Сборник задач по высшей математике 2 том
.pdfQ a) TaK KaK
h(x) = fx(x) = {2(1- x), x E [0,1]'
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x ¢ [0,1]' |
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IIpH X ~ °HMeeM |
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Fl(X)= |
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Fl(X) = jOdx+ j2(I-x)dx+ jOdu=1.
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TaKHM 06pa30M, |
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x ~ 0, °< x ~ 1,
1 < x.
AHaJIOrH'IHOHaXO)l,HM, 'ITO
Y ~ 0,
0< Y ~ 1, 1 < y.
6) IIcIIOJIb3Y5I <P0PMYJIY
P{XI ~ X ~ X2, Yl ~ Y ~ Y2} = F(X2,Y2)-F(Xl,Y2)-F(X2,Yl)+F(xl,yt},
HaXO)l,HM HCKOMYIO Bep05lTHOCTb:
P{0,7 ~ X ~ 3, °~ Y ~ 0,3} =
=F(3; 0,3) - F(0,7j 0,3) - F(3; 0) + F(0,7j 0) = |
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= (0,3)2 - (2.0,7.0,32 - 0,72 .0,32) - 02 + (2·0,7· |
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= 0,09 - 0,126 + 0,0441 = 0,0081. |
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IIoJIY'IHMTaKoil: :lKe OTBeT, KaK H B 3a)l,a'le6.12.21. |
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6.12.23. 3a)l,aHa IIJIOTHOCTb COBMecTHoro pacIIpe)l,eJIeHH5I CHCTeMbI HeIIpe-
pbIBHbIX c. B. (X, Y): |
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y) = {c. xy, |
eCJIH (x, y) ED, |
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eCJIH(x,y)¢D, |
400
r,n:e D = {(x,y): x ~ 0, y ~ 0, x +y ~ I}. Haitnl:
a) K03<pqHIIl;HeHT Cj
6) nJIOTHOCTH pacnpe,n:eJIeHHH OT,n:eJIbHbIX KOMnOHeHT X H Y j
B)<PyHKIJ;HH pacnpe,n:eJIeHHH OT,n:eJIbHbIX KOMnOHeHTj
r)BepOHTHOCTb C06bITHH A = {X > ~, Y ~ I}.
6.12.24.,IJ,BYMepHM CJIY'IaitHMBeJIH'IHHa (X, Y) HMeeT nJIOTHOCTb pacnpe,n:eJIeHHH BepoHTHocTeit
!(x,y) = ( 1+ x2)C (3 + y2) |
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x E JR, |
Y E JR. |
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HaitTH:
a) 3Ha'IeHHeBeJIH'IHHbICj
6) <PYHKIJ;HIO pacnpe,n:eJIeHHH F(x,y)j
B) nJIOTHOCTH pacnpe,n:eJIeHHH OT,n:eJIbHbIX KOMnOHeHT X H Y j r) BepOHTHOCTb C06bITHH A = {X < 1, Y < V3}.
6.12.25. HCnOJIb3YH YCJIOBHe 3a,n:a'IH6.12.21, npOBepHTb, 3aBHCHMbI JIH CJIy- 'IaitHbleBeJIH'IHHbIX H Y.
a npoBepHM, BbIllOJIHHeTCH JIH YCJIOBHe He3aBHCHMOCTH ,n:BYX HenpepbIBHbIX CJIY'IaitHblxBeJIH'IHHX H Y: !(x,y) = II (x) . h(y).
B xo,n:e peIIleHHH 3a,n:a'IH6bIJIH nOJIY'IeHbICJIe.n:yIOIu;He pe3YJIbTaTbI:
1) C = 4 H, 3Ha'IHT,nJIOTHOCTb pacnpe,n:eJIeHHH BepoHTHocTeit !(x, y) HMe-
eT BH,n: ° °
!(x, y) = { ~~(1- x), npH ~ x ~ 1, ~ y ~ 1,
BOCTaJIbHbIX CJIY'IMXj
2)nJIOTHOCTH pacnpe,n:eJIeHHH OT,n:eJIbHbIX KOMnOHeHT X H Y HMelOT BH,n:
lI(x) = {2(1- x), |
npH °~ x ~ 1, |
0, |
B OCTaJIbHbIX CJIY'IMX. |
h(Y) = {2Y, |
npH °~ y ~ 1, |
0, |
B OCTaJIbHbIX CJIY'IMX. |
KaK BHMM, paBeHcTBo !(x, y) = II (x)· h(y) BbIllOJIHHeTCH. CJIe,n:oBaTeJIbHO, CJIY'IaitHbleBeJIH'IHHbIX H Y He3aBHCHMbI. B 3TOM }l{e MO}l{HO y6e,n:HTbcH,
npoBepHB BbIllOJIHeHHe paBeHcTBa F(x,y) = F1 (x)· F2(y). •
6.12.26. HCnOJIb3YH YCJIOBHe 3a,n:a'IH6.12.23, BbIHCHHTb, HBJIHIOTCH JIH CJIy-
'IaitHbleBeJIH'IHHbIX H Y He3aBHCHMbIMH.
6.12.27. HCnOJIb3YH YCJIOBHe 3a,n:a'IH6.12.24, nOKa3aTb, 'ITOCJIY'IaitHbleBeJIH'IHHbIX H Y He3aBHCHMbI.
6.12.28. He3aBHcHMble CJIY'IaitHbleBeJIH'IHHbIX H Y HMelOT COOTBeTCTBeH-
HO nJIOTHOCTH:
II (x) = {3e- 3Z , |
npH x |
~ 0, |
npH y ~ 0, |
0, |
npH x |
< 0, |
npH y < 0. |
401
Haihu:
a) n~OTHOCTb pacnpe~e~eHU~ ~BYMepHO~ c~yqa~HO~ Be~UqUHhl
(X,Y);
6) <PYHKIJ;UIO pacnpe~e~eHU~ Fxy(x,y).
6.12.29. IIcno~b3Y~ yc~oBue 3Maqu 6.12.21, H~TU yc~oBHbIe n~OTHOCTU
a |
OT~e~bHbIX KOMnOHeHT X U Y. |
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.n~~ H~XO)K~eHU~ yC~OBHO~ n~OTHOCTU pacnpe~e~eHU~ f(x I y) BocnO~b |
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3yeMc~ <POPMY~O~ f(x I y) = |
f(x,y) |
npu Bcex y E [0,1]. TaK KaK |
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h(y) |
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f (x, y) = {4Y(1 |
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B OCT~bHbIX c~yqMX , |
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h(Y) = {2Y, |
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B OCT~bHbIX c~yqMX, |
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f(x I y) = {2(1- x), |
o:::;; x :::;; 1, 0:::;; y :::;; 1, |
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B OCT~bHbIX C~yqMX. |
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f(y Ix) |
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Y o:::;; x :::;; 1, 0 :::;; y :::;; 1, |
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B OCT~bHbIX c~yqMX. |
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OTMeTUM, qTO 6e3yc~oBHbIe n~OTHOCTU pacnpe~e~eHu~ KOMnOHeHT X U Y paBHbI cooTBeTcTBYIOIIJ;UM yC~OBHbIM n~OTHOCT~M. 9TO ~oKa3bIBaeT, qTO c~y qa~HbIe Be~UqUHbI X U Y He3aBUCUMbI. •
6.12.30.IIcno~b3y~ yc~oBue 3Maqu 6.12.23, Ha~Tu yc~oBHbIe n~OTHOCTU
KOMnOHeHT X U Y.
6.12.31. .nBYMepHM c. B. (X, Y) UMeeT paBHOMepHoe pacnpe~e~eHue Bepo-
~THocTe~ B Tpeyro~bHo~ o6~acTu D (.0.ABC), T. e.
f(x,y) = {~' |
ec~u |
(x,y) E D, |
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ec~u |
(x, y) f/. D, |
r~e S - n~OIIJ;Mb 06~acTU D. Koop~uHaTbI BepWUH Tpeyro~b
HUKa ABC TaKOBbI: A(-1; 1), B(l; 1), C(O; 0). Hahu n~OTHOCTU pacnpe~e~eHu~ KOMnOHeHT X U Y, yc~oBHbIe n~OTHOCTU pacnpe-
~e~eHU~ c. B. X U Y . .HB~~IOTC~ ~u c. B. X U Y He3aBucuMbIMH?
6.12.32. IIcno~b3y~ yc~oBue 3Maqu 6.12.21, Ha~Tu:
a) M(X), M(Y); |
6) D(X), D(Y). |
a a) IIcno~b3y~ <POPMY~y |
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M(X) = |
!00 !00 x . f(x, y) dxdy |
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- 00 - 00 |
402
Ji! yqllTbIBaH, qTO BHe 06JIaCTll D llMeeM f(x, y) |
= 0, |
HaXO,Il.llM MaTeMaTllqe- |
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CKOe mKll,Il.aHlle KOMnOHeHTbI X: |
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M(X) = II X· 4y(1- X) dxdy = 4 I x(l- x) dx I ydy = |
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X) 11 |
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AHaJIOrllqHO HaXO,Il.llM M (Y): |
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M(Y) = Ily· f(x,y) dxdy = Ily· 4y(1- x) dxdy = |
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D |
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D |
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= 4 j(1 - x) dx jy2 dy = 4 } |
1- x) dx· y; I: = ~ (X - ~2) I: = ~.!= i. |
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OTMeTllM, qTO M(X) II M(Y) MO}l{HO TaK }l{e HaitTll, llCnOJIb3YjI <POPMYJIbI:
00 |
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M(X) = I X· h(x) dx, |
M(Y) = I y. /2(Y) dy. |
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A llMeHHO: |
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M(X) = jx . 2(1 - x) dx = 2 (X; -X;) I: = 2 . i = ~; o
M (Y) = I1 y . 2y dy = 2 . y; I: = i· o
6) )l;JIjI HaXO}l{,Il.eHlljI ,II.llCnepCllll c. B. X MO}l{HO BOCnOJIb30BaTbCjI O,Il.HOit
113 CJIe,Il.yIOIII"X <P0PMYJI
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D(X) = I |
I (X - a:IY .f(x, y) dxdy; |
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- 00 - 00 |
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D(X) = I |
I X 2 . f(x,y) dxdy - (a:lY; |
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- 00 - 00 |
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D(X) = I (X - a:lY· h(x) dx = I X 2 . h(x) dx - |
(a:lY· |
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3,I1.eCb ax = M(X). Hait,Il.eM D(X), llCnOJIb3YjI nepBYIO <P0PMYJIY: |
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D(X) = II(x-~) |
2 |
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121 |
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.4Y(1-x)dxdY=4/(1-x)(x-~) |
dx Iydy= |
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D |
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0 |
0 |
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= 4 j(1- x) (X2 - |
ix + ~) dx· (y; I:) = |
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= 21 (X2 - |
~x + 1 - x3 + ~X2 - |
Ix) dx = |
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= 2 (Q .x 3 _ |
1 . x 2 + Ix _ X4) 11 = 2 (Q _1.. + 1 _ 1) = 1... |
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18 . |
Tenepb Hait,Il.eM D(X) ,Il.pyrHM cnoco6oM, HCnOJIb3YH TpeTblO <P0PMYJIY:
D(X) = [ 1x 2 . /I (x) dx - a~] = jx 2 ·2(1- x) dx - G)2 =
-00 0
= 2 (X; -~4) I:-~ = i-~ = l8'
OTCIO,Il.a BH,Il.HO, 'ITOBTOPOit cnoco6 OKa3aJICH npow;e. HaxO,Il.HM 9THM cnoco-
60M D(Y):
00 |
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(i) |
2 |
D(Y) = 1y2 . h(Y) dy - (a y)2 =1y2 . 2ydy - |
= |
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0 |
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= 2 • Y44 1: - ~ = ~- ~ = l8' •
6.12.33. IIcnoJIb3YH YCJIOBHe 3a,I1.a'IH6.12.21, HaitTH KOppeJIHUHOHHblit MoMeHT Kxy (HJIH: COY (X, Y)) H K09<P<PHuHeHT KOppeJIHUHH TXY.
Q KOppeJIHUHOHHblit MOMeHT c. B. X H Y MO:>KHO HaitTH, HCnOJIb3YH <pop-
MyJIbI |
00 |
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Kxy = 11(x - ax)(Y - |
ay)f(x, y) dxdy |
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- 00 - 00 |
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HJIH |
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Kxy = 11xy' f(x,y) dxdy - |
axay. |
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- 00 - 00 |
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3,I1.eCb ax = M(X), ay = M(Y). BOCnOJIb3yeMcH BTOPOit <P0PMYJIOit. |
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Kxy = 11xy . 4y(1 - |
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1 |
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x) dxdy -1'i = 41(x - x 2) dx 1y2 dy - ~ = |
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D |
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0 |
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= 4 ·1(x; - x;) I: -~ = ~. i -~ = o. |
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Kxy |
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TXY = a(X) . a(Y) = O. |
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2horo CJIe,Il.OBaJIO O:>KH,Il.aTb, Be,Il.b X H Y - |
He3aBHCHMble CJIY'IaitHbleBeJIH- |
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'IHHbd(eM. pemeHHe 3a,I1.a'IH6.12.25.) |
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404
6.12.34. lIcnoJIb3YH YCJIOBHe 3a,Il;aqH 6.12.31, HaitTH:
a) MaTeMaTHqeCKHe O)KH,Il;aHHH M(X) H M(Y); 6) ,Il;HCnepCHH D(X) H D(Y);
B)K09<P<PHIJ;HeHT KOppeJIHIJ;HH r x y .
6.12.35.lIcnoJIb3YH YCJIOBHe 3a,Il;aqH 6.12.23, HaitTH:
a) M(X) H M(Y); |
6) D(X) H D(Y); |
B) COV (X, Y); |
r) rXY. |
6.12.36. ITJIOTHOCTb COBMeCTHoro pacnpe,IJ;eJIeHHH CJIyqaitHblx BeJIHqHH X
H Y 3a,Il;aHa <P0PMYJIOit
f(x,y) = {00',25(1- xy3), npH -1 ~ x ~ 1, -1 ~ Y ~ 1,
B OCTaJIbHbIX CJIyqMX.
HaitTH:
a) K09<P<PHIJ;HeHT KOppeJIHIJ;HH c. B. X H Y;
6) 6e3YCJIOBHble H YCJIOBHble nJIOTHOCTH pacnpe,Il;eJIeHHH c. B. X
H Y;
B)YCJIOBHOe MaTeMaTHqeCKOe O)KH,Il;aHHe M (Y IX = x).
6.12.37.CJIyqaitHble BeJIHqHHbI X H Y He3aBHcHMbI, HMelOT nJIOTHOCTH pacnpe,Il;eJIeHHH COOTBeTCTBeHHO
hex) = {Al e- AiX , |
x ~ 0, |
y ~ 0, |
0, |
x < 0, |
y < O. |
(AI> 0, A2 > 0). HaitTH:
a) nJIOTHOCTb COBMeCTHoro pacnpe,Il;eJIeHHH f (x, y);
6)P{X > Y};
B)3HaqeHHH M(X) H M(Y).
6.12.38.ITo IJ;eJIH npOH3BO,Il;HTCH ,Il;Ba He3aBHCHMbIX BbICTpeJIa. BepoHTHoCTb nona,Il;aHHH B IJ;eJIb npH nepBOM BbICTpeJIe paBHa 0,8, npH BTOPOM
0,9. CJIyqaitHM BeJIHqHHa X - qHCJIO nOna,Il;aHHit npH nepBOM BbICTpeJIe, Y - qHCJIO nOna,Il;aHHit npH BTOPOM BbICTpeJIe. HaitTH: a) 3aKOH pacnpe,Il;eJIeHHH CHCTeMbI CJIyqaitHblx BeJIHqHH
6) 6e3YCJIOBHble 3aKOHbI pacnpe,Il;eJIeHHH OT,Il;eJIbHbIX KOMnOHeHT X
H Y H HX <PyHKIJ;HH pacnpe,Il;eJIeHHH;
B) <PYHKIJ;HIO pacnpe,Il;eJIeHHH
6.12.39. <l>yHKIJ;HH pacnpe,Il;eJIeHHH CHCTeMbI ,Il;HCKpeTHblx c. B. (X, Y) HMeeT
BH,Il; |
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npH |
y ~-4 |
-4 < y ~ 1 |
1<y~8 |
8<y |
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x ~-2 |
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-2 < x ~ 3 |
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3<x |
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405
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HaihH Ta6JIHUY pacnpe,n:eJIeHHjI CJIyqaitHoro BeKTOpa (X, Y)j pjl.JJ: |
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pacnpe,n:eJIeHHjI C. B. Yj BepOjITHOCTb C06bITHjI {X > Y}. |
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6.12.40. |
CHMMeTpHqHYID MOHeTY no,n:6pacbIBaIDT 3 pa3a. IIycTb c. B. X |
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KOJIHqeCTBO rep6oB, BbIIIaBIIIHX B nepBOM H BTOPOM HcnbITaHHjIX, |
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c. B. Y - KOJIHqeCTBO rep6oB, BbIIIaBIIIHX BO BTOPOM H TpeTbeM |
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HCnbITaHHjIX. HaitTH: COBMeCTHoe pacnpe,n:eJIeHHe c. B. X H Y j |
Be- |
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POjITHOCTb C06bITHjI {X f; Y}. |
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6.12.41. |
3a,n:aHO pacnpe,n:eJIeHHe ,n:BYMepHoit CJIyqaitHoit BeJIHqHHbI (X, Y) |
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X\Y |
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YCTaHOBHTb, 3aBHCHMbI JIH KOMnOHeHTbI X H Y. HaitTH |
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P{XY > 2}. |
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6.12.42. |
lIcnoJIb3YjI YCJIOBHe 3a,n:aqH 6.12.6, HaitTH: |
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a) YCJIOBHblit 3aKOH pacnpe,n:eJIeHHjI c. B. Y npH X = 2,5j |
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6.12.43. |
6) P{X = Xi IY = 2}j |
B) P{v'X2 + y2 ~ V2}. |
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CHcTeMa CJIyqaitHblx BeJIHqHH (X, Y) 3a,n:aHa Ta6JIHIreit pacnpe,n:e- |
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JIeHHjI
X\Y |
-1 |
0,2° |
1 |
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0,1 |
0,1 |
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0,1 |
0,3 |
0,2 |
HaitTH:
a) 6e3YCJIOBHblit 3aKOH pacnpe,n:eJIeHHjI c. B. Yj
6) |
3aKOH pacnpe,n:eJIeHHjI |
c. B. Y npH YCJIOBHH, qTO X = OJ |
B) |
BepOjITHOCTb C06bITHjI |
{X = 0, Y ~ O}. |
6.12.44. Cpe,n:H 10 JIOTepeitHblx 6HJIeTOB eCTb 2 BbIHrpblIIIHbIX. CHaqaJIa ,n:e-
BylliKa BbITjIrHBaeT O,n:HH 6HJIeT, 3aTeM O,n:HH 6HJIeT BbITjIrHBaeT IDHOllia. OnHcaTb 3aKOH pacnpe,n:eJIeHHjI CHCTeMbI CJIyqaitHblx Be-
JIHqHH (X, Y), r,n:e X - |
qHCJIO BbIHrpbIIIIHbIX 6HJIeTOB y ,n:eByIIIKH, |
Y - Y IDHOIliH. HaitTH: |
6) P{Y = Yi IX = I}. |
a) P{X > Y}j |
6.12.45. lIcnoJIb3YjI YCJIOBHe 3a,n:aqH 6.12.6, HaitTH M(X), D(X), a(X).
6.12.46. lIcnoJIb3YjI YCJIOBHe 3a,n:aqH 6.12.11, HaitTH MaTeMaTHqeCKHe Q}KH-
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,n:aHHjI H ,n:HcnepCHH CJIyqaitHblx BeJIHqHH X H Y. |
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6.12.47. |
lIcnoJIb3YjI YCJIOBHe 3a,n:aqH 6.12.6, HaitTH rXY. |
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6.12.48. lIcnoJIb3YjI YCJIOBHe 3a,n:aqH 6.12.11, HaitTH Kxy, rXY' |
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6.12.49. |
,!1;BYMepHM CJIyqaitHM BeJIHqHHa 3a,n:aHa Ta6JIHrreit pacnpe,n:eJIe- |
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HHjI |
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X\Y |
4 |
5 |
6 |
7 |
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1 |
0,08 |
0,10 |
0,10 |
0,03 |
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2 |
0,08 |
0,14 |
0,16 |
0,05 |
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3 |
0,04 |
0,06 |
0,14 |
D |
406
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RaitTlI BeJIHqHHY D, O,Il.HOMepHbIe paCnpe,Il.eJIeHHH COCTaBJIHIOru:HX; |
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npOBepHTb He3aBHCHMOCTb CJIyqaitHbIx BeJIHqHH X H Y; |
BbIqH- |
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CJIHTb M(X), M(Y), D(X), D(Y), a(X), a(Y), cov (X, Y) =Kxy , |
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rXY· |
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6.12.50. |
CHcTeMa CJIyqaitHbIx BeJIHqHH (X, Y) nO,Il.qHHeHa 3aKoHY pacnpe- |
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,Il.eJIeHHH C nJIOTHOCTblO |
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f(x, y) = { ~:sin (x + y), |
B 06JIacTH D, |
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BHe 06JIaCTH D, |
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r,Il.e D = {(x,y) : x;::: 0, x ~ i, y;::: 0, y ~ i}. RaitTH: |
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a) K03<P<PHIJ;HeHT C; |
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6) nJIOTHOCTH pacnpe,Il.eJIeHHH OT,Il.eJIbHbIX KOMnOHeHT X H Y; |
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B) BepOHTHOCTb nOna,Il.aHHH CJIyqaitHoit TOqKH (X, Y) B npHMO- |
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yrOJIbHHK, OrpaHHqeHHbIit npHMbIMH x = 0, x = ~, y = 0, |
y = i' |
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6.12.51. |
CHcTeMa HenpepbIBHbIX c. B. (X, Y) paBHoMepHo pacnpe,Il.eJIeHa |
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(T.e. f(x,y) = C = const) BHyTpH 3JIJIHnCa 9x2 + 16y2 |
~ 144, |
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BHe 3JIJIHnCa f(x, y) = 0. RaitTH: |
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a) cOBMecTHYIO nJIOTHOCTb f(x, y); |
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6) nJIOTHOCTH KOMnOHeHT X H Y (T.e. /I(x) H h(Y»; |
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B) BepOHTHOCTb C06bITHH A = {-1 ~ X ~ 1, °< Y < 1}. |
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6.12.52. ~aHa nJIOTHOCTb pacnpe,Il.eJIeHHH BepoHTHocTeit ,Il.BYMepHoit C.B.
(X,Y) |
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f(x, y) = {OC,' e- z - y , x;::: 0, |
y ;::: 0, |
x < 0, |
y < 0. |
RaitTH:
a) napaMeTp C;
6) <PYHKIJ;HIO pacnpe,Il.eJIeHHH BepoHTHocTeit F(x, y);
B) BepOHTHOCTH C06bITHit: A = {X < 0, Y < 2}, B = {O ~ X ~ 1,
-X~Y~X}.
6.12.53. ~BYMepHaH CJIyqaitHaH BeJIHqHHa (X, Y) HMeeT nJIOTHOCTb pac-
npe,Il.eJIeHHH BepoHTHocTeit
f(x, y) = {c, |
B 06JIacTH D, |
0, |
BHe 06JIacTH D, |
r,Il.e D = {(x, y): y;::: 0, |
x + y ~ 1, 2y - x ~ 2}. RaitTH |
a) BeJIHqHHY C ; |
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6) nJIOTHOCTb pacnpe,Il.eJIeHHH CJIyqaitHoit BeJIHqHHbI X;
B) <PYHKIJ;HIO pacnpe,Il.eJIeHHH Fx (x) = P {X < x};
r) BepOHTHOCTb C06bITHH {X ;::: O}.
407
6.12.54. IIJIOTHOCTb pacnpe,n:eJIeHIHI BepoHTHocTeit CHCTeMbI CJIyqatiHblx BeJIHqHH (X, Y) HMeeT BH,n:
!(x, y) = {Xo,+ y, XE [0,1], Y E [0,1],
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B OCTaJIbHbIX CJIyqMX. |
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HaitTH |
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6) P{X + Y < 1}. |
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a) !x(x) H Jy(y)j |
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51BJIHIOTCH JIH c. B. X H Y He3aBHcHMbIMH? |
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6.12.55. |
,!1;BYMepHM c. B. (X, Y) 3a,Il;aHa nJIOTHOCTblO COBMeCTHOI'Opacnpe- |
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,n:eJIeHHH |
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!(x, y) = {c. xy4, |
(x,y) ED, |
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0, |
(x,y)~D, |
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r,n:e D 06JIaCTb Ha nJIOCKOCTH Oxy, onpe,n:eJIHeMM CHcTeMoit He- |
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paBeHCTB: {y > -1, x > 0, |
y < -x3 }. HaitTH 6e3YCJIOBHOe H |
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YCJIOBHOe pacnpe,n:eJIeHHH COCTaBJIHlOm;eit X. Y6e,n:HTbCH, qTO C. B. |
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X H Y 3aBHCHMbI. |
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6.12.56. |
HCnOJIb3YH YCJIOBHe 3a,Il;aqH 6.12.51, HaitTH YCJIOBHble nJIOTHOCTH |
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!(x I y) H !(y I x) |
KOMnOHeHT X H Y ,n:BYMepHoit CJIyqaitHoA |
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BeJIHqHHbI (X, Y). |
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6.12.57. |
3a,n:aHa nJIOTHOCTb |
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!(x, y) |
= *e-<x2 +4X Y+8y2 ) |
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COBMeCTHOI'Opacnpe,n:eJIeHHH ,n:BYMepHoit c. B. (X, Y). HaitTH 6e3- |
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YCJIOBHble H YCJIOBHble nJIOTHOCTH pacnpe,n:eJIeHHH CJIyqaitHblx Be- |
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JIHqHH X H Yj BbIHCHHTb HBJIHIOTCH JIH c. B. X H Y He3aBHCHMbIMH |
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(H3BecTHo, qTO |
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!00e-u2 du =..;:rr). |
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- 00 |
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6.12.58. |
HCnOJIb3YH YCJIOBHe 3a,Il;aqH 6.12.50, HaitTH: |
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a) M(X), M(Y)j |
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6) D(X), D(Y)j |
6.12.59. |
0) cov(X,Y), T.e. Kxyj |
r) rXY. |
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3a,Il;aHa nJIOTHOCTb COBMeCTHOI'O pacnpe,n:eJIeHHH CHCTeMbI ,n:BYX |
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C.B. (X, Y) |
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!(x, y) = {900,x2 y2, |
(x, y) E D, |
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(x,y) ~ D, |
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r,n:e D = {(x, y): Ixl + Iyl < 1, |
y < o}. HaitTH: |
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a) K09<pqmUHeHT KOppeJIHUHH rXyj |
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6) fI(x)j |
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0) !(y Ix). |
408
t<OHTponbHble BOnpOCbl M 60nee CnO)l(Hbie 3aI\ilHM,.
6.12.60. 3aKoH paCIIpe,!l;eJIeHIUI ,!l;HCKpeTHoit ,!l;BYMepHoit CJIY'faitHOitBeJIH- 'fHHbI3a.,n;aH Ta6JIHu;eit
X\Y |
-2 |
0,05° |
4 |
10° |
0,15 |
0,10° |
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0,10 |
0,20 |
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20 |
0,05 |
0,10 |
0,25 |
COCTaBHTb <PYHKU;HIO pacIIpe,!l;eJIeHHH Fx,y(x,y). RaitTH YCJIOBHblit 3aKOH paCIIpe,!l;eJIeHHH c. B. Y IIpH X = 20. BbIHCHHTb, 3aBHCHMbI JIH CJIY'faitHbleBeJIH'fHHbIX H Y.
6.12.61. B ypHe CO,!l;ep:>KHTCH 5 6eJIbIX H 3 'fepHbIXwapa.lh Hee H3BJIeKaIOT
2 wapa 6e3 B03Bparu;eHHH. IIycTb c. B. X - 'fHCJIO6eJIbIX wapOB B BbI6opKe, c. B. Y - 'fHCJIO'fepHbIXwapOB B BbI6opKe. CocTaBHTb
3aKOH COBMeCTHoro pacIIpe,!l;eJIeHHH CJIY'faitHoroBeKTopa
RaitTH: |
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a) P{X ~ 2, Y = 1}j |
6) D(X) H D(Y)j |
0)K09<P<PHU;HeHT KOppeJIHU;HH r x y .
6.12.62.3a.,n;aHa CHCTeMa CJIY'faitHblxBeJIH'fHH(X, Y). IhBecTHo, 'ITO:
M(X) = 1, M(Y) = -2, |
.j2 |
D(X) = 4, D(Y) = 2, rXY = T. |
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RaitTH: |
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a) M(2X + Y)j |
6) D(X - 3Y). |
6.12.63. CJIY'faitHbleBeJIH'fHHbIX H Y CBH3aHbI 3aBHCHMOCTbIO Y =-X + 1.
IIoKa3aTb, 'ITOrXY = -1.
6.12.64. ,IJ,aHa IIJIOTHOCTb pacIIpe,!l;eJIeHHH BepoHTHocTeit ,!l;BYMepHoit CJIY-
'faitHoitBeJIH'fHHbI(X, Y) |
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f(x,y) = {c. cos x cosy, |
B 06JIaCTH D, |
0, |
BHe 06JIaCTH D, |
r,!l;e D = {(x,y): x E (Oj~), Y E (Oj~)}. RaitTH:
a) <PYHKU;HIO pacIIpe,!l;eJIeHHH Fxy(x,y)j 6) IIJIOTHOCTb fx(x)j
0) BepOHTHOCTb C06bITHH A = {Y < 2X}.
6.12.65. <l>YHKU;HH paCIIpe,!l;eJIeHHH ,!l;BYMepHoit CJIY'faitHoitBeJIH'fHHbIHMeeT
BH,!l;
FXy (x,y) ={ |
1 - e-z - e- Y + e- z - y, x ~ 0, |
y ~ 0, |
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0, |
x < 0, |
Y < 0. |
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RaitTH:
a) ,!l;BYMepHYIO IIJIOTHOCTb BepoHTHocTH CHCTeMbI (X, Y)j 6) BepOHTHOCTb C06bITHH A = {X < 1, Y < 1}?
409