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

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[НЕСОРТИРОВАННОЕ]

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Сборник задач по высшей математике 2 том

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6.9.3.	3a,L1;aHa <PYHKIJ;IUI paCnpe,ll;eJIeHHjf H. C. B. X
	O'	x < -11",
	F(x) = { a(cosx + c),	-11" :::;; X :::;; 0,
	1,	0< x.

Hat!:TH:

a) 3HaqeHHjf nOCTOjfHHbIX a H C;

6)f(x);

B) P1 {X E [- ~, ~]}, P2 { X = 2;03}.

6.9.4.CJIyqai\HM BeJIHqHHa X 3a,L1;aHa <PYHKIJ;Hei\ pacnpe,ll;eJIeHHjf

O'	x :::;;A,
F(x) = { ~v'X- 1,	A<x:::;;B,
1,	B <x.

Hai\TH: 3HaqeHHjf A H B, nJIOTHOCTb pacnpe,ll;eJIeHHjf H. c. B. X,

BepOjfTHOCTb C06blTHjf C = {X E (3; 5)}.

6.9.5. IIpH KaKOM 3HaqeHHH napaMeTpa C <PYHKIJ;Hjf

			f(x) = {xS'X		~ 1,
			0,	x < 1
	MO:>KeT 6bITb nJIOTHOCTbIO pacnpe,ll;eJIeHHjf HeKoTopoi\ HenpepbIB-
	HOi\ c. B. X? Hai\TH P{1 < X < 5}.
Q OqeBH,ll;HO, 'ITOf(x)			> °npH C > 0,		lim	f(x) = 0.		lknoJIb3Yjf CBOi\-
			00	x-+±oo
CTBO HOpMHpOBaHHOCTH (			f f(x) dx = 1), Hai\,ll;eM 3HaqeHHe napaMeTpa C:
			- 00
00	1		00		b
f	f(x) dx = f	°dx + f ~ dx = C· b~~ f x-4 dx =
- 00	- 00		1		1	lib = _ C (0 -1) = C,
			= _C . lim			lib = _ C (0 -1) = C,
				3	b-+oo x 3 1			3	3
T. e. ~ = 1, OTCIO,ll;a C = 3. TaKHM o6pa30M, <PYHKIJ;Hjf
			3 ,	X	~ 1,
			f(x) = { x4	x < 1
			0,	x < 1
jfBJIjfeTCjf nJIOTHOCTbIO pacnpe,ll;eJIeHHjf HeKoTopoi\ c. B. X.
Hai\,ll;eM HCKOMYIO BepOjfTHOCTb, HCnOJIb3Yjf <POPMYJIY
					b
		P{a:::;;X:::;;b}=		ff(x)dx.
IIoJIyqaeM:			5	a
			3			1	15	124	= 0,992.	•
P{1 :::;; X :::;; 5} = P{1 < X < 5} = f x4 dx = 3· -3x3							1	= 125	= 0,992.
			1

350

6.9.6.HerrpephIBHaJI C. B. X HMeeT rrJIOTHOCTh pacrrpe)J;eJIeHHH BepOHTHO-

cTeii

34 , X ~ 1, f(x) = { x

0, x < 1.

HaiiTH ¢YHKIIHIO pacrrpe)J;eJIeHHH BepoHTHocTeii F(x); rrOCTpOHTh

rpa¢HKH f(x) H F(x).

o MCrrOJIh3yeM ¢OpMyJIy

F(x) = !x f(t) dt.

-<Xl

I1PH X E (-00,1) HMeeM:

F(x)= !O.dt=O.

-<Xl

TIPH X E [1, +00) rrpOMe:>KYTOK HHTerpHpOBaHHH pa36HBaeTcH Ha )J;Ba:

f(x)	F(x)
3 ----	26 1 ---------------------
	27 ~
	8"

o	1 2 3	x	o	1 2	x

Puc. 82		Puc. 83
TaKHM o6pa30M,
	O,	X < 1,
F(x) =	{ l _ l	x~1.
	X 3	x~1.
	X 3	'
rpa¢HKH ¢YHKIIHii f(x)	H F(x) rrpe)J;cTaBJIeHhI COOTBeTCTBeHHO Ha
puc. 82 H pHC. 83.		•

351

6.9.7. HeIIpepbIBHaH CJIY'IaiiHaH BeJIlI'IIIHa X paCIIpe,11,eJIeHa «IIO 3aKoHY IIpHMoyroJIbHOrO TpeyroJIbHIIKa» Ha IIHTepBaJIe (0,4); Ha pllC. 84

H306pIDKeHa IIJIOTHOCTb pacIIpe,11,eJIeHIIH 9TOii c. B. HaiiTII: a) 3Ha'leHlIe Yo;

6) aHaJIIITII'IeCKOe BbIpIDKeHlIe ,11,JIH IIJIOTHOCTII f(x) II <PYHKIIIIII

paCIIpe,11,eJIeHIIH F(x). llOCTPOIlTb rpa<PIIK F(x).
a a) TaK KaK IIJIOIII8,!I.b S <PlIrYPbI, OrpaHH'IeHHOii cBepxy KPIlBOii pacIIpe-
,11,eJIeHIIH (T. e. rpa<PIIKoM <PYHKIIIIII	f(x», a	CHlI3Y -	OCbIO Ox, paBHa 1, TO
1	1
S = S.6.AOB = 2" ·4· Yo = 1. OTCIO,11,a Yo = 2".
6) YpaBHeHlIe IIPHMOii AB Haii,11,eM KaK ypaBHeHlIe IIPHMOii, IIpOXO,11,HIIIeii
'Iepe3 TO'lKII A (O;~) II B(4;0»: 'J/= -kx+~. AHaJIIITII'IeCKOe BbIpIDKeHlIe
,l1,JlH IIJIOTHOCTII paCIIpe,11,eJIeHIIH c. B. X TaKOBO:
II		xE[0,4],
f(x)= { -8 x	+2"'	xE[0,4],
0,		x fi. [0,4].
f(x)
yo A
o		B
o		4	x

Puc. 84

TeIIepb Haii,11,eM <PYHKIIIIIO paCIIpe,11,eJIeHIIH F(x): x

eCJIII	x E (-00,0), TO F(x) = j Odt = 0;
	- 00
eCJIII	x E [0,4], TO

o	x						+~.
F(x) = j	Odt + j(_lt + 1) dt = (-le + i) IX = _x2						+~.
	8	2	16	2	0	16	2'
- 00	0
eCJIII x E (4,+00), TO
o	4		x
F(x)= jOdt+ j(-kt+~) dt+ jOdt=
- 00	0		4

= 0 + ( - ~~ + ~)I:+ 0 = -1 + 0 + 2 - 0 = 1.

352

TaKHM 06pa3oM,

	O,		x < 0,
F(x) = {	X 2	X	°~ x ~ 4,
F(x) = {	-16 + 2'		°~ x ~ 4,
	1,		x> 4.	•
fpa<pHK <PYHKlI.HH F(x) H306p8.)KeH Ha pHC. 85.
F(x)
F(x)
	-------------		:.:.-----
			4	x
	Puc. 85

6.9.8.,naHa nJIOTHOCTb pacnpe,11,eJIeHHg c. B. X:

O,	npH x < 0,
f(x) = { b·x,	npH °~ x < 5,8,
0,	npH 5,8 ~ x.

Onpe,11,eJIHTb nOCTOgHHYIO b, HaiiTH <PYHKlI.HIO pacnpe,11,eJIeHHg F(x)

H nOCTpOHTb ee rpa¢HK, BbI'IHCJIHTbBepOgTHOCTb Toro, "ITO C. B.

X npHMeT 3Ha"leHHe, Y,11,OBJIeTBOpgIOm;ee YCJIOBHIO:

a) X < 3,3;

6) 3,3 < X < 7,8.

6.9.9.llJIOTHOCTb BepOgTHOCTH c. B. X HMeeT BH,11,

f(x) = eX :e-x ' x E III

HaiiTH 3Ha"leHHe napaMeTpa a, <PYHKIJ.HIO pacnpe,11,eJIeH~ F(x).

6.9.10.CJIY"IaiiHM BeJIH"IHHa X HMeeT nJIOTHOCTb pacnpe,11,eJIeHHg

f(x) = {v'4~x 2 '	Ixl < 2,
0,	Ixl ~ 2.

HaiiTH:

a) 3Ha"leHHe napaMeTpa C;

6) <PYHKlI.HIO pacnpe,11,eJIeHHg F(x);

0)P{l < X < 5}.

6.9.11.3a,I1;aHa nJIOTHOCTb pacnpe,11,eJIeHHg H. c. B. X:

o,	x ~ 1,
f(x) = { 2x -	2, 1 < x ~ 2,
0,	2 < x.
12 CooPHH. _qno ....ew.A NBTCNlTH... 2 KypC	353

qTO BepmlTHee: nona,.u,rume CJIY'IaiiHOii Beml'IllHbI B IIHTepBaJI

(1,6; 1,8) IIJIII B (1,9; 2,6)?

6.9.12. 3a,.u,aHa nJIOTHOCTb pacnpe,II.eJIeHIIH H. c. B. X:

a,	x< -A,
f(x) = { -x,	-A::::; x <0,
2x,	o::::;x < A,
0,	A::::;x.
HaiiTII A, F(x), P{ -2 < X	< I}.

6.9.13. rpa<PIIK nJIOTHOCTII pacnpe,II.eJIeHIIH H. c. B. X IIMeeT BII,II., 1I306pa- :>KeHHblii Ha pllC. 86. 3anllcaTb aHaJIlITlI'IeCKOe Bblpa:>KeHlIe ,II.JIH nJIOTHOCTII pacnpe,II.eJIeHIIH f (x) .

F(x)

f(x)	1	-----------------------	r--------
f(x)			I
		:;:	I
	0,5		I
	0,5
x	o		x
Puc. 86		Puc. 87

	F(x) = {3X ,	npll x ::::; 0,
	1,	npll x > 0.
	HaiiTII:
	a) nJIOTHOCTb f(x);
	6) BepOHTHOCTb Toro, 'ITOC. B. X B pe3YJIbTaTe onbITa npllMeT 3Ha-
	'IeHlIeB IIHTepBaJIe (-1, 1).
6.9.15.	Ha pllC. 87 3a,.u,aH rpa<pIIK <PYHKIJ;1I11 pacnpe,II.eJIeHIIH c. B. X. HaiiTII
	aHaJIlITlI'IeCKOeBblpa:>KeHlIe ,II.JIH:
	a) F(x);	6) f(x).
	IIOCTPOIlTb rpa<pIIK nJIOTHOCTII pacnpe,II.eJIeHIIH c. B. X.
6.9.16.	cDYHKIJ;IIH pacnpe,II.eJIeHIIH H. c. B. X 3a,.u,aHa Bblpa:>KeHlIeM

F(x)= {a.sin(x+~)+b,

354

	Ha:lhH:	6) rrJIOTHOCTb f{x);
	a) K09<P<lmIJ;HeHTbI a, b;	6) rrJIOTHOCTb f{x);
	B)P{O~X<~}.
	IIOCTPOHTb rpa¢HK <PYHKIJ;HH f{x).
6.9.17.	3a,n;aHa <PYHKIJ;Hg
	O,	x ~ 0,
	F{x)= { a·x2 ,	O<x~l,
	1,	1 < x.
	Orrpe,n;eJIHTb:
	a) rrpH KaKOM 3Ha'leHHHa <PYHKIJ;Hg F{x) 6y,n;eT <PYHKIJ;He:li pac-
	rrpe,n;eJIeHHg HeKoTopo:li c. B. X;
	6) rrJIOTHOCTb BepOgTHOCTH c. B. X;
	B) BepOgTHOCTb C06bITHg D = {- ~ ~ x ~ ~}.
6.9.1S.	cDYHKIJ;Hg pacrrpe,n;eJIeHHg C. B. X HMeeT BH,n; F{x) = a +b· arctg~,
	x E lR. Ha:liTH:
	a) 3Ha'leHHerrapaMeTpOB a H b;
	6) rrJIOTHOCTb BepOgTHOCTH.
6.9.19.	cDYHKIJ;Hg pacrrpe,n;eJIeHHg H. c. B. X - BpeMeHH 6e30TKa3Ho:li pa6o-
		x
	TbI HeKoToporo rrpH60pa - paBHa F{x) = 1- e- T , x ~ 0. Ha:liTH
	P {X > T}, T. e. BepOgTHOCTb 6e30TKa3Ho:li pa60TbI rrpH60pa 3a
	BpeMg, 60JIblIIee T.
6.9.20.	)J;aHa <PYHKIJ;Hg
	f{X)={A.x.e- x , x~O,
	0,	x < 0.
	IIPH KaKOM 3Ha'leHHHrrapaMeTpa A 9Ta <PYHKIJ;Hg gBJIgeTCg rrJIOT-
	HOCTbIO pacrrpe,n;eJIeHHg HeKoTopo:li H. c. B. X? Ha:liTH F{x).
6.9.21. 3a,n;aHa rrJIOTHOCTb BepOgTHOCTH CJIY'Ia:liHo:liBeJIH'IHHbIX:
	O,	x < -4,
	f{x) = { -Ax,	-4 ~ x < 0,
	Ay'x,	°~ x < 4,
	0,	4 ~ x.
	Ha:liTH A, F{x), P{ -1 < X < 5}.
6.9.22.	IIJIOTHOCTb BepOgTHOCTH H. c. B. X HMeeT BH,n;
	0,	x< -A,
	f{x) = ;~'	-A ~ x < 0,
	12'	°~ x < 2A,
	0,	2A ~x.

Ha:liTH A, F{x), P{ -0,5 < X < 2}.

355

KOHTponbHble BOnpOCbI M 60nee CnO)l(Hble 3aACIHMH

6.9.23. 3a,rJ;aHa HeKOTOpaJI <PYHKIJ;IUI
a,	x ~ -1,
F(x)= { i,(X+l)3,	-1 < x ~ 1,
1,	1 < x.

HBJIHeTCH JIH OHa <PYHKIJ;Heii pacrrpe,11,eJIeHHH HeKOTopoii CJIyqaiiHoii BeJIHqHHbI X? qeMY paBHa BepOHTHOCTb C06bITHH

A={O~X<I}?

6.9.24.	3HaqeHHH c. B. X HaxO,11,HTCH B	rrpOMe:>KYTKe	(OJ i). MO:>KeT JIH
	<PYHKIJ;HH pacrrpe,11,eJIeHHH F(x) paBHHTbCH Ha 3TOM yqacTKe:
	a) sinx;	6) X 2 j
	B) 1,lj	r) COSXj
	.n.) 2:?
6.9.25.	HerrpephIBHaJI c. B. X 3a,rJ;aHa <PYHKIJ;Heii pacrrpe,11,eJIeHHH
	O'	x < -i,
	F(x)= { a.sin(x-i)+b, _i~x~3;,
	C,	371"
	C,	4" <x.
	a) 3HaqeHHH a, b, Cj
	6) rrJIOTHOCTb ! (x) pacrrpe,11,eJIeHHH c. B. X. ITOCTPOHTb rpa¢HKH
	<PYHKIJ;Hii F(x) H !(x).
6.9.26.	cDYHKIJ;HH pacrrpe,11,eJIeHHH H. c. B. X, paBHaJI
	a,	x ~ 1,
	F(x) = {ax2 + bx + c,	1 < x ~ 2,
	1,	2 < X.
	cDYHKIJ;HH F(x) = ax2 + bx + C rrpH x = 2 HMeeT MaKCHMYM. HaiiTH:
	a) rrapaMeTpbI a, b, Cj		= {I ~ X < 3},
	6) BepOHTHOCTH C06bITHii A = {X > 3}, B		= {I ~ X < 3},

E = {X E ( - lj 1,5)}.

6.9.27.HerrpepbIBHaJI c. B. X pacrrpe,11,eJIeHa rro 3aKOHY JIanJIaca:

!(x) = A· e->"Izl, r,11,e A > 0.

HaiiTH K03<P<PHIJ;HeHT A H <PYHKIJ;HIO pacrrpe,11,eJIeHHH F(x). IToCTPOHTb rpa¢HKH !(x) H F(x).

6.9.28. HerrpepbIBHaJI c. B. X 3a,rJ;aHa <PYHKIJ;Heii pacrrpe,11,eJIeHHH

O'		x ~ y'3,
F(x) = { x6 -	X4 - 18	. '3<	~ 2	,
	30	,v.,	x"	,
1,		2 < X.

HaiiTH P{I,8 < X < 2,8}, !(x)j rrocTpoHTb rpa¢HKH F(x) H !(x).

356

xapaICmepucmuICu nOArotCeH'U.fI

fx(x).

6.9.29. CJIY"IaftHruI Beml"lHHa X nO,!l;"IHHHeTCH 3aKOHY raycca:

f(x) =	(:I: +2)2
f(x) =	1 . e -- l - S -
	3· v'21T

HaftTH ¢YHKIJ;HIO pacnpe,!l;eJIeHHH F(x), nOCTpOHTb rpa¢HKH ¢YHK-

IJ;Hft f(x) H F(x).

6.9.30. IIPH KaKOM 3Ha"leHHH napaMeTpa A ¢YHKIJ;HH

{~ Ixl ~ 1, f(x) = . (1 - Ix!), Ixl < 1

HBJIHeTCH nJIOTHOCTbIO pacnpe,!l;eJIeHHH HeKoTopoft H. c. B. X? IIo-

CTPOHTb rpa¢HK HaftTH ¢YHKIJ;HIO pacnpe,!l;eJIeHHH 9TOft

c.B. HaftTH

§10. l.IVlC110BbIE XAPAKTEPVlCTVlKVI C11Yl.IAtiHbIX BE11 Vll.I VI H

TIPH peIlIeHHH MHorHX 3a,n;a'lTeopHH BepOlITHOCTH BOBce He061I3aTeJIbHO 3HaTb 3aKOH pacnpe,1l;eJIeHHlI ,1I;aHHOit CJIY'IaitHOitBeJIH'IHHbI, nOJIHOCTbIO ee OnHCbIBaIOru;eil:. 3a'lacTYIO,1I;OCTaTO'lHOHMeTb no,1l; PYKOit JIHilib HeCKOJIbKO "'UCA061>tX xapaICmepucmuIC aToit CJIY'IaitHoitBeJIH'IHHbI,T. e. '1HCJIOBbIXnapaMeTpOB, xapaKTepH3YIOru;HX HaH60JIee BalKHbIe '1epTbIee 3aKOHa pacnpe,1l;eJIeHHlI. BalKHeitIlIHMH Cpe,1l;H HHX lIBJIlIIOTClI (MaTeMaTH'IeCKOeOlKH,1I;aHHe, Me,1l;H- aHa H T. ,11;.) H xapaICmepucmuICu paCCeslH'U.fI (,1I;HCnepCHlI, Cpe,1l;HeKBa,n;paTH'IeCKOeOTKJIOHeHHe H ,1I;p.).

~MameMamu",eCICuM rotCuiJaHueM (HJIH Cpe,1l;HHM 3Ha'leHHeM)M(X) (HJIH MX)

AHCKpeTHoit CJIY'IaitHoitBeJIH'IHHbI X Ha3bIBaeTClI CYMMa npOH3Be,1l;eHHit Bcex ee B03MOlKHbIX 3Ha'leHHitXi Ha HX COOTBeTCTBYIOIIIHe BepOlITHOCTH:

M(X) = L XiP.·

ECJIH ,11;. C. B. X npHHHMaeT KOHe'lHOe'1HCJIO3Ha'leHHitXl, X2, ••• , X", TO ee MaTeMaTH'IeCKOeOlKH,1I;aHHe HaxO,1l;HTClI no <popMYJIe

M(X) = L XiPi. i=l

ECJIH lKe ,11;. c. B. X npHHHMaeT C'IeTHOe'1HCJIO3Ha'leHHit,TO

M(X) =L :riPi;

i=l

npH aTOM M&TeMaTH'IeCKOeOlKH,1I;aHHe CYIIIecTByeT, eCJIH plI,1I; B npasoit '1acTHaToit <P0PMYJIbI a6COJIIOTHO CXO,1l;HTClI.

357

MaTeMaTll'leCKHMOlKH)l;aHHeM HenpepbIBHolt CJIY'IaltHoltBeJIH'IHHbIX C ITJIOT-

HOCTbIO BepOlITHOCTH I(x) HaxO)l;HTCli no <popMYJIe

M(X) = !00X· I(x) dx.

- 00

TIPH 9TOM MaTeMaTH'IeCKOeOlKH)l;aHHe cym;ecTByeT, eCJIH HHTerpaJI B npaBolt '1aCTH

<P0PMYJIbI a6COJIIOTHO CXO)l;HTClI (9TO 3Ha'lHT,'ITOCXO)l;HTCli HHTerpaJI !00 Ixl/(x) dx).

- 00

CBOMCTBa MaTeMan,...eCKoro O)l(MA'lHM5I

1.M(G) = G, r)l;e G = constj

2.M(GX) = G· M(X)j

3.M(X ± Y) = M(X) ± M(Y) (npaBHJIO CJIOlKeHHlI MaTeMaTH'IeCKHXOlKH)l;a-

HHlt)j

4. M(X . Y) = M(X) . M(Y), eCJIH X H Y - He3aBHCHMble CJIY'IaltHbleBeJIH-

'1HHbI(npaBHJIO YMHolKeHHlI MaTeMaTH'IeCKHXOlKH)l;aHHlt).

0603Ha'lHMMaTeMaTH'IeCKOeOlKH)l;aHHe c. B. X '1epe3a.

~ ,l(ucnepcueti (paccelIHHeM) CJIY'IaltHoltBeJIH'IHHbIX Ha3b1BaeTClI MaTeMaTH'IeCKoe OlKH)l;aHHe KBa)l;paTa OTKJIOHeHHll c. B. OT ee MaTeMaTH'IeCKOrOOlKH)l;aHHll a:

D(X) = M(X - a)2.

Cpa3Y H3 onpe)l;eJIeHHll BblTeKaeT '1acTOHCnOJIb3yeMaH <p0pMYJIa

ECJIH X -	)l;HCKpeTHalI CJIY'IaltHaHBeJIH'IHHa,TO ee )l;HCnepCHlI BbI'IHCJIlIeTClI
no <popMYJIe
	n			n
D(X) = ~)Xi - a)2 . Pi,			T. e.	D(X) = L x~ . Pi - a2
	i=l			i=l
B CJIY'IaeKOHe'lHOrO'1HCJIa3Ha'leHHlt,npHHHMaeMbIX C. B. X, H nc;> <p0pMYJIe
			00
		D(X) = L(Xi - a)2pi
			i=l
(T. e. D(X) =	00
(T. e. D(X) =	~ X~Pi -	a2 ) B CJIY'IaeC'IeTHOrO'1HCJIa3Ha'leHlIlt.
	i=l
ECJIH X -	HenpepblBHalI CJIY'IaltHaHBeJIH'IHHaC nJIOTHOCTbIO I(x), TO
	00			00
D(X) = !(x -		a)2 . I(x) dx,	HJIH	D(X) = ! x 2 • I(x) dx - a2 •
	- 00			- 00

358

CBoiiiCTBa AMCnepCMM

1.D(C) = 0, r,n;e C = constj

2.D(CX) = C2 . D(X)j

3. D(X ± Y) = D(X) + D(Y), eCJIH X H Y - He3aBHCHMble CJIY'IaitHbleBeJIH-

qHHbI (npaBHJIO CJIOlKeHHX ,n;HcnepcHit)j

4.D(X + C) = D(X).

~Cpea'HUM 7I:6aapamU"I.eC7I:UM om1CJlO'He'HUeM CJIY'IaitHoitBeJIH'IHHbIX Ha3bIBaeT-

Cfl '1HCJIOu(X), onpe,n;eJIXeMOe paBeHCTBOM u(X) = ..jD(X).

BeJIH'IHHau(X) HeOTpHn;aTeJIbHa H HMeeT TY lKe pa3MepHOCTb, 'ITOH C. B. X.

~ Ha"l.aJl'b'H'btM MOMe'HmOM nopx,n;Ka k (k = 0,1,2, ... ) CJIY'IaitHoitBeJIHqHHbI X Ha3bIBaeTCfl '1HCJIOOk, onpe,n;eJIXeMOe no <popMYJIe

eCJIH X - ,n;. C. B., H	00
Ok =	! xk. f(x)dx,
eCJIH X - H . C. B.	- 00
eCJIH X - H . C. B.

~ Lfe'HmpaJl'b'H'btM MOMe'HmOM nopJ/,a7l:a k C. B. X Ha3bIBaeTCfl '1HCJIOI-'k, onpe,n;e-

JIXeMOe no <popMYJIe

eCJIH X - ,n;. C. B., H

				00
			I-'k=	!(x-a)k·f(x)dx,
			- 00
eCJIH X - H . C. B.
~	K03rjjifJUv,ue'Hm	aCUMMempuu		(<<c7I:owe'H'Hocmu»), HJIH, KOpO'le, aCHMMeTpHX
c. B. X eCTb BeJIH'IHHa				A=~.
				A=~.
				u 3 (X)
~	K03rjjrjjuv,ue'Hm	371:cv,ecca (<<ocmpo6epWU'H'Hocmu»), HJIH,			nporn;e,	9KCn;ecc
C. B. X, eCTb BeJIH'IHHa
			E=~-3.
				u 4 (X)
~ Moaa a. C. 6. X		-	eCTb ee HaH60JIee BepOXTHoe 3Ha'leHHeMo(X). Moaa 'H. c. 6.
X	C nJIOTHOCTbIO f(x)		eCTb TO ee 3Ha'leHHeMo(X), npH KOTOPOM <PYHKn;Hx f(x)
)J,OCTHraeT MaKCHMYMa.						$
~ Meaua'Ha CJIY'IaitHoitBeJIH'IHHbI X (o603Ha'leHHe Me(X» -					eCTb	TaKoe ee

3Ha'leHHex p, ,n;JIX KOToporo o,n;HHaKOBO BepoflTHo, OKalKeTCfl JIH c. B. X MeHbIIIe xp

HJIH 60JIbIIIe x p , T. e.

PiX < x p } = PiX > Xp} = "2.

359

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