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sol05 - y EIY VARIY E liejmw— chug/2 jdw AcosWt c...

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Unformatted text preview: y EIY] VARIY] E liejmw— chug/2 jdw AcosWt + c E[A]cosWt+c= mcosm +c coszwt VAR[A}— _ 0' coszfit I} il '3 ll II A w. 3 | q ll 3 _H [_0_2ejmw—0202[2 + (1m _ 02w)2ejmw—azw2/2] 1 F[_02 +j2m2] = 02 +1712 E[X2] — ELY]2 = 02 Gm 1" 91:0 VARLX] —- 1- qz d 10 P9 6.! EX = —G 2: = — = —— [ ] dz XMI 1 (1-q2)2(g)2_1 (1-902 P (P pg 2199'? 2‘12 2 ._.. __ ._... _. E[X]“E[X] "' dz”GX( )221 ‘ (l—qz)32q 2-1 (1 —q)3 _ 2 m’rhe entropy is a. function of probabilities and it does not depend on the values taken by the RV. Thus Hy 2 Ex. 0% K H); —ZPk10ng k=1 K --1 — Z P), 1(3ng -—- PK log PK I:-1 —(1_— PK) log(1 — PK) + (1 — PK) Iog(l — PK) K—l But (1 —~ PK) = 2 Pk. Therefore, k=1 K—l K—I H}; 2 —-PK 11)ng - (1 — PK) 10g(1 —- PK) -- 2: Pk 10g Pk + E P); 10g(l — PK) K=l k=1 K—l Pk 2—P1P—1—P11—P-— P1! K05 K ( K)0g( A) El k°g(1——PK) We finish the proof by noting that K—~1 _ Pk . _ , __ Pk Hy _ I; PhlogdH—fl—(1_PK)smce P[Y—kfX 3£ I1] — 1 —PK I P[ Output 000, Input 000] PI t0000t 12000 = —~--~—-————— [ npu I u pu ] P[ Output 000] _ (1 —P)3 6 — (I—P)3’%+P3'% _ (1 —P)3 ‘ (1 — P)3 + P3 P3 P[Input llll Output 000] = m 1 — P 3 1 — P 3 P3 P3 HX|A=_ ( 3) 3103 ( 3) 3— 3 310g 3 3 (1—P) +P (1—p)+P (l—P) +P (1—P) +P If the output is 010, _ P(1 —P)2 P(1_P)2 3"" ‘ _P(1 — P)2 + P2(1 _ P) “5 P(1~ P)2 + mu m P) 132(1— P) 1 P(1— P)2 _P(1~ P)2 + P2(1 — P) 0g P(1-— P)2 + P2(1 -— P) ...
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