=g *X / ;O;5) l m [ Y U Q=7gSQ[7gS U &’7=g U &7= >;2 K< U ;O;52 2E.2 / R l U < K’ UW / U l U & 1, K’ /1 5,2 /5G;M;5G;52 ,B ;/2E;+ G/; J506 K8 /1 /5G;M;5G;52 ,B ;/2E;+ G/; FHK IJK FJKVj 456 ’ ,?2 ,B ]&) ]’) .5G K8 .+; /5G;M;5G;52 -?2 2E; 2E+;; 2,P;2E;+ .+; 5,2j $^._M0; =C[3) *./+ ,B ]/3; :;;< = !"#$ &’() *+,-.-/0/12/3 45.061/1 A & C C C g & C C C g " $^._M0; =C[-n) KN, h,/5 a0/M1 KN, /5G;M;5G;52 3,/5 B0/M1 *X V&) V’ Y U *X V& Y *X V’ Y U Q&7’SQ&7’S U &7[ "?MM,1; :/5 /B 3,/51 _.23E) : U V&V’ H K&K’ KE;5 *X : c V& Y U &7’ U *X : c K& Y U *X :Y 45G *X : c V’ Y U &7’ U *X : c K’ Y U *X :Y 456 ’ ,?2 ,B V&) V’) .5G : .+; /5G;M;5G;52 -?2 2E; 2E+;; 2,P;2E;+ .+; 5,2j
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This note was uploaded on 10/14/2008 for the course MS&E 120 taught by Professor Shatcher during the Fall '08 term at Stanford.

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week3 - #$/0/12/3E.061/1(787(9 F.001.5G%H...

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