Сборник задач по высшей математике 2 том
.pdf.n.) lIcnoJIb3Y.H <POPMYJIY
P{a < X < (3} |
(3 - a) |
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(a - a) |
= «)0 ( -0'- |
«)0 -0'- , |
P{! <X < 3) = ~oh,!)-~ob·,!)=
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= «)0(2,82) - |
«)(0) = 0,4976 - |
0= 0,4976. |
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3Ha'IeHHe«)0(2,82) Hafi.n;eHo no Ta6JIHIJ;e 3Ha'IeHHfi<PYHKIJ;HH |
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x |
t 2 |
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«)o(x) = .~ Je-'2 dt |
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V 211' 0 |
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(CM. npHJIO)KeHHe 2 B KOHIJ;e KHHrH). |
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6.11.25. |
CJIY'IafiHbleomH6KH H3MepeHH.H .n;eTaJIH no.n;'IHHeHbIHOPMaJIbHOMY |
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3aKoHY C napaMeTpOM 0' = 20 MM. HafiTH BepO.HTHOCTb TOro, 'ITO |
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H3MepeHHe .n;eTaJIH npOH3Be.n;eHO C omH6Kofi, He npeBocxo.D:.Hmefi no |
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MO.n:yJIIO 25 MM. |
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a BOCnOJIb3yeMC.H <popMYJIofi P{IX - |
al |
< c} = 2«)0 (~) . B |
HameM CJIY'Iae |
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0' = 20, c = 25, n09TOMY |
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P{IX - al < 25} = 2«)0 (;~) = |
2«)0(1,25) = 2·0,3944 = 0,7888. |
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6.11.26. |
IIycTb X ""'" |
N(5; 0,5). HafiTH BepO.HTHOCTb Toro, |
'ITOnpH Tpex |
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He3aBHCHMbIX |
HcnbITaHH.HX C. B. X |
XOT.H 6bI B O.n;HOM H3 HHX X |
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npHMeT 3Ha'IeHHeB HHTepBaJIe (2; 4). |
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6.11.27. IIJIOTHOCTb BepO.HTHOCTefi C.B. X HMeeT BH.n; |
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2 |
4 |
1 |
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/(x) = c· e-2x |
-'3 x +'3. |
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HafiTH: c, M(X), D(X), F(x), p{-1 < x < i}· |
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6.11.28. |
1I3BecTHo, 'ITOX ""'" N(50,0'), P{X E (40;60)} = 0,7888. HafiTH |
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D(X). |
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6.11. 29. POCT B3POCJIbIX MY)K'IHH.HBJI.HeTC.H CJIY'IafiHOfiBeJIH'IHHOfiX, pacnpe.n;eJIeHHOfi no HOPMaJIbHOMY 3aKOHY: X ""'" N(175; 10). HafiTH: nJIOTHOCTb BepO.HTHOCTH, <PYHKu;H1O pacnpe.n;eJIeHH.H 9TOfi CJIy~aA
Hofi BeJIH'IHHbI;BepO.HTHOCTb Toro, 'ITOHH O.n;HH H3 3 Hay.n;a'IYBbl6paHHbIx MY)K'IHHHe 6y.n;eT HMeTb POCT MeHee 180 CM.
AononHMTenbHble 3aAilHMfI
6.11.30. KOHTPOJIbHM pa60Ta no TeopHH BepO.HTHOCTH COCTOHT H3 6 3a.n;a'I. BepO.HTHOCTb pemHTb npaBHJIbHO Ka:>K.n:y1O 3a,u.a'IY .D:JI.H .n;aHHoro
380
CTY.D:eHTa paBHa 0,7. RaitTH MareMaTHqecKoe O>KH.D:aHHe H .D:HCnep-
CHIO c. B. X - qHCJ1a npaBHJ1bHO pemeHHbIX 3a,n;aq.
6.11.31.CTpeJ1b6a no MHmeHH Be.D:eTCjf .D:O BToporo nona,n;aHHjf. RaitTH
M(X), r.D:e c. B. X - qHCJ10 nona,n;aHHit, eCJIH BepOjfTHOCTb nona,n;aHHjf npH O.D:HOM BblCTpeJ1e paBHa 0,25.
6.11.32.IhBecTHo, qTO: cpe.D:Hee qHCJ10 nona,n;aHHit B MHmeHb B cepHU U3 n
BbICTpeJ10B paBHO 192; BepOjfTHOCTb nona,n;aHUjf npH KaJK.D:O¥ BbI-
CTpeJ1e pasHa p; a(X) = 8, r.D:e c. B. X - qUCJ10 nona,n;aHuit. RaitTH n H p.
6.11.33. B 6oeBOit onepaU;UH YQacTByIOT 30 CaMOJ1eTOB. BepOjfTHOCTb ru6Mu CaMOJ1eTa B pe3YJ1bTare 06cTpeJ1a npOTHBHHKOM paBHa l5.
RaitTu M(X) U a(X), r.D:e C. B. X - QUCJ10 C6HTbIX CaMOJ1eTOB.
6.11.34.YcneBaeMOCTb CTY.D:eHTOB I Kypca COCTaBJ1jfeT 80%. RaitTU MaTeMaTHQecKoe O:>KH.D:aHUe H .D:UCnepCHIO QUCJ1a ycneBaIOIIJ;Hx cTY.D:eHTOB cpe.D:H 50 HaY.D:aQy oT06paHHblx nepBOKypCHUKOB.
6.11.35. lIcn0J1b3Yjf YCJ10BHe 3a,n;aQU 6.6.4, HaitTH M(X) U a(X), r.D:e
c. B. X - QHCJ10 nona,n;aHuit B MHmeHb.
6.11.36.lIcn0J1b3Yjf YCJ10BUe 3a,n;aQU 6.6.12, HaitTu cpe.D:Hee QUCJ10 npueMOB pa.n;uOCUrHaJIa.
6.11.37. lIcn0J1b3Yjf YCJ10BHe 3a,n;aQH 6.7.19, HaitTH M(X) U a(X), r.D:e
c. B. X - QUCJ10 ::meKTpo::meMeHTOB, Bblme.D:WHX U3 CTPOjf.
6.11.38.lIcn0J1b3Yjf yCJ10BHe 3a,n;aQU 6.7.21, HaitTH cpe.D:Hee QUCJ10 CeMjfH,
KOTOPbIe He npopacTyT.
6.11.39.lIcn0J1b3Yjf YCJ10BUe 3a,n;aQH 6.7.23, HaitTH:
a) M(X) U D(X), r.D:e c. B. X - QUCJ10 KHur c6pomIOpoBaHHblx
HenpaBHJ1bHO;
6)BepOjfTHOCTb Tom, QTO THPaJK CO.D:ep:>KHT 10 6paKOBaHHbIX KHHr.
6.11.40.Cpe.D:Hee QHCJ10 Bbl30BOB, nocTynaIOIIJ;UX Ha ATC B MUHYTY, paBHO
180. KaKOBa BepOjfTHOCTb C06bITHjf:
a) A = {3a 2 ceKyH.D:bI Ha ATC He nocTynuT HH O.D:HOrO Bbl30Ba};
6) B = {3a 2 ceKYH.D:bI Ha ATC nocTynHT MeHee 2-x BbI30BOB}?
6.11.41.BepOjfTHOCTb nona,n;aHHjf B u;eJ1b npH O.D:HOM BblCTpeJ1e paBHa 0,6. KaKOBa BepOjfTHOCTb TOm, QTO nepBoe nona,n;aHue B U;eJ1b npoH30it-
.D:eT npH QeTBepTOM BblCTpeJ1e?
6.11.42.IIpou3Bo.IJ:jfTCjf nOCJ1e.D:OBaTeJ1bHbIe He3aBHCHMbIe ucnbITaHHjf njf-
TU npH60poB Ha Ha,n;e:>KHOCTb. Ra,n;e:>KHOCTb KaJK.D:OrO H3 npu60poB pasHa 0,8. KaJK.D:blit CJ1e.D:YIOIII,Uit npu60p ucnbITbIBaeTCjf J1Umb B CJ1YQae, KOr.D:a npe.D:bI.D:YIII,Uit OKa3aJICjf Ha,n;e:>KHbIM. CocTaBuTb 3a-
KOH pacnpe.D:eJ1eHHjf .D:. c. B. X - QHCJ1a HCnblTaHHbIX npH60poB. RaitTH M(X).
6.11.43.CTY.D:eHT 3HaeT 30 H3 40 BonpOCOB. 9K3aMeHaTop 3a,n;aeT BonpOCbI cTY.D:eHTY.D:O rex nop, nOKa 06Hap)')KuTHe3HaHHe Bonpoca. RaitTu BepOjfTHOCTb TOm, QTO QHCJ10 3a,n;aHHbIX BonpOCOB 6oJ1bme .D:ByX.
381
6.11.44. IIPOH3BO)J,HTCjf CTpeJIb6a no IJ;eJIH )J,O nepBoro nona)J,aHHjf. BepOjfT-
HOCTb nona)J,aHHjf npH O)J,HOM BbICTpeJIe paBHa 0,1. HaitTH:
a) M(X), D(X) H a(X), r)J,e c. B. X - '1HCJIOC)J,eJIaHHbIX BbICTpeJIOB;
6) BepOjfTHOCTb Tom, 'ITOnOTpe6yeTCjf C)J,eJIaTb He 60JIee Tpex BbICTPeJIOB.
6.11.45. ABT06YCbI )J,aHHom MapmpyTa H)J,yT C HHTepBaJIOM 30 MHH. IIacca-
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:>KHp nO)J,XO)J,HT K aBT06ycHoit OCTaHOBKe B npOH3BOJIbHbIit MOMeHT |
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BpeMeHH. BpeMjf O:>KH)J,aHHjf aBT06yca eCTb HenpepblBHM CJIY'lait- |
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HM BeJIH'IHHaX, HMeIOIII;M paBHOMepHoe pacnpe)J,eJIeHHe. HaitTH: |
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nJIOTHOCTb BepOjfTHOCTH; <PYHKIJ;HIO pacnpe)J,eJIeHHjf; MaTeMaTH'Ie- |
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CKoe O:>KH)J,aHHe H )J,HcnepCHIO 9TOit CJIY'laitHoitBeJIH'IHHbI;BepOjfT- |
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HOCTb nOjfBJIeHHjf nacca)lmpa He paHee '1eM'1epe317 MHHyT nOCJIe |
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yxo)J,a npe)J,bI)J,yIII;ero aBT06yca, HO He n03)J,Hee '1eM3a O)J,HY MHHyTy |
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)J,O OTXO)J,a CJIe)J,yIOIII;ero aBT06yca. |
6.11.46. |
HaitTH MaTeMaTH'IecKoe O:>KH)J,aHHe H )J,HcnepCHIO npOH3Be)J,eHHjf |
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)J,ByX He3aBHCHMbIX HenpepbIBHbIX CJIY'laitHbIxBeJIH'IHHX H Y C |
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paBHOMepHbIMH 3aKOHaMH pacnpe)J,eJIeHHjf: |
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X '"R[O; 1], Y '"R[I; 3]. |
6.11.47. |
1I3BecTHo, 'ITOHenpepbIBHM CJIY'laitHMBeJIH'IHHaX HMeeT nJIOT- |
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HOCTb BepOjfTHOCTH |
0,375, |
xE (a - ~;a+ ~) , |
f(x) = { |
xtt (a-~;a+~). |
0, |
HaitTH M(X) H D(X).
6.11.48. lI,eHa )J,eJIeHHjf mKaJIbI H3MepHTeJIbHOrO npH60pa paBHa 0,1. IIoKa3aHHjf npH60pa oKpyrJIjfIOT )J,O 6JIH:>Kaitmero )J,eJIeHHjf. IIoJIara&, 'ITOomH6Ka oKpyrJIeHHjf pacnpe)J,eJIeHa no paBHOMepHOMY 3aKOHY B npOMe:>KYTKe OT 0 )J,O 0,1, HaitTH:
a) BepOjfTHOCTb Tom, 'ITOomH6Ka oKpyrJIeHHjf 60JIee 0,03;
6) BepOjfTHOCTb Toro, 'ITOomH6Ka oKpyrJIeHHjf MeHbme 0,02;
B)cpe)J,Hee 3Ha'leHHeomH6KH.
6.11.49.HenpepbIBHM CJIY'laitHMBeJIH'IHHaX pacnpe)J,eJIeHa no nOKa3aTeJIbHoMY 3aKOHy, a HMeHHO,
f(x) = {A. e-4x , |
npH x ~ 0, |
0, |
npH x < o. |
HaitTH: |
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a) 3Ha'leHHenapaMeTpa A; |
6) M(X) H D(X). |
IIOCTPOHTb rpacPHK <PYHKIJ;HH pacnpe)J,eJIeHHjf F(x). HaitTH Bepo-
jfTHOCTb Toro, 'ITOC'.B. X npHMeT 3Ha'leHHe,MeHbmee, '1eMM(X).
382
6.11.50. |
RaitTH MaTeMaTHqeCKOe O>KH,Il;aHHe c. B. X, pacnpe,Il;eJIeHHoit no no- |
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Ka3aTeJIbHoMY 3aKoHY, eCJIlI ee <PYHKIJ;IUI pacnpe,Il;eJIeHlIH HMeeT |
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F(x) = {1 - e- 5x , |
npH x ~ 0, |
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0, |
npH x < O. |
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RaitTH P{IX - M(X)I < 3a(X)}. |
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6.11.51. |
90% JIaMnOqeK neperopaIOT nOCJIe 800 qacOB pa60TbI. RaitTH Be- |
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POHTHOCTb Toro, qTO JIaMnOqKa neperopHT B npOMelKYTKe OT 100 |
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,Il;0 200 qacOB pa60TbI (c. B. T - |
BpeMH 6e30TKa3Hoit pa60TbI JIaM- |
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nOqKH). |
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6.11.52. |
HcnbITbIBaIOTcH ,Il;Ba He3aBHCHMO pa6oTaIOIIIHx ::meMeHTa. .r:LrrH- |
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TeJIbHOCTb BpeMeHH 6e30TKa3Hoit pa60TbI 9JIeMeHTa HMeeT nOKa- |
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3aTeJIbHOe pacnpe,Il;eJIeHHe co Cpe,Il;HHM 3HaqeHHeM ,Il;JIH 1-ro 9JIe- |
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MeHTa 20 qacOB, 2-ro - 25 qaCOB. RaitTH BepOHTHOCTb TOro, qTO |
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3a npOMe:>KYTOK BpeMeHH )1;JIHTeJIbHOCTbIO 10 qacOB: |
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a) 06a 9JIeMeHTa 6Y)1;yT pa6oTaTb; |
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6) OTKaJKeT TOJIbKO O,Il;HH 9JIeMeHT; |
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B) XOTH 6bI O,Il;HH 9JIeMeHT OTKaJKeT. |
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6.11.53. |
H3BeCTHO, qTO BpeMH peMOHTa TeJIeBH30pOB eCTb C. B. T, pacnpe- |
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,Il;eJIeHHaH no nOKa3aTeJIbHOMY 3aKOHY; npH 9TOM Cpe,Il;Hee BpeMH |
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peMOHTa TeJIeBH30pa COCTaBJIHeT )];Be He,Il;eJIH. RaitTH: |
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a) D(T) H a(T); |
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6) BepOHTHOCTb Toro, qTO Ha peMOHT TeJIeBH30pa nOTpe6yeTcH Me- |
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Hee ,Il;eCHTH ,Il;Heit. |
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6.11.54. |
H3BeCTHO, qTO X '" |
N (a, a), a MaKCHMaJIbHOe 3HaqeHHe llJIOTHOCTH |
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BepOHTHOCTH paBHo |
1~. RaitTH D(X). |
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3· v77r |
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6.11.55.X - HOPMaJIbHO pacnpe,Il;eJIeHHaH C. B., npHqeM M(X) = 6,2 H a(X) = 4,4. RaitTH P{IX - M(X)I < 5,7}.
6.11.56.OmH6Ka H3MepeHHH nO,Il;qHHeHa HOPMaJIbHOMY 3aKOHY C napaMe-
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TpaMH a = 50 ,Il;M H a = |
10 ,Il;M. RaitTH BepOHTHOCTb Toro, qTO |
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H3MepeHHoe 3HaqeHHe 6Y,Il;eT OTKJIOHHTbCH OT HCTHHHOro He 60JIee |
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qeM Ha 20 ,Il;M. |
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6.11.57. |
YCTaHOBJIeHO, qTO c. B. X '" N (a, a), |
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P{X > 20} = 0,02, |
P{X < 10} = 0,31. |
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RaitTH M(X) H D(X). |
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6.11.58. |
CPOK 6e30TKa3Hoit pa60TbI TeJIeBH30pa npe,Il;CTaBJIHeT co60it c. B. |
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X", N(12; 3). RaitTH BepOHTHOCTb TOro, qTO TeJIeBH30p npopa6o- |
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TaeT |
6) OT 6,Il;0 9 JIeT, |
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a) He MeHee 15 JIeT; |
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B)OT 9 ,Il;0 15 JIeT.
6.11.59.OTKJIOHeHHe pa3Mepa ,Il;eTaJIH OT CTaH,Il;apTa npe,Il;CTaBJIHeT c060it c. B. X, pacnpe,Il;eJIeHHyIO HOPMaJIbHO, C MaTeMaTHqeCKHM O)l(H- ,Il;aHHeM M(X) '=4 H co Cpe,Il;HeKBa,n.paTHqeCKHM OTKJIOHeHHeM
383
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a = 0,2. |
RaftTH npOl1,eHT ,ll;eTa.JIeft, OTKJIOH»IOIIJ,HXC» OT M(X) no |
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MO.rr.yJIIO He 60JIee '1eMHa 0,05. |
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6.11.60. |
,I1;eTa.JIb H3rOTaBJIHBaeTC» Ha CTaHKe C cHCTeMaTlI'leCKoftomH6Koit |
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3, Cpe,ll;HeKBa,rr,paTlI'leCKoftomH6Koft 4 H |
C'IHTaeTC» rO,ll;Hoft, eCJIH |
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ee OTKJIOHeHHe OT HOMHHa.JIa MeHee 12. RaftTH BepO»THOCTb Toro, |
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'ITOTpH HaY,ll;a'lYB3»TbIe ,ll;eTa.JIH H3 n»TH |
6y.rr.yT rO,ll;HbIMH. |
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KOHTponbHble BonpOCbI M 60nee CnO)l(Hbie 3aAClHMH |
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6.11.61. |
CJIY'laftHMBeJIH'IHHaX pacnpe,ll;eJIeHa no 6HHOMHa.JIbHOMY 3aKo- |
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Hy. RaftTH: |
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a) Ha'la.JIbHbleMOMeHTbI ,ll;0 4-ro nOp»,ll;Ka BKJIIO'IHTeJIbHOj |
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6) l1,eHTpa.JIbHbIe MOMeHTbI ,ll;0 4-ro nop»,ll;Ka BKJIIO'IHTeJIbHOj |
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B) acHMMeTpHIO (A = |
It:) |
H 9KCI1,ecC (E = It: -3) CJIY'laftHOit |
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BeJIH'IHHbIX. |
a |
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a |
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6.11.62. RaftTH MaTeMaTH'IeCKOeOlKH,ll;aHHe H ,ll;HCnepCHIO OTHOCHTeJIbHOft |
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nA |
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'1acTOTbIn |
B n He3aBHCHMbIX HcnbITaHH»X, B KroK,ll;OM H3 KOTOPbIX |
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c06b1THe A MOlKeT HacTynHTb C BepO»THOCTblO p. |
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6.11.63. ,I1;oKa3aTb peKyppeHTHylO <t>OpMyJIy ,ll;JI» |
6HHOMHa.JIbHbIX BepO»T- |
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HocTeft |
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Pn(m + 1) |
P |
n-m |
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= q . m + 1 . Pn(m). |
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6.11.64. ,I1;oKa3aTb, |
'ITOcyMMa ,ll;ByX He3aBHCHMblX CJIY'laftHbIxBeJIH'IHH, |
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pacnpe,ll;eJIeHHblX no 3aKOHY IIyaccoHa C napaMeTpaMH al H a2, |
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TaKlKe pacnpe,ll;eJIeHa no 3aKOHY IIyaccoHa C napaMeTpOM a = al + a2·
6.11.65. KaKM H3 BeJIH'IHHB 3aKOHe IIyaccoHa 60JIbme: MaTeMaTH'IeCKOe
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OlKH,ll;aHHe, '1HCJIOHe3aBHCHMbIX HcnbITaHHft HJIH ,ll;HCnepcH»? |
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6.11.66. |
Bepo»THoCTb 6paKa napTHH ,ll;eTa.JIeft paBHa 0,2. CKOJIbKO B cpe,ll;- |
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HeM HYlKHO npOBepHTb ,ll;eTa.JIeft,ll;O nepBoro 06HapYlKeHH» 6paKa? |
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6.11.67. ,I1;oKa3aTb, |
'ITOBepO»THOCTH Pi OT,ll;eJIbHbIX 3Ha'leHHft,ll;. C. B. X, |
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HMelOlIJ,eft reOMeTpH'IecKoepacnpe,ll;eJIeHHe, y,ll;OBJIeTBop»IOT YCJIo- |
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BHIO |
00 |
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LPi = 1. |
6.11.68. |
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i=l |
IhBecTHo, 'ITOHenpepbIBHM CJIY'laftHMBeJIH'IHHaX """ R[a, b]. |
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RaftTH '1eTBepTbIftl1,eHTpa.JIbHblft MOMeHT H 9KCI1,eCC 9TOft CJIY'laft- |
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HOft BeJIH'IHHbI.
6.11.69. C. B. X HMeeT nOKa3aTeJIbHOe pacnpe,ll;eJIeHHe. RaftTH:
a) l1,eHTpa.JIbHbIe MOMeHTbI TpeTbero H '1eTBepTOronop»,ll;KOBj
6) acHMMeTpHIO H 9KCl1,ecc. |
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6.11.70. ,I1;oKa3aTb, 'ITO<t>YHKI1,H» |
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F(x) = {1 - e-AX , |
x ~ 0, |
0, |
x < 0, |
384
r,ll;e A > 0, HBJUleTCH <PYHKIJ;Heit pacnpe,ll;eJ1eHH;H HeKoTopoit CJ1YqaitHoit BeJ1HqHHbI X. IIocTpoHTb ee rpa<PHK npH A = 2.
6.11.71. Bo CKOJ1bKO pM YMeHbmHTCH MaKCHMaJIbHOe 3HaqeHHe Op,ll;HHaTbI KPHBOit raycca, eCJ1H ,ll;HCnepCHH CJ1yqaitHoit BeJ1HqHHbI YBeJ1HqHTb
B 16 pM?
6.11.72.BblqHCJ1HTb IJ;eHTpaJIbHbIe MOMeHTbI BTopom, TpeTbero H qeTBepToro nOpH,ll;KOB CJ1yqaitHoit BeJ1HqHHbI, pacnpe,ll;eJ1eHHoit no HOpMaJIbHOMY 3aKOHY.
6.11.73.qeMY paBHbI MO,ll;a H Me,ll;HaHa c. B. X '" N(a, O')?
6.11.74.HabH <PYHKIJ;HIO pacnpe,ll;eJ1eHHH F(x) c. B. X '" N(a, 0'). YqeCTb,
|
00 |
t 2 |
IF |
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! |
-"2 dt - |
Y..!:!!. |
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e |
- |
2 . |
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o |
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6.11.75. |
M3BecTHo, 'ITO:c. B. X'" N(lj 0'), P{X < 2} = 0,99. HaitTH: |
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a) O'j |
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6) M(X2). |
6.11.76. |
CJ1yqaitHaH BeJ1HqHHa X pacnpe,ll;eJ1eHa no 3aKOHY N(a,O'). Hait- |
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TH P{XI ~ X ~ X2}, |
r,ll;e Xl |
H X2 - a6CIJ;HCCbI TOqeK neperH6a |
COOTBeTCTBYIOrn;eit KPHBOit raycca.
6.11.77. B HOPMaJIbHO pacnpe,ll;eJ1eHHoit cOBoKynHocTH 15% 3HaqeHHit X
MeHbme 12 H 40% 3HaqeHHit X 60J1bme 16,2. HaitTH Cpe,ll;Hee 3HaqeHHe H Cpe,ll;HeKBa,n,paTHqHOe OTKJ10HeHHe ,ll;J1H ,ll;aHHOrO pacnpe,ll;e- J1eHHH.
6.11. 78. CJ1yqaitHaH BeJ1HqHHa X pacnpe,ll;eJ1eHa HOPMaJIbHO, ee nJ10THOCTb
BepOHTHOCTH HMeeT BH,ll;
+s
f(x) = !.:-e(x--2)2-.
v87f
HaitTH MaTeMaTHqeCKOe O:lKH,ll;aHHe c. B. Y = 3X - 1, 3HaH, 'ITO
Y'" N(a, 0').
§12. ClilCTEMbl CllY"'A~HbIX BElllll ... lIIH
QlYHK4MR pacnp~eneHMR CMCTeMbl cnY'iaMHbiXBenM'iMH
Bo MHorHX rrpaKTH'IeCKHX3a,IJ;a'laxpe3YJ1bTaT orrbITa orrHCbIBaeTCH He O,n;HOit,
a ,n;BYMH (HJIH 60JIee) CJIY'IaitHbIMHBeJIH'IHHaMHX H Y. B 3TOM CJIY'IaerOBopHT 0
CUcme.Me asyx cJlY"I,a'fl:H'btX SeJlU"I,UH (X, Y) (HJIH aSY.MepHoiJ, CJlY"I,aiJ,HoiJ, SeJlU"I,UHe
(X, Y)).
reOMeTpH'IeCKHCHCTeMY ,n;BYX CJIY'IaitHbIxBeJIH'IHH(X, Y) MOlKHO HHTeprrpeTlrpoBaTb KaK CJIY'IaitHYIOTO'lKYHa rrJIOCKOCTH.
Ha ,n;BYMepHble CJIY'IaitHbleBeJIH'IHHbIrrpaKTH'IeCKH6e3 H3MeHeHHit rrepeHOCHTcx OCHOBHbIe rrOHHTHH .n;JIH o,n;HOMepHbIX CJIY'IaitHblXBeJIH'IHH,B '1aCTHOCTH3aKOH pacnpe,n;eJIeHHH, <PYHKIIHH pacrrpe,n;eJIeHHH, rrJIOTHOCTb pacrrpe,n;eJIeHHH H T. ,n;.
3aKoH pacrrpe,n;eJIeHHH CHCTeMbl (X, Y) ,n;BYX ,n;HCKpeTHbIX CJIY'IaitHbIxBeJIH'IHH B CJIY'IaeKOHe'lHOrO'1HCJIa3Ha'leHHitMOlKHO 3a,IJ;aTb <P0PMYJIOit
Pij=P{X=Xi,Y=Yi}, i=l, ... ,n, j=l, ... ,m
13 C60PHHK 31U1aq no ow.meA MaTeMaTH". 2 rypc |
385 |
HJIH C 1I0MOIW>IO Ta6JIH~1 C ,n:BofiHhlM BXO,n:OM:
X\Y |
Yl |
Y2 |
... |
Ym |
Xl |
pn |
P12 |
... |
plm |
X2 |
P21 |
P22 |
... |
p2m |
Xn |
Pnl |
Pn2 |
... |
pnm |
n m
r,n:e E E Pij = 1. i=lj=l
Ba)l{HefiIIlefi H3 HC'IepllhlBaIOIIIHXXapaKTepHCTHK (3aKOHOB paCllpe,n:eJIeHHH) CH-
CTeMhl CJIY'IafiHhlXBeJIH'IHHHBJIHeTCH <PYHKIIHH paCllpe,n:eJIeHHH.
~qJy'/t'/C'4Uei1. pacnpeiJeJle'/tWl. (HHa'le: HHTerpaJIbHOfi <PYHKIIHefi) CHCTeMhl C. B.
(X, Y) Ha3blBaeTCH <PYHKIIHH F(x,y), KOTOpaH ,n:JIH JII06hlX ,n:efiCTBHTeJIbHbIX '1HCeJI X H Y pasHa BepOHTHOCTH COBMeCTHoro BblIIOJIHeHHH ,n:BYX C06hlTHfi {X < x} H {Y < y}, T.e. F(x,y) = PiX < X, Y < y} (C06bITHe {X < X, Y < y} 03Ha'iaeT
IIpOH3Be,n:eHHe C06hlTHfi {X < x} H {Y < y}). |
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reOMeTpH'IeCKHKalK,n:oe 3Ha'leHHe<PYHKIIHH F(x, y) 03Ha'iaeTBepOHTHOCTb 110- |
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lIa,n:aHHH CJIY'IafiHOfi TO'lKH (X, Y) B 3aIIlTpHXOBaHHofi IIPHMOfi yrOJI R""y |
(KBa- |
,n:paHT) C BepIIlHHofi B TO'lKe(x,y) (pHC. 91). |
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y
Puc. 91
BepOHTHOCTb 1I0Ila,n:aHHH CJIY'IafiHofiTO'lKH(X, Y) B IIpHMoyrOJIbHHK D co CTopOHaMH, lIapaJIJIeJIbHblMH Koop,n:HHaTHhlM OCHM, Haxo,n:HTCH 110 <p0pMYJIe:
CBOMCTBa ABYMepHoM 4>YHKI..\IIIIII pacnp«!AeneHIIIR
1.0 ~ F(x,y) ~ 1;
2.F (x, y) - He y6hlBaeT 110 KalK,n:OMY H3 CBOHX aprYMeHTOB (lIpH <pHKCHPO-
BaHHOM ,n:PyroM aprYMeHTe):
F(X2,y) ~ F(Xl,y) IIpH X2 > Xl;
F(x, Y2) ~ F(x, yd IIpH Y2 > Yl.
3. F(x, y) HellpepblBHa CJIeBa 110 KalK,n:OMY H3 CBOHX aprYMeHToB;
386
4. F(x, -00) = F( -00, y) = F( -00, -00) = 0, r.ll:e, HarrpHMep, F(x, -00) 03Ha-
qaeT lim F(x,y); y-+-oo
5.F(+oo,+oo) = 1;
6.F(x,+oo) = H(x) = Fx(x), F(+oo,y) = H(y) = Fy(y), me H(x) H
F2 (y) - <PYHKI.r;HH pacrrpe.ll:eJIeHHH c. B. X H Y COOTBeTCTBeHHO.
3Ha'leHHe F(x, y) <PYHKI.r;HH pacrrpe.ll:eJIeHHH B CJIY'Iae CHCTeMbl (X, Y) .lI:BYX
,UHcKpeTHhlx c. B. HaxO.ll:HTCH CYMMHpoBaHHeM Bcex BepoHTHocTeil: Pij C HH.lI:eKCOM i,j,.IVIH KOTOPhlX Xi < X, Yj < y, T.e.
F(x,y) = L L Pij.
Xi<XYj<Y
nllOTHOCTb paCnp~elleHIIIH CIIICTeMbl cllY'"IaiiiHblXBelllll'"lIllH
B CJIY'IaeCHCTeMhl HerrpepblBHblX CJIY'Iail:HhlxBeJIH'IHH(X, Y) ee 3aKOH pac-
npe.ll:eJIeHHH Y.lI:06HO 3a.ll:aBaTb C rrOMOII.r;bIO rrJIOTHOCTH pacrrpe.ll:eJIeHHH.
~II.!Iom'ltocm'b'lO pacnpeiJe.!le'ltUR aepo.f!m'ltocmeiJ. (HJIH rrpOCTO n.!lom'ltocm'b'lO) CH-
CTeMbl (X, Y) ,lI;Byx HerrpephlBHhlX CJIY'Iail:HblxBeJIH'IHHHa3blBaeTCH BTopaH CMe-
waHHaH rrpOH3BO.ll:HaH ee <PYHKI.r;HH pacrrpe.ll:eJIeHHH, T. e.
{)2 F(x, y) |
/I |
f(x,y)= {)x{)y |
= Fxy(x,y). |
CBOMCTBa ABYMepHOM nnOTHOCTIiI pacnpt!AeneHIiIH BepOHTHOCTeM
1.f(x, y) ;;:: 0; +00 +00
2.! ! f(x, y) dxdy = 1;
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- 00 - 00 |
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3. |
P{(X, Y) E D} = !!f(x, y) dxdy, r.ll:e D - rrpOH3BOJIbHaH 06JIacTb; |
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F(x,y) = !'" !y |
D |
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4. |
f(u,v)dudv; |
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- 00 - 00 |
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+00 |
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+00 |
5. |
! f(x,y)dy = /1 (x) = fx(x), |
! f(x,y)dx = /2(y) = fy(y)· |
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-00 |
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He3aBIIICIIIMbie cllY'"IaiiiHbleBelllll'"lIllHbl
~CJIY'Iail:HhleBeJIH'IHHblX H Y Ha3hlBaIOTCH 'lte3aaUCU.M'bI..MU, eCJIH He3aBHCHMbl-
MH HBJIHIOTCH C06blTHH {X < x} H {Y < y} .IVIH JII06hlX .lI:eil:cTBHTeJIbHbIX 'IHCeJIX
H y. B rrpOTHBHOM CJIY'IaeCJIY'Iail:HbleBeJIH'IHHblHa3blBaIOTCH 3aaUCU.M'bI..MU. ~
Cpmy H3 orrpe.ll:eJIeHHH He3aBHCHMOCTH c. B. X H Y BblTeKaeT CJIe.ll:YIOII.r;ee paaeHcTBO, KOTopoe MOlKHO rrOJIOlKHTb B oCHoay paBHOCHJIbHOrO orrpe.ll:eJIeHHH:
F(x,y) = H(x)· F2 (y).
387
B CJIY'IaeCHCTeMbI ,D;BYX ,D;HCKpeTHbIX CJIY'Iail:HbIxBeJlH'IHH(X, Y) Heo6xo,D;H- MbIM H ,D;OCTaTO'IHbIMYCJIOBHeM HX He3asHcHMOCTH HBJIHeTCH paseHcTBo
PiX = Xi, Y = Yj} = PiX = xd' P{Y = Yj},
BbInOJIHHIOm;eeCH )J;JIH JII06bIX i = 1, ... ,n, j = 1, ... ,m.
Heo6xo,D;HMbIM H ,D;OCTaTO'lHbIMYCJIOBHeM He3asHcHMOCTH .D:BYx HenpepbIBHbIX c. B. X H Y, 06pa3YIOm;Hx cHcTeMY (X, Y), HBJIHeTCH paseHcTBo
f(x,y) = /I(x)· h(Y)·
YCIlOBHble 3aKOHbi paCnp~elleHMH
~YC.IIOB1t"btM 3alw'ttOM pacnpeae.lle'ttUR O,D;HOil: H3 c. B., BXO,D;Hm;HX B CHCTeMY (X, Y),
Ha3bIBaeTCH 3aKOH ee paCnpe,D;eJleHHH, Hail:,D;eHHbIiI: npH YCJIOBHH, 'ITO,D;PyraH c. B. '
npHHHJIa Onpe,D;eJleHHOe 3Ha'leHHe(HJIH nOnaJIa B HeKHiI: HHTepBaJI). |
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B '1acTHOCTH,B CJIY'IaeCHCTeMbI ,D;BYX ,D;HCKpeTHbIX CJIY'Iail:HbIXBeJlH'IHH(X, Y) |
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YC.IIOB'tt'btM 3a1CO'ttOM pacnpeae.lle'ttUR |
C. B. Y npu YC.IIOBUU X = Xi Ha3bIBaeTCH COBo- |
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KynHocTb BepOHTHocTeil: |
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P{Y = Yj IX = xd = PiX = Xi, Y = Yj} , j = 1, ... , mj i = 1, ... ,n. |
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PiX =xd |
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AHaJIOrH'IHOOnpe,D;eJlHeTCH YCJIOBHbIiI: 3aKOH pacnpe,D;eJleHHH ,D;HCKpeTHoil: c. B. , |
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X npH YCJIOBHH Y = Yj. |
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YC.IIOB'ttaJI n.IIom'ttocm'b 'ttenpep'btB'ttoit C. B. Y npu YC.IIOBUU X = X |
(0603Ha'leHHe' |
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f(y I X)) Onpe,D;eJlHeTCH paseHcTBoM |
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f(y IX) = |
f(x,y) |
, |
me /I(x) 1= o. |
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/I (X) |
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f(x I y) = |
f(x,y) |
, |
r,D;e h(Y) 1= O. |
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h(Y) |
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TeopeMa YMHOlKeHHH nJIOTHocTeil: pacnpe,D;eJIeHHH: |
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f(x,y) = /I (x) . f(y I X) = h(y)' f(x I y). |
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MaTeMaTM'"IeCKOeO)l(MAaHMe M AMcnepCMH CMCTeMbl |
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CIlY'"IaMHbIXBeIlM'"IMH |
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MameMamU"'teC1CUM O~Uaa'ttueM aBYMep'ttoit C. B. (X, Y) Ha3bIBaeTCH cOBoKyn- |
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HOCTb ,D;BYX M. o. M(X) H M(Y) (T. e. ynopH,D;O'leHHaHnapa (M(X), M(Y))), onpe- ,D;eJlHeMbIX paseHcTBaMH:
n m |
n m |
M(X) = L L XiPij |
H M(Y) = LLYiPij, |
i=l j=l |
i=l j=l |
388
eCJIH X H Y - )l;HCKpeTHbie C. B.;
M(X) = !00 !00x . f(x, y) dxdy H |
M(Y) = !00 !00 y. f(x, y) dxdy, |
- 00 - 00 |
- 00 - 00 |
eCJIH X H Y - HenpepblBHble C. B. |
|
MaTeMaTH'IeCKOeOlKH)l;aHHe C. B. cp(X, Y), HBJIHIOrn;eitcH <PYHKD:Heit KOMnOHeHT
X H Y )l;BYMepHoit C. B. (X, Y), HaxO)l;HTCH aHaJIOrH'IHOno <P0PMYJIaM:
M(cp(X,y)) = !00 !00cp(x,y)·f(x,y)dxdY)l;J1HHenpepblBHOrOCJ1Y'IaH;
- 00 - 00 |
|
n |
m |
M(cp(X, Y)) = L |
L cp(Xi, yj) . Pij )l;J1H )l;HCKpeTHoro CJ1Y'Iax. |
i=l j=l |
|
.II:HcnepcHH CHCTeMbi c. B. (X, Y): |
|
n m |
n m |
D(X) = LL(Xi -ax)2pij H D(Y) = LL(Yj -ay )2pii>
eCJ1H (X, Y) -
=M(Y));
i=lj=l |
i=lj=l |
CHCTeMa )l;HCKpeTHbiX |
CJIY'IaitHbixBeJIH'IHH (ax = M(X), ay |
00 |
00 |
00 |
00 |
|
D(X) = ! ! |
(x - ax)2 f(x, y) dxdy = |
! ! x 2f(x, y) dxdy - a~ |
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- 00 - 00 |
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- 00 - 00 |
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H |
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00 |
00 |
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00 |
00 |
D(Y)= ! |
!(y-ay )2 f (x,y)dxdy= |
! |
!y2f(x,y)dxdy-a~, |
|
-00 -00 |
|
-00 -00 |
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eCJIH (X, Y) - CHCTeMa HenpepblBHblX CJIY'IaitHbixBeJIH'IHH.
IIycTb (X, Y) - CHCTeMa )l;HCKpeTHblx CJIY'IaitHbixBeJIH'IHH. YC.!IOB'HOe MameMamU'l.eCICOe o;)tCuaa'Hue aUClCpem'Hoi1 C. B. Y npu YC.!IOBUU X = Xi Onpe)l;eJIHeTCH
paaeHCTBOM:
m
M(Y I X = Xi) = M(Y Ixd = LYjp(Yj I Xi),
j=l
p(Yj I xd = p{Y = Yj I X = Xi}.
n
M(X IY = Yj) = M(X IYj) = LXi· p(Xi IYj)·
i=l
IIycTb Tenepb (X, Y) - CHCTeMa HenpepblBHblX CJ1Y'IaitHbixBeJIH'IHH.B aTOM CJIY'IaeYC.!IOB'HOe MameMamU'l.eCICOe O;)tCuaa'Hue C. B. Y npu YC.!IOBUU X = X onpe- )l;eJIHeTCH paaeHCTBOM:
M(Y I X) = !00 y. fey I X) dy.
- 00
389