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20071ee131A_1_hw10sol

20071ee131A_1_hw10sol - EE 131A Homework#10 DO NOT TURN IN...

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Unformatted text preview: EE 131A Homework #10 DO NOT TURN IN Winter 2007 K. Yao E[X] == / z2e"'(1 — e") dz: = 2/ we“ dz —/ 22:5“ a’dzt: = -. o o 0 ~ 2 W 1* b‘ * °° 1 EV] = / 3/26"” dy = —. o 2 Val-[X] = g — (-)2 = 3—, Var[Y] = 22 %. E[XY] = /0 A my2e‘”e"’dydw =/o 2xe"(1—e"’——ze'”)dm 0 h—--——-—’ ‘-~—-v-—-—’ l 1 % 00 2/ we“"'dx—- o \——v—¥ 1 3.82 Y = X/n E[Y] = E[X]/'n : np/n =1; VARlY] = VAR[X]/112 = upq/n2 2;)(1/71, q z 1 - p z: W PHIY ‘11” > (113 = __ a2 na2 as n a co P[{|Y—p|} > a] —+ 0 for any fixed a > 0 4.64 EIXI = HY] = 0 is all cases, so £[XY] ”XO’Y va — a) 00 00 f 326"” da: —/ x2262” dz = 1. 0 i) SLYY] = ("1)22‘3+ 2( | )(—~ l.)é+ 12% = 0 => [)Xy := 0 uncorrelated and orthogonal ii) £[XY] = %[(—-l)2 + 2(l)(—~l)+ 12] = 0 :> pxy = 0 uncorrelated and orthogonal iii) EIXY :%(|)(—-l)+("l)(l))=_§ , J l 2 (73‘! = £14le = (—l)2§ +12 (5) = 5 = 0?, —2/3 =—1 .01 5.22 The relevant parameters are n = 1000, m 2 up = 500, a2 = npq = 250. The Central Limit 'I‘lu-xn‘cm then gives: 400 — 500 N —— m 600 —- 500 —-~— 5 ~—~— s ————— m (7 «250 11400 g N g 000] : P 22 Q(—6.324) —- Q(6.324) = 1 —- 2Q(6.324) 1 - 2.54(10—’°) 1 0(0) — Q(3.162) = E -— 7.300”) 11 P[500 s N s 550] 22 E[S,,] = 12E[X,'] = n -1 = n. Var[Sn] = no?“ = n -12 = h. Assuming 5,. approximately Gaussian: P[S,. > 15] = .P [bi/E." > 12"] z 06;") = 0.99. From Table 3.4: $57? = -—2.3263 => n - 2.3263‘fii— 15 = 0 => n = 27.04 => buy 28 pairs. S, = X + --. X 00 1 + 100 # crrom iid Bernoulli RVs 1:15.100] = 100]) = 15, vaI'ISwo] = lOOpq = 12.75. ”51005201 = 1*P[3100>20] __ Slog—15 20—15] _ 1-4—...) «12.75 «12.75 g 1—Q(1.4)=0.92. Total error is 3100 = X1+ X2 + .. - + X100, where X.- is uniform in [—%, fl, E[X;] = 0, and Var[X.'] :i’i" 3,00 6 P5 = N .... —2 [ups] q>[ loo[12> loom]_Q(2.078)_1.79(1o ). 8 . Obs. Expected (0 — €)2/€ 0 0 10.5 10.5 1 0 10.5 10.5 2 24 10.5 17.36 # degrees of freedom 2 9 3 2 10.5 6.88 1% significance level :> 21.7 4 25 ' 10.5 20.02 D2 > 21.7 5 3 10.5 5.36 :> Reject hypothesis 6 32 10.5 44.02 that #’s are 7 15 10.5 1.93 unif. dist. in {0,1,...,9} 8 2 10.5 6.88 9 2 10.5 6.88 105 m Obs. Expected (0 — €)2/€ 2 24 105/8 9.01 3 2 105/8 9.43 # degrees of freedom 2 9 4 25 105/8 10.74 1% significance level 2? 21.7 5 3 105/8 7.81 D2 > 21.7 6 32 105/8 77.41 :;‘> Reject hypothesis 7 15 105/8 0.27 that #’s are 8 2 105/8 9.43 unif. dist. in {0,1,...,9} 9 2 105/8 9.93 105 83.26 J" "L /\ /\ 4 7") e 5 f1 mmft Y 49 \l/ : a X+ h 507 ’7’} 19" j CVL,\I7£{M 4/1) '2. 1 1-, j “W :2. “q ""“ : /MX /¢1r7¢/MZM 4+6 0;, 71:2” 7-1—4 /3 A (101/ [x y) A A 4 .1. .. ._.. .......... ”2......” ~ ' _._. ‘X’Z , g2)~—‘/(/t Y ci/fl/l \ ..,,/ f .1. . V E g“ g l 74%} 4‘ 1:37.137 f 34/1 f) f “1““ \K' 1:: if»? “) m.” r W t \ 1...... H t (/Y 7‘ it...“ ;\t _, ”/4 3213/1 1.... f/Afif‘f/z W K? ..... 3 7 A I /\ 0‘ '1 '3“ ~ _. 3.... -3: 1 4 g? [3 ) fia 2 .43 x": 3 Z lK/N/l’: ~JKT§37§7 ...
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