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# Сборник задач по высшей математике 2 том

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HOMY 3aKOHY:

U,eHTpanbHaH npE!AenbHaH TeOpeMa

C<p0PMYJIHpyeM u;eHTPaJIbHyIO rrpe.n;eJIbHyIO TeOpeMY .n;JIH CJIY'laHO.n;HHaKOBO pacrrpe.n;eneHHblX CJIaraeMblX.

TeopeMa 6.12. nYCTb He3aBlllCillMbie c. B. Xl, X2, X 3 , ••• , X n , ... OAIilHaKOBO pacnpeAeneHbl C MaTeMaTIII'IeCKIllMO>KIIIAaHllleM a III AlIIcnepcllleA 0'2. TorAa c:PYHKIIIII" pacnpeAeneHIII" lIeHTplllpOBaHHoA III HOpMlllpOBaHHoA CYMMbl Zn 3TIIIX cnY'IailiHblX

Bem1'H1H

n

EXina

i=l

CTpeMIIITC" nplll n -+ 00 K c:PYHKIIIIIIII pacnpeAeneHIII" CTaHAapTHoili HopManbHoili cnY'IaAHoABenlll'lIllHbl:

 FZn(X) = P{Zn < x} --+ ~(x) = ~ x t 2 je- 2 dt. n-+oo V 271' - 00

1h U;eHTpaJIbHoil: rrpe.n;enbHoil: TeopeMbI, B '1acTHOCTH,CJIe.n;yeT, 'ITOrrpH 60JIbIlIHX n cyMMa Sn = Xl + X 2 + ... + Xn rrpH6JIHJKeHHO pacrrpe.n;eneHa rro HOPMaJIb-

Sn '" N(na, VnO')·

HarrOMHHM, 'ITO:

1.C. B. X Ha3b1BaeTCH u;eHTpHpOBaHHOfi H HOpMHpoBaHHoil: (T. e. CTaH.n;apTHoil:),

eCJIH M(X) = 0 H D(X) = 1.

2.IfIYHKU;HH JIarrJIaca

CBH3aHa C HOpMHpOBaHHofi <pYHKU;Hefi JIarrJIaca

paBeHCTBOM

~(X) = ~ + ~o(x).

1h TeopeMbI 6.12 TaKJKe CJIe.n;yeT, 'ITOrrpH .n;OCTaTO'lHO60JIbIlIHX n (yJKe rrpH n > 10) BblrrOJIHHeTCH COOTHOIlIeHHe

 Pia < Sn < (3} ~ ~o ( (3 - M(Sn») - ~o (a -M(Sn») . (14.4) O'(Sn) O'(Sn)

qacTHblM CJIY'laeMu;eHTpaJIbHoil: rrpe.n;eJIbHoil: TeopeMbI HBJIHIOTCH paccMoTpeH-

Hble paHee JIOKaJIbHaH H HHTerpaJIbHaH TeopeMbI Myaapa-JIaIIJIaca.

430

6.14.1. ,II:HcKpeTHaH c. B. X 3a,n:aHa PMOM pacnpege.TIeHHH

HCnO.TIb3YH HepaBeHcTBo qe6blIIIeBa, ou;eHHTb BepOHTHOCTb Toro,

 'ITOIX - M(X)I < 5. o HaiigeM CHa'Ia.TIaMaTeMaTH'IeCKOeO:lI<:HgaHHe H gHcnepcHIO C. B. X: M(X) = °.0,2 + 2 . 0,3 + 6 . 0,4 + 10 . 0,1 = 4; D(X) = 02 ·0,2 + 22 . 0,3 + 62 . 0,4 + 102 ·0,1 - 42 = 25,6 - 16 = 9,6. COr.TIaCHO
 HaiiTH BepoHTHoCTb C06bITHH A = {IX - M(X)I < 1,5}. Ou;eHHTb 3TY BepOHTHOCTb, nO.TIb3YHCb HepaBeHCTBOM qe6blIIIeBa. 6.14.4. HenpepbIBHaH c. B. X '" R(2,8). HaiiTH BepOHTHOCTb C06bITHH A = {3,5 < X < 6,5}; ou;eHHTb BepOHTHOCTb C06bITHH A, HCnO.TIb- o 3YH HepaBeHCTBO qe6blIIIeBa. C.TIY'IaiiHMBe.TIH'IHHaX HMeeT n.TIOTHOCTb pacnpege.TIeHHH f(X)={~' xE(2,8), 0, x ¢ (2,8). I1cnO.TIb3YH

431

6) MeHee

TaKKaK

A = {3,5 < X < 6,5} = {-1,5 < X - 5 < 1,5} = {IX - 51 < 1,5},

TO, HCIIOJIh3YH HepaBeHCTBO qe6hlIIIeBa (14.1), IIOJIyqaeM (3.n:ech c = 1,5)

HCKOMYIO ou;eHKy:

P(A) = P{IX - 51 < 1,5} ~ 1- ~ = -l.

1,5

IIoJIyqHJIH HeHHTepecHYIO (rpy6ylO) ou;eHKY; BepoHTHoCTh JII06oro C06hITHH Bcer.n:a HeoTpHu;aTeJIhHa! •

6.14.5. llcIIOJIh3YH HepaBeHCTBO qe6hlIIIeBa, ou;eHHTh BepOHTHOCTh Toro, qTO c. B. X OTKJIOHHTCH OT CBoero MaTeMaTHqeCKOrO Q)lm.n:aHHH M(X) MeHee, qeM Ha:

 a) a; 6) 3a; B) 9a, r.n:e a = VD(X) - cpe.n:Hee KBa.LJ:paTHqeCKOe OTKJIOHeHHe C.B. X.

6.14.6.X - HeIIpephIBHM c. B. C IIJIOTHOCThlO paCIIpe.n:eJIeHHH

f(x) = Qe -3Ixl, x E Ilt

2

Ha:liTH:

a)P{IXI < 2};

6)ou;eHKy BepOHTHOCTH c06hITHH {IXI < 2}, HCIIOJIh3YH HepaBeHCTBO qe6hIIIIeBa.

6.14.7.BCXO:>KeCTh ceMHH HeKoTopo:li KyJIhTyphI paBHa 0,85. Ou;eHHTh BePOHTHOCTh Toro, qTO H3 400 IIoceHHHhlX ceMHH qHCJIO B30IIIe.n:IIIHX 6y.n:eT 3aKJIIOqeHO B IIpe.n:eJIax OT 300 .n:o 380.

6.14.8.YCTPO:liCTBO COCTOHT H3 400 He3aBHCHMO pa6oTalOru;Hx 3JIeMeHTOB. BepoHTHoCTh OTKa3a JII06oro H3 HHX 3a BpeMH T paBHa 0,01. C IIO-

MOru;hlO HepaBeHCTBa qe6hlIIIeBa ou;eHHTh BepOHTHOCTh Toro, qTO MO.n:yJIh pa3HOCTH Me:>K.n:y qHCJIOM OTKa3aBIIIHX 3JIeMeHTOB H cpe.n:- HHM qHCJIOM OTKa30B 3a BpeMH T OKroKeTCH He MeHee 5.

6.14.9.QHCJIO .n:O:>K.n:JIHBhIX .n:He:li B ro.n:y .n:JIH .n:aHHO:li MeCTHOCTH HBJIHeTCH c. B. X C M(X) = 100. Ou;eHHTh BepoHTHoCTh TOro, qTO B CJIe.n:y- lOru;eM ro.n:y B .n:aHHo:li MeCTHOCTH 6y.n:eT MeHhIIIe 140 .n:O:>K.n:JIHBhIX

.n:He:li.

6.14.10.IIapHKMaxepcKaH 06CJIJ)KHBaeT B cpe.n:HeM 120 KJIHeHTOB B .n:eHh. Ou;eHHTh BepoHTHoCTh Toro, qTO cero.n:HH B .n:aHHo:li IIapHKMaxepCKO:li 6y.n:eT o6CJIY:>KeHo:

a) He MeHee 150 KJIHeHTOB; 160 KJIHeHTOB.

6.14.11. Ou;eHHTh BepOHTHOCTh TOro, qTO IIpH 15000 IIo.n:6pachIBaHHHx MoHeThI OTHOCHTeJIhHM qacToTa IIOHBJIeHHH rep6a OTKJIOHHTCH OT BepoHTHoCTH IIOHBJIeHHH rep6a IIpH O.n:HOM IIo.n:6pacbIBaHHH IIO Mo-

.n:yJIIO MeHhIIIe, qeM Ha 0,01.

432

a PaccMaTpHBaeMble HcnbITaHHjI ygoBJIeTBOpjllOT cxeMe BepHYJIJIH. Boc- nOJIb3yeMcjI HepaBeHCTBOM (14.2).I1MeeM p = ~, q = ~, n = 15000, c = 0,01, n03ToMY

 1 1 P {115~00 - ~I < 0,01} ~ 1 - 150g0·.~,012 ~0,83, T. e. P ~ 0,83. • 6.14.12. I1rpaJIbHM KOCTb nog6pacbIBaeTcjI 1200 pa3. Ou;eHHTb BepOjlT-

HOCTb OTKJIOHeHHjI OTHOCHTeJIbHOfi qaCTOTbI Bblna)J;eHHjI 6 OqKOB OT BepOjiTHOCTH 3Toro C06b1THjI (no MO)J;yJIIO) Ha BeJIHqHHY, MeHb-

WYIO, qeM 0,02.

6.14.13. B ypHe HaxogHTcjI 20 6eJIbIX H 80 qepHbIX wapOB. 113 Hee H3BJIeKa-

lOT, C B03Bparn;eHHeM, 40 wapOB. Ou;eHHTb BepOjiTHOCTb Toro, qTO KOJIHqeCTBO 6eJIbIX wapOB B BbI60pKe 3aKJIIOqeHO Me}K)J;y 4 H 12.

6.14.14. B aBTonapKe 200 aBTOM06HJIefi. KroKgblfi H3 HHX 3a BpeMjI 3KCnJIyaTau;HH t MO}KeT BblfiTH H3 CTPOjl, He3aBHCHMO OT gpyrHx, C BepOjiTHOCTblO 0,04. Ou;eHHTb BepOjiTHOCTb Toro, qTO gOJIjI Ha)J;e}K- HbIX aBTOM06HJIefi OTJIHqaeTCjI no MO)J;yJIIO OT BepOjiTHOCTH 6e30TKa3Hofi pa60TbI JII060ro H3 HHX He 60JIee qeM Ha 0,1.

6.14.15. I1rpaJIbHM KOCTb nOg6paCbIBaeTcjl400 pa3. Ou;eHHTb BepOjiTHOCTb Toro, qTO cpegHee apH<pMeTHqeCKOe qHCJIa BblnaBWHX OqKOB OTKJIOHHTCjI OT MaTeMaTHqeCKOrO O}KHgaHHjI qHCJIa OqKOB, BblnaB-

 WHX npH ogHoKpaTHoM nog6pacbIBaHHH KOCTH, no MO)J;yJIIO MeHb- a we, qeM Ha 0,1. 0603HaqHM qepe3 Xi(i = 1,2, ... ,400) - qHCJIO OqKOB, BbmaBWHX Ha

rpaHH KOCTH B i-M HcnbITaHHH.9TH CJIyqafiHble BeJIHqHHbI He3aBHCHMblj HMelOT OgHO H TO}Ke MaTeMaTHqeCKOe O}KHgaHHe, paBHoe ~ (CM. 3a)J;aqy 6.10.2) H

 OrpaHHqeHHble B cOBoKynHocTH gHcnepcHH, paBHble ~~ (CM. 3a)J;aqy 6.10.2). II03ToMY K gaHHofi nOCJIegoBaTeJIbHOCTH CJIyqafiHbIX BeJIHqHH Xl, X 2, ... •.. , X400 npHMeHHM 3aKOH 60JIbWHX qHCeJI (TeopeMa Qe6bIweBa). I1cKOMYIO ou;eHKy nOJIyqHM, HCnOJIb3Yjl HepaBeHcTBo (14.3), rge n = 400, 35 7 c = D(Xi ) = 12'M(Xi ) = 2'c = 0,1: 400 400 P{14~0tr Xi - 4~0trM(Xi)1 < 0,1} = 400 = !~ ~0,271. = P{14~0tr Xi - ~I < 0,1} ~ 1- 12. 45g. 0,01 •

6.14.16. ,lJ;HcnepCHjI KroKgofi H3 2000 He3aBHCHMbIX c. B. He npeBblwaeT 2. Ou;eHHTb BepOjiTHOCTb Toro, qTO OTKJIOHeHHe cpegHero apH<pMeTHqeCKOrO 3THX c. B. OT cpegHero apH<pMeTHqeCKOrO HX MaTeMaTHqeCKHX O}KHgaHHfi MeHbwe 0,04.

433

Xl, X 2,.

6.14.17. TIpHMeHHMa JIH K nOCJIe.n:OBaTeJIbHOCTH He3aBHCHMbIX C. B.

X 3, • •• TeOpeMa lJe6bIIlIeBa, eCJIH 3aKOH pacnpe.n:eJIeHHH K8.)K.n:Oil: H3 C. B. Xn (n = 1,2,3, ...) HMeeT BH.LJ::

a) !-.:..:2..:-+--;;-;:=-+--:::--°=-+--:::-:::c::-i, r.n:e a > 0;

6.14.18. CKOJIbKO pa3 HY)l{HO H3MepHTb .LJ:JIHHy .n:eTaJIH, HCTHHHoe 3HaqeHHe KOTOPOit a, qTo6bI C BepOHTHOCTbIO He MeHbIlIeil:, qeM 0,95, MO)l{HO

6bIJIO YTBep)l{.n:aTb, qTO cpe.n:Hee apmpMeTHqeCKOe 3THX H3MepeHHiI: OTJIHqaeTCH OT a no MO.n:yJIIO MeHbIlIe, qeM Ha 1, eCJIH McnepCHH K8.)K.n:oro H3MepeHHH MeHbIlIe 16?

6.14.19. Ha oTpe3Ke [o;~] CJIyqai!:HbIM 06pa30M BbI6paHbI 162 qHCJIa, T. e. paCCMaTpHBaIOTCH 162 He3aBHCHMbIX H paBHOMepHO pacnpe.n:eJIeHHbIX CJIyqai!:HbIX BeJIHqHH XI, X 2 , X 3, ... , X 162 • Hail:TH BepOHTHOCTb Tom, qTO HX cyMMa 3aKJIIOqeHa Me)l{.n:y 22 H 26.

a TIYCTb Xl +X 2 +X3 +... +X 162 = 8 162. CJIyqail:Hble BeJIHqHHbI He3aBHCHMbI, o.n:HHaKOBO pacnpe.n:eJIeHbI C MaTeMaTHqeCKHM O)l{H.n:aHHeM M(Xi ) =

 a+b] 1 = [(b-a)2] 1 = [-2- = 8 H .n:HcnepcHeit D(Xi ) 12 = 192' YCJIOBHH IJ,eHT-

PaJIbHOil: npe.n:eJIbHoil: TeopeMbI C06JIIO.n:eHbI, n03ToMY CJIyqaitHaH BeJIHqHHa

8 162 npH6JIHLKeHHO pacnpe.n:eJIeHa no HOPMaJIbHOMY 3aKOHY C nJIOTHOCTbIO

 f8162(S)~ (8 - M(8 162))2 1 e- 2.".2(8,62 ) 0'(8162 ) • ../'Fi MTaK, 8 162 '" N (162. ~,v'162· J1~2)' TaK KaK M(8162) = M(~Xi) = ~M(Xi)= ~·162 = 81,4 D(8 162 ) = D(~Xi) = ~D(Xi)= 1~2 ·162 = ;~, TO, npHMeHHH

~ 410(6,26) - 41(1,91) ~ 0,5 - 0,4719 = 0,0281.

MTaK, C BepOHTHOCTbIO, npH6JIH3HTeJIbHO paBHoit 0,03, MO)l{HO YTBep)l{.n:aTb, qTO cyMMa 162 CJIyqai!:HbIX qHCeJI, BbI6paHHblx Ha OTpe3Ke [0; l] ,3aKJIIOqeHa

 Me)l{.n:y 22 H 26. •

434

300
2: Xi;
i=l
HafiTH:
. .. , X 300 ,
[0; 0,4].
BepofiTHoCTb IIOIIa,rr,amm B u;eJIb IIpH KroK.n;OM BbICTpeJIe .n;JIfi .n;aHHOI'OCTpeJIKa paBHa 0,7. HafiTH BepOfiTHOCTb TOI'O,'ITO'lHCJIOIIoIIa,rr,aHHfi B u;eJIb 6y.n;eT 60JIbWe 52, eCJIH OH IIpOH3BeJI 84 BbICTpeJIa. CKJIa,rr,bIBaIOTcfI 300 He3aBHCHMbIX CJIY'lafiHbIXBeJIH'lHHXl, X 2 , • ••
paBHOMepHO paCIIpe.n;eJIeHHbIX Ha OTpe3Ke

6.14.20.

6.14.21.

a) IIpH6JIIDKeHHOe BblproKeHHe IIJIOTHOCTH C. B. Y =

6) BepOfiTHOCTb C06bITHfi A = {56 < Y < 65}.

6.14.22. HaIIpIDKeHHfI Ha BbIxo.n;ax 40 KaHaJIOB pa,rr,HOTeXHH'leCKOI'OYCTpofiCTBa eCTb He3aBHCHMble c. B. C MaTeMaTH'leCKHMH O:lKH.n;aHHflMH, paBHbIMH 5 B H .n;HCIIepCHflMH, paBHbIMH 10 B. HafiTH BepOfiTHOCTb TOI'O,'ITOHaIIpIDKeHHe Ha BbIxo.n;e YCTpoficTBa, CYMMHpYIOIIJ;eI'OHaIIpIDKeHHfI KaHaJIOB:

a) 6y.n;eT Haxo.n;HTbCfI B IIpe.n;eJIax OT 140 B .n;o 200 B; 6) IIpeBbICHT 180 B.

6.14.23. IIPH CTaTHCTH'leCKOMOT'leTeCKJIa,rr,bIBaeTCfI 900 'lHCeJI, KroK.n;oe H3 KOTOPbIX OKpYI'JIeHOC TO'lHOCTblO.n;o 0,001. IIpe.n;IIOJIaraeTCfI, 'ITOOWH6KH OKpYI'JIeHHfIHe3aBHCHMbI H paBHOMepHO paCIIpe.n;eJIeHbI B HHTepBaJIe (-0,5.10- 3 ; 0,5 .10-3 ). HafiTH HHTepBaJI, CHMMeTPH'lHblfiOTHOCHTeJIbHO MaTeMaTH'leCKOI'OO:lKH.n;aHHfI, B KOTOPOM C BepOfiTHOCTblO, He MeHbwefi, 'leM0,996, 6y.n;eT Haxo.n;HTbCfI CYMMapHaf! oWH6Ka.

6.14.24. HI'PaJIbHaf!KOCTb IIo.n;6pacbIBaeTcfI 120 pa3. Ou;eHHTb BepOfiTHOCTb TOI'O,'ITO:

a) 'lHCJIOIIOflBJIeHHfi 6 O'iKOB6y.n;eT He MeHbwe 30;

6)6 O'iKOBIIOflBHTCfI OT 12 .n;o 28 pa3.

6.14.25.MOHeTa IIo.n;6pacbIBaeTcfI 100 pa3. C IIOMOIIJ;blO HepaBeHcTBa qe6blweBa ou;eHHTb BepofiTHoCTb TOI'O,'ITO'lHCJIOBbIIIaBWHX I'ep60B 6y.n;eT Haxo.n;HTbCfI B IIpe.n;eJIax OT 40 .n;o 60. HafiTH BepOfiTHOCTb 3TOI'O:lKeC06bITHfi C IIOMOIIJ;blO HHTeI'paJIbHOfi<P0PMYJIbI MyaBpaJIaIIJIaca.

6.14.26.CJIY'lafiHaf!BeJIH'lHHaX 3a,rr,aHa <pYHKU;Hefi paCIIpe.n;eJIeHHfI

 o, IIpH X :::; 1, F(x) = {(x - 1)2, IIpH 1 < x :::; 2, 1, IIpH 2 < x.

C IIOMOIIJ;blO HepaBeHCTBa qe6blweBa ou;eHHTb BepOfiTHOCTb C06bI-

THfi A = {IX - M(X)I < ~}. HafiTH P {IX - M(X)I < ~}.

6.14.27. H3I'OTOBJIeHaIIapTHfi .n;eTaJIefi. Cpe.n;Hee 3Ha'leHHe.n;JIHHbI .n;eTaJIH

PaBHO 20 CM, a cpe.n;Hee KBa,rr,paTH'leCKOeOTKJIOHeHHe paBHO 0,1 CM.

435

 Ou;eHUTh CHU3Y BepOJlTHOCTh TOro, 'ITO.n:JIUHa CJIyqafiHO oT06paH- Hofi .n:eTaJIU HaxO.n:UTCJI B npe.n:eJIax OT 19,6.n:0 20,4 CM. 6.14.28. IhBecTHo, 'ITO: X U Y - HeoTpUu;aTeJIhHhle He3aBUCUMhle CJIy- qafiHhle BeJIUqUHhI; M(X) = 2, M(Y) = 7. Ou;eHuTh BepOJlTHOCTh 6.14.29. C06hITUfi A = {X + Y < 16}, B = {XY < 42}. Bcxo}KecTh CeMJlH HeKoToporo paCTeHUJI COCTaBJIJleT 80%. Ou;e- HUTh BepOJlTHOCTh Toro, 'ITOnpu noceBe 4000 CeMJlH: a) OTKJIOHeHUe qUCJIa B30Ille.n:umx CeMJlH (c. B. X) OT M(X) He npeB30fi.n:eT no MO.n:yJIIO 100; 6) OTKJIOHeHUe .n:OJIU B30Ille.n:IllUX CeMJlH OT BepOJlTHOCTU BCXO}Ke- CTU JII060ro U3 HUX He npeB30fi.n:eT no MO.n:yJIIO 0,03. 6.14.30. Ou;eHuTh C nOMOIu;hIO HepaBeHCTBa qe6hIllieBa BepOJlTHOCTh Toro, 'ITOcpeM 800 HOBOPO}K.n:eHHhIX .n:eTefi MaJIhqUKOB 6y.n:eT OT 370 .n:o 430 BKJIIOqUTeJIhHO. CqUTaTh BepOJlTHOCTh PO}K.n:eHUJI MaJIhqUKa paBHofi 0,5. 6.14.31. CKOJIhKO pa3 Ha.n:O no.n:6POCUTh MOHeTY, qT06hI BepOJlTHOCTh OT- KJIOHeHUJI OTHOCUTeJIhHOfi qaCTOThI nOJlBJIeHUJI rep6a OT BepOJlT- HOCTU ero nOJlBJIeHUJI npu O.n:HOM no.n:6pachIBaHuu Ha BeJIUqUHY, MeHhIllYIO 0,1, 6hIJIa: a) 60JIhIlle 0,90; 6) 60JIhIlle 0,98? 6.14.32. IbBecTHO, 'ITOC. B. X UMeeT nJIOTHOCTh pacnpe.n:eJIeHUJI x ~ 0, x < O. llcnoJIh3YJl HepaBeHCTBO Qe6hIllieBa, ou;eHUTh BepOJlTHOCTh C06hI- TUJI A = {X E (0; 6)}. 6.14.33. ,II:ucnepcuJI K~.n:ofi U3 He3aBUCUMhIX c. B. Xi, 03HaqaIOIu;efi npo.n:OJl}KUTeJIhHOCTh rOpeHUJI 9JIeKTpOJIaMnOqKU, He npeBhIlliaeT 30 qacOB. CKOJIhKO JIaMnOqeK Ha.n:O B3J1Th .LJ:JIJI ucnhITaHufi, qT06hI BepOJlTHOCTh Toro, 'ITOOTKJIOHeHUe cpe.LJ:Hefi npo.n:OJl}KUTeJIhHOCTU rOpeHUJI JIaMnOqKU OT cpe.n:Hero apmpMeTUqeCKOrO UX MaTeMaTU- qeCKUX O}Ku.n:aHufi MeHhIlle (no MO.n:yJIIO) o.n:Horo qaca, 6hIJIa He MeHhIlle 0,90? 6.14.34. Y.n:oBJIeTBOpJleT JIU nOCJIe.n:OBaTeJIhHOCTh Xl, X 2 , • •• , X n , ... He3a- BUCUMhIX C. B., UMeIOIu;UX nJIOTHOCTh Ix; (x) = 1 2 2' 3aKOHY 60JIhIIIUX quceJI? (1 + x ) 6.14.35. BepoJlTHoCTh UCK~eHUJI o.n:HQrO CUrHaJIa paBHa 0,02. IIoJIh3YJlCh

u;eHTpaJIhHOfi npe.n:eJIhHofi TeopeMofi, HafiTU BepOJlTHOCTh Toro, 'ITO U3 1000 nepe.n:aHHhIX CUrHaJIOB 6y.n:eT UCK~eHO:

 a) 60JIhIlle 22; 6) MeHhIlle 40. 6.14.36. IIoe3.n: COCTOUT U3 49 BarOHOB. Bec BaroHa - CJIyqatiHruI BeJIUqU- Ha X, .n:JIJI KOTOpofi M(X) = 60 T, a(X) = 7 T. JIOKOMOTUB M0-

}KeT Be3TU noe3.n:, eCJIU Macca nOCJIe.n:Hero He npeBocxo.n:UT 3000 T.

436

B IIPOTHBHOM CJIY'IaeIIO.n:QeIIJIHIOT .n:OIIOJIHHTeJIbHbIii JIOKOMOTHB.

KaKOBa BepOHTHOCTb TOrO, 'ITO3TOrO .n:eJIaTb He IIpH.n:eTCH?

6.14.37.MrpaJIbHaH KOCTb IIo.n:6pacbIBaeTcH 360 pa3. HaiiTH BepOHTHOCTb

Toro, 'ITOCYMMapHoe 'IHCJIOO'IKOB6y.n:eT HaxO.n:HTbCH B IIpe,n;eJIax OT 1200 .n:o 1298 (T. e. P{1200 ::; 8360 ::; 1298}).

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6.14.39. IIycTb Xl, X 2 , .•. , X 100 - IIOCJIe.n:OBaTeJIbHOCTb He3aBHCHMbIX CTaH.n:apTHbIX CJIY'IaiiHbIXBeJIH'IHH(T.e. Xi '" N(O; 1)). MCIIOJIb-

3yH u;eHTPaJIbHYIO IIpe.n:eJIbHYIO TeopeMY, HaiiTH BepoHTHoCTb Toro, 'ITOC. B. 8100 = Xl + X? + ... + Xloo IIpHMeT 3Ha'IeHHe60JIbIIIe,

'IeM125,8.

KOHTponbHble Bonpocbl M 60nee CnO)l(Hbie 3aAaHMH

6.14.40.Ou;eHHTb BepoHTHoCTb Toro, 'ITOC. B. X OTKJIOHHTCH OT CBoero MaTeMaTH'IeCKOrOO)KH.n:aHHH M(X) MeHee 'IeMHa 3a(X). YKa3aTb 3TH BepOHTHOCTH .LJ:JIH C. B. X, HMelOrn;eii:

a) IIOKa3aTeJIbHOe paCIIpe.n:eJIeHHe;

6) paBHOMepHoe pacIIpe,n;eJIeHHe;

B)HOpMaJIbHOe paCIIpe.n:eJIeHHe.

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a) 6y.n:eT He MeHee 400;

6)OTKJIOHHTCH OT MaTeMaTH'IeCKOrOO)KH.n:aHHH M(X) MeHbIIIe, 'IeMHa 25.

6.14.42.06rn;aH CTOHMOCTb Bcex 6YKeToB B U;BeTO'IHOMKHOCKe COCTaBJIHeT 18000 py6. BepoHTHoCTb Toro, 'ITOCTOHMOCTb B3HToro Hayra,IJ; 6yKeTa He IIpeBbIIIIaeT 300 py6JIeii, paBHa 0,7. QTO MO)KHO CKa3aTb 0 KOJIH'IeCTBe6YKeToB B KHocKe?

6.14.43.BepoHTHoCTb BbIxo.n:a H3 CTPOH 3JIeMeHTa pa,IJ;HOTeXHH'IeCKoro YCTpoiicTBa 3a BpeMH T paBHa 0,1. Ou;eHHTb BepOHTHOCTb Toro, 'ITO3a BpeMH T H3 100 3JIeMeHTOB BbIii.n:eT H3 CTPOH MeHee 20.

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6.14.45. ,lJ;oKa3aTb TeopeMY MapKoBa:

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n

lim ~ L D(Xi ) = O.

n-too n

i=l

Tor.n;a gJIH JIIo6oro c > 0

 6.14.46. HafiTH a m -a lim '"' ~ a-too L..J m! ' m=O r.n;e a - rreJIoe IIOJIOlKHTeJIbHOe qHCJIO. 6.14.47. IbBecTHo, qTO CJIyqafiHble BeJIHqHHbI X1 ,X2 , .•• ,Xn HMeIOT paB-

HOMepHoe pacIIpe.n;eJIeHHe COOTBeTCTBeHHO Ha IIpOMe}KYTKax (0; 1), (0; 2),... , (0; n). KaK 6y.n;eT MeHHTbCH c. B.

n

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o

§1. OCHOBHblE 3flEMEHTAPHbiE Cl»YHKLJ.L-1L-1 KOMnflEKCHoro nEPEMEHHoro

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ECJIH Z = X + iy, W = u + iv, TO MOlKHO 3aIIHcaTb J(z} B BH,n;e

J(z} = u(x, y} + iv(x, y},

f,!l;e u(x,y} = ReJ(z} - ,n;eil:cTBHTeJIbHaH qacTb J(z}, v(x,y} = ImJ(z} - MHHMaH

qaCTb J(z}.

YKalKeM HeKOTopbIe 3JIeMeHTapHble <PYHKIIHH KOMIIJIeKCHOfO IIepeMeHHOfO:

 eiz =cosz+isinz, zEIR (,popMYJl.a 9iiltepa) , eiz _ e- iz (1.1) sin z = "---;:-:-=--- 2i + e- z shz = eZ _ e- z chz = eZ 2 ' 2 ' In z = In Izl + i arg z (arg z E (-7r, 7r]),

Lnz = Inlzl +iArgz = Inlzl + iargz +i· 27rk = Inz + i· 27rk (k E Z).

3aMeTHM, qTO Ln z - MHOr03HaqHaa <PYHKIIHH, KOTopaa KalK,!I;OMY qHCJIY z I- 0

CTaBHT B COOTBeTCTBHe 6eCKOHeqHOe MHOlKeCTBO 3HaqeHHiI: {Ln z}.

 7.1.1. )J,JIf! .n;aHHoii

Q a) Z2 = (X+iy)2 = x 2+2x·iy+(iy)2 = x 2+i·2xy_ y2 = (X 2_ y2)+i·2xy,

 T.e. u(x,y) = x 2 - y2; v(x,y) = 2xy. 6) ~ = 1 = _1_._ = ~ + iy . x + iy = x + z x+iy x-zy (x-zy)(x+zy) X2 +y2 X2 +y2 +i· 2 Y 2,T.e.u(x,y)= 2 X 2'V(X,y)= 2 Y 2' X +y x +y x +y B) eZ = ex +iy = eX. eiy = eX. (cosy + isiny) = eX cosy + i· eX siny, T.e. u(x,y) = eX cosy; v(x,y) = eXsiny.

439

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