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W07MATH323_Solutions-A1

# W07MATH323_Solutions-A1 - —.2 —.— 5 2 The Isamplpm...

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Unformatted text preview: -- —'..-_.._. .2 —. -....—:... ._..,-._-_ 5 2 The Isamplpm Space for the toss of three balanced coins, the values for 1’3 and Y2 a: eaCh -' .i outcome, and the probability of each outwme are g1 ven below: ______5 OUTCOMES {g}, y-g) PROBABILITY . HH‘H '(3, 1) § H—J HHT . (3, 1') g . I HTH -- (2. 1) § I 'E 2”? . HTT (1,1) . g— .1 ° . THH (2,2) g ' THT (1,2) g —-‘--—'1 'TTH (I, 3) ﬁ- ' {I‘TT - ' (0, — I) g; g . —'—._—J y! I' I 0 l 2 3 . Tm”; —1 2 o . o 0 . j _2_.___i 22 1 '. 0 —;—.- 2 g. . i‘ :_ g ' . 2 0 '2 2 ' 0 - . —.«I- -- 3- o . -§— 0 o - I 12- 21)~Pm<2Y2<r)—(0~1)+2(11)+2C21)—- .... —. .L... Then . _ 1 K ;_ K 1 I Ixy‘yzdyldyFKf (2’2 [22‘] dyz=2ufwdm=2[ use that If: 4 w m ‘ ”’ 4:“ 3d _ b_ Ftyhygpf' 1f 42122221 dt2_nf[ l: dtg— =af2yl 2 t2. .1322 < <ﬂland0<y2<L Recallthat _ . fore yl - 0, fory1<00ryz<0 ‘F{y1_.y_2)‘ -'H 1, form 2 I anting 1. .5. '2 ‘._ _.—_.. ..__.—_.... __..‘__.. ____ .. — .--... . .— --- _ .._l._ ..._., . ;. __ _ _ _ 5.1 0 The region over which the joint Mdensity function . is poSitiv‘e is the triangular region shown in :Figure 5. 5._ The shaded area is the region in . which Y1__ _. 321de < 3. " P(Yl<4=Y3 Fig) HM“: 'j3H.|-1r: __ ' f‘ If 2 dyl dye + f f 2 din dill ' m u MW _ (I a.“ I U" _ '__‘_"""" ——!-‘-—- -—-.H_I_...,.....-___. .____H_... 5.14:1. P(Y1<§.Y2>i—=) .£!.(yl+y2)dyldy2= I(%+%de2=% b Refer to Figure 5. 3 Integrating over jhe shaded region, we obtain ' PIYI+Yé S 1)=f- f (91+y2deldya _ I: II ' Ii hill—- 1 §. 1'1 Figure 5.8 15' 22 :3. By deﬁnition I 111111~ ”1111,11111111—1411111111 (.1111) (1%)];=211 1111-0511151 and . .' 1 1 ' '_ 13(92): f 4-313}? 6531— (4.112) (3’ )1“ == 231: for 0 S y; 5 l b. By the deﬁnition ofconditionai probability, “1151111th I}; 1 ' 1,1; Poll 5% Yi a): f 1:! 4311929!“ dill = f 29'! [9213,14 dim: 0 4 . ._. '1'" '— 6‘4 Now and P(Y-_1_ ~)— =f_ faith) din: f 231’? {192:3331314 *1 "T1? 314 ' .1114 Pt _-1iY1>11= 11111. Notice this the same probability as P(Y_ < 2) 1:. By Deﬁnition _.5 7, iflJ 1: yg_ < l ' Hwy?)— —' 13611): = \$113222”, Notice this' Is the same as Hm)- a. 1ft) < y;_ < l f (11111) 1323:”)— Notice this IS the sanie‘ as ﬂyg) Hence 5261. PI(Y1_ 2,13%} and l-{d . = f 2(1 ~y2) dye =1?— _ (l Heme l __ _ P(Y1 __iIY1< i)= igi=i Notice we could have this prohabiiity without integration as the joint density is conétaiit If 0 <2 3;; < l, the conditional distribution of '5’; given 1"; 15: {£311.31 _ 2 Iain) " W" or- __ fimtyz)=ﬁ 05.111 S 1 ~112- IfY§=4lthem -1 - 1 'f (lyliiﬂ = “1%) forO 5 y; S 1 and hence, ' ' 1r Y1Y< 111111111=1e1 ”2 Again notice we could have this probability without integration as the conditional density' 15 cwonstmt _ ._ f2{y2)= Calculate- . 1 ..--Ip0ﬁ2%%2§ gﬁyég 1-“ 31 1 . ) . , ' . ) .r -. c.- First consider Hamlin): AWE 1“] .5393 \$111329 *3?“ ' - - - f(21122)—‘“-’i‘ _ f2§iwéij ﬂ Theo - {mﬁﬁﬁﬂéﬁ' ..._.~__.—.._, .—-__._ ..-v - '____._.__ﬂ;#_#————.l.—-—-1--I—---..__..u____,___,_ ' . I - - I "fl. '_ " -___._, ..____ ..__ ,. . . _,,,._'-—- 5. 32 I113 given that f1('y{) -+ i for 0 < 3,115 l ,whereYl = amount stocked. Further for a '1‘”- ﬁxed value °f_Y1153)’ Yi= 1111 f(y'2l=y1) - “for 0 S '1}: S in, “harsh 2.3111011111501111 __- _I ‘- I .11. By definition, " ' . ‘ - - -' ~"-' . ' _-' ' ,forO < y;_ < y] < 1 fig-11:02): .(f1 (y1)f(y2ly1)={21‘-' D, elsewhere - ..: b. Given 1ha1 Y1: i , it- is__ necessary to f' 11d P (141>i12|Y_= %) Using the conditional density of 1’3 given Y1: g,’which 15 _ ' 2 0< -<— -1 2 1 = . 2112- . f(y?iy-1 '1‘: 2) 7G)" { ﬂ, olsewhore ‘ Wehave ' ' _ 3" _ ' u: 1 ' 23.03 3’ ili’i = i): f 2 {£92 =2yzli£=§ . m ' _ - c., The probability of interest is P (Y1 2 gm = 1—).1-1211122' it, is necessary to calculate ﬂyalyzi- Note th'at ' ' ' ' I I 1 f2f1f2)=f Tdyl=:nyl] =—'lﬂy2 05y2<1- SF! ' Then'if03y2_ <1 ' ffyzlyz)— — ”"1." 5. 4-0 No. bonsidering P(Y1 =3 Yg=11andp(Y1-=31p}’- =1)” 11.3 11=11é (1) (11 5121131122111 Thus,Y1and 1? are not Independent. -T_ __£ 111a rage of- y1 values on 1111111111 f( (1111,1111: 15 deﬁned depends on y; -_~-.p-—.--.— --—-—---- _-.._..__ —._._.....__..._______ —-—‘——....'.‘_ 5. 50 Depehdent as the range of 1,11 values 011 which f_(y1,y'g1 is deﬁned depends 1111 312. 1 More rigorously One couid verify mm the epiytitm to exercise .5. 28 that f(y1|}’1= ”1111 56 f( 112}. ﬁ—I k5.58lt11'1g111'r-111 111atu— “H h - I I“ I. I I .1 '1. 1.; - . ‘ 1 ‘ 12-01(91)? .18 1(7-11’11313‘“ 31:0 _1 2;. -P2('y2)'-=( 1131"(?11,f_ 112—0 11:. J ' 11- 131111.112)= m (111111211121— — (f -1 (EN 3)“(3)2‘“(7)‘"‘” fpry1§0,1,2-andyg=0,'._l ' _ 118111113 Y; 15 the number of customers in line'1,i= 1_,2 purchasing more 1111111 \$50 11:1 groceries, the probability of interest 13-- ﬁ(¥1+3’1<11-PY1—0 16=01+P(Y1= 1.16:0) 8+P(Y1=_B.Y2= (312(7)+2(2)(8)(7)+1319}!—...._,_._ _..___—-'y—- ...
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W07MATH323_Solutions-A1 - —.2 —.— 5 2 The Isamplpm...

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