Добавил:

Upload Опубликованный материал нарушает ваши авторские права? Сообщите нам.

Вуз:

Санкт-Петербургский государственный университет телекоммуникаций им. проф. М.А. Бонч-Бруевича

Предмет:

[НЕСОРТИРОВАННОЕ]

Файл:

Сборник задач по высшей математике 2 том

.pdf

Скачиваний:

Добавлен:

15.03.2015

Размер:

27.38 Mб

Скачать

☆

<<< < Предыдущая 32 33 34 35 36 37 38 39 40 41 42 4344 / 6044 45 46 47 48 49 50 51 52 53 54 55 56 > Следующая >>>

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

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)
Pia < Sn < (3} ~ ~o		O'(Sn)	- ~o	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 <popMY.TIe (14.1), nO.TIY'IaeMC.TIe)J;yIOIII;yIO ou;eHKY BepOHTHOCTH:
	5	•
	P{IX - 41 < 5} ~ 1- 9,~ = 1 - 0,384 = 0,616.
6.14.2.	CpegHHii CpOK C.TIY:lK6bI npH60pa 10 .TIeT. HCnO.TIb3YH HepaBeHcTBo
	MapKoBa, ou;eHHTb BepoHTHoCTb Toro, 'ITOgaHHblii npH60p He npo-
	C.TIY:lKHT 6o.TIee 15 .TIeT.
6.14.3.	,II:HcKpeTHaH c. B. X 3agaHa 3aKOHOM pacnpege.TIeHHH:

	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.
o	C.TIY'IaiiHMBe.TIH'IHHaX HMeeT n.TIOTHOCTb pacnpege.TIeHHH
				f(X)={~'		xE(2,8),
					0,	x ¢ (2,8).
I1cnO.TIb3YH <POPMY.TIY
				P{a < X <,B} = !f3 f(x)dx,
						a
HaxogHM BepOHTHOCTb C06bITHH A. HMeeM:
		6,5	1	1	6	3	1
	P(A) =	!	1	1	1,5	3	1	T. e.	P(A) = 0,5.
	P(A) =		- dx = - .X		3,5	= - = -		T. e.	P(A) = 0,5.
			6	6	3,5	6	2'
		3,5
	MaTeMaTH'IeCKOeO:lKHgaHHe H gHcnepcHH c. B. X TaKOBbI:
	M(X) = [a ~b] = 8 ~2 = 5,					D(X) = [(b ~2a)2] = ~~ = 3.

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

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.
a	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		= !~ ~0,271.	•

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
	f8162(S)~		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 = ;~,
	162
TO, npHMeHHH <POPMYJIY (14.4), nOJIyqHM
P{22 < S,,' < 26) '"~o(2~~)				~o( 2/¥.~)'"

~ 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}.
6.14.29.	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}).

6.14.38.IIpIDKHBalOTcH B cpe,n;HeM 70% OT 'IHCJIaIIOCruKeHHbIX CruKeHu;eB. CKOJIbKO HY)KHO IIOCa,IJ;HTb CruKeHQeB, 'IT06bIC BepOHTHOCTblO, He MeHbIIIeii 0,9, O:llm.n:aTb, 'ITOOTKJIOHeHHe 'IHCJIaIIpIDKHBIIIHXCH ca- )KeHu;eB OT MaTeMaTH'IeCKOrOO)KH.n:aHHH He IIpeBbIIIIaeT IIO MO.n:yJIIO 40? PeIIIHTb 3a,IJ;a'IYC IIOMOrn;blO HepaBeHCTBa Qe6bIIIIeBa.

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.

6.14.41.MrpaJIbHaH KOCTb IIo.n:6pacbIBaeTcH 100 pa3. Ou;eHHTb BepOHTHOCTb Toro, 'ITOCYMMapHoe 'IHCJIOO'IKOB(c. B. X):

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.

6.14.44.IIpHMeHHMa JIH K IIOCJIe.n:OBaTeJIbHOCTH He3aBHCHMbIX c. B. Xl,'"

... ,Xn,"" r.n:e

Xi '" R(a, b),

437

6.14.45. ,lJ;oKa3aTb TeopeMY MapKoBa:

IIycTb X1 ,X2 , ..• ,Xn - IIOCJIe.n;OBaTeJIbHOCTb He3aBHCHMbIX C. B., .n;JIH KOTOPbIX

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.

X = ft LXi

i=l

....

rnaBa 7. TEOPLIIH Q>YHK4L11L11 KOMnflEKCHOrO nEPEMEHHOrO

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

~ITYCTb D - HeKOTopoe IIo,n;MHOlKeCTBO KOMIIJIeKcHoil: IIJIOCKOCTH c. Ko~mJl.e1CC

tloiJ. ,pY'H.1CtJ,ueii J(z} C 06JIacTbIO OIIpe,n;eJIeHHH D Ha3bIBaeTCH oTo6palKeHHe, KOTOpoe KalK,n;Oil: TOqKe zED CTaBHT B COOTBeTCTBHe KOMIIJIeKCHOe qHCJIO W = J(z}. ~

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 <PYHKIIHH J(z)	= u(x, y) + iv(x, y), r.n;e z = x + iy,
	HaiiTH .n;eiicTBHTeJIbHYIO qaCTb u(x, y) H MHHMYIO qacTb v(x, y):
	a) J(z) = Z2;	6) J(z) = t
	B) J(z) = eZ ;	z
	B) J(z) = eZ ;	r) J(z) = sin z.

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

<<< < Предыдущая 32 33 34 35 36 37 38 39 40 41 42 4344 / 6044 45 46 47 48 49 50 51 52 53 54 55 56 > Следующая >>>

Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]

#
15.03.20158.82 Кб16Самостоятельная работа 2.docx
#
15.03.2015605.18 Кб8Самоубийцы.doc
#
15.03.2015205.24 Кб63саодисс3.docx
#
29.03.2016401.92 Кб83САПР (методичка).doc
#
04.09.2019147.46 Кб8сацо-30.doc
#
15.03.201527.38 Mб58Сборник задач по высшей математике 2 том.pdf
#
15.03.20151.11 Mб86сборник иб .doc
#
15.03.201513.6 Кб47сводная таблица налоги.docx
#
15.03.2015140.33 Кб21Связи с общественностью.docx
#
04.09.201995.23 Кб4СГП.doc
#
23.11.2019207.16 Кб0СДЭ.docx