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20071ee131A_1_hw9sol

20071ee131A_1_hw9sol - EE 131A Homework#9 Winter 2007 Do...

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Unformatted text preview: EE 131A Homework #9 Winter 2007 Do not need to turn in K. Yao I 4.12 a) The probability is obtained by integrating the joint pdf over the region indicated below. HX+YS$ (E 00 y-HO - = “d 2 ”lid (3) P[X Y S 10] A; 0 e 1: e y a: = /°°(1—e;‘”+‘°’) 0 "1—2‘8—10 31—13—10 3 “ 5” "A” P“; a; writ-mi} ) ‘ /°°xe”‘e ””dyr—xe'” (16“)00—5‘” 2.4.13 fX(‘” “ o z 0 x °° ’x“+"’((1 + y)x 1* 1) °” _. -r(1+y) _e fY(J) —- j; are da: “+302 0 __ 1 — (H31)2 3. 4.16 My) = PIYSy,X=+1}+P[Ysg,X=—1] 1 1 = Pi},Sin=+115+PIYSy|X=_1].2. 0 y<—3 0 y<—3 $05,453) —3Sy<—-1 %—3Sy<_1 = §(y+3)+ i(3/+1) —lSySl fy(y)= i _lsy_<_1 5(y+1)+;15yss 3 I<y_<_3 1 3<y 0 3<y 4‘ 4.17 a) P[X = i,Y s y] = P[Y S M = 1PM = 1'1 IfX= 1: —1 P[Y g le =1]P[X :1} = P[N+1< ylé- = if 3e 0|:le 180(y—l) y < 1 _ 2(2 — VOW—1)) y >1 X = —1 is obtained in similar fashion: 1 a(y+l) < =' = _ = e y < ‘1 P[Y —- yIX ”P[X 1] { 2(2 __ e—a(y+1)) y 2 _1 b) My) - Fly 5 le -1P{x -11+ P[Y s yIX = -1]p[x = _11 1 1 P[N+1£y]§+P[N'—1Sy]§ d 1 d 1 d fY(y) @FYW) - EEFNW — 1) + EEFNW +1) gm -1)+§f~(y +1) ._ _ W — __.s_.1‘ 16—" c) P[X —1iY > 01— my > 0] ‘ 2P1[Y > 0] e _ W _ £1.11le P[X — fllY > 01— p[}’ > 011 _ 2P[Y2> 0] -1- -° P[X:1|Y>0]—P[X=—~1|Y>0]=§%D—[-}%—b%>0 => X = 1 is more likely f(m,y) = ame‘flznbye’bya/2, 2:,y,a,b,>0 me) = /: f(w,y’)dy’ = axing/2 foo bye‘by2/2dy = axe—”2’2 0 M!) = bzze‘b””2 f(x,y) = f(:c)f(y) for all a: and y X and Y are independent. 6. 4.23 a) P[a<Xsb,ygd]=P[a<ng]P[Ygd] = (FX(b) “ FX(a))FY(d) b) W s X 5 17,6 3 Y s a1=(Fx(b)— Fx(a*)><Fy(d) — men) C) 13(le > 0’0 S Y S d] = (1 - Fx(a) + Fxéa‘))(Fy(d) - Fx(€‘)) _ fxy(x,y) _ $+y 7‘ 4'31fY(”'””)“ fx(w) n+1 °<y<1 2 3- lf.34X=cosG X7+Y2 = 1 Y=sine =>Y = im 1 1 fYMx) = 55(y+v1—x3)+—2—6(y_‘/1_$2) “Vi-“”1 = %(-V1-z2)+%x/1—T£7=0 alla: 7. 3.81 a) For a uniform random variable in [~b,b] we have —b O b Exact: 1- 9- 0 < < b P[|X—m|>c]={ 0 1’ c;;~ Chebyshev Bound gives 0’2 b2 P X — < i = ~— n ml > c1_ C: 362 b) For the Laplacian random variable 8[X] = 0 and VAR[X] = 2/0:2 Exact: P[IX — ml > c] = P[|X| > c] = e‘” 2 Bound: P[IX ~ ml > c] S W c) For the Gaussian random variable 8 [X] = O and VAR[X] = 02 Exact: P[IX -— m| > c] = 2Q (5—) 2 Bound: P[IX — ml > c] S E2— /0. 4.51 Z = 172(2) = fz(z) = I, d) P[X2<Y] = /°° /:°e e‘"’2e2”dyda: $.41 = [me "e "2” d1: 0 [00 e--2(x’+-‘-ac+-L )dm ooe— (1'+ L)2/2('}) (ah/Zia: 1)/0 dw 00 ”/2‘Z2 $+.l. e%\/:/\/— 6 dz Letz: 1.4 27l- ‘ 2 WQG) 1| 60/ ...
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