Сборник задач по высшей математике 2 том
.pdf~K6aHmUJlb70 yp06'HJ1 p C. B. X Ha3hIBaeTClI qHCJIO X p , y,!l;OBJIeTBOplIIO~ee ypas-
HeHHIO P{X < Xp} = p. ~
TaxHM 06pa30M, Xp lIBJIlIeTClI pemeHHeM ypaBHeHHlI F(xp) = p. B qacTHOCTH,
XO,5 = Me(X).
6.10.1. 3a,u.aH 3aKOH pacrrpe,ll;eJIeHHH,lJ,. c. B. X:
HaitTH MaTeMaTH"!eCKHe O>KH,lJ,aHHH H ,lJ,HCrrepCHH CJIy"!aitHblx BeJIH"!HH X, -2X, X2.
<) CHa"!aJIa Hait,ll;eM MaTeMaTH"!eCKHe O>KH,lJ,aHHH ,lJ,aHHbIX BeJIH"!HH. IIcrroJIb- n
3yH <P0PMYJIY M(X) = E XiPi, HaXO,lJ,HM i=l
6
M(X) = L XiPi = -2 ·0,1-1·0,2 +0 ·0,25+ 1·0,15+ 2 ·0,1 + 3·0,2 =0,55.
i=l
,IVrH HaxO>K,ll;eHHH M( -2X) BOCrrOJIb3yeMcH CBOitCTBOM MaTeMaTH"!eCKOrO O>KH,ll;aHHH: M(CX) =C·M(X).IIMeeM: M(-2X) = -2·M(X) =-2·0,55 =
= -1,1.
3aKOH pacrrpe,lJ,eJIeHHH c. B. X 2 3arrHweM B BH,lJ,e Ta6JIHn;bI pacrrpe)J,eJIeHHH:
rro orrpe,ll;eJIeHHlO ,lJ,HCrrepCHH HMeeM:
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D(X) = L(Xi - M(X))2 . Pi =
i=l
= (-2 - 0,55)2. 0,1 + (-1- 0,55)2·0,2 + (0 - 0,55)2. 0,25+ + (1 - 0,55)2 ·0,15 + (2 - 0,55)2 ·0,1 + (3 - 0,55)2 ·0;22 = 2,6475.
9TY >Ke BeJIH"!HHY MO>KHO HaitTH rrpOln;e, HCrrOJIb3YH <P0PMYJIY
D(X) = M(X2) - (MX)2.
,ll;eitcTBHTeJIbHO, M(X2) =2,95, (M(X))2 =0,552 =0,3025. CJIe)J,oBaTeJIbHO,
D(X) =2,95 - 0,3025 =2,6475. ,ll;aJIee,
D( -2X) = (_2)2 D(X) =4 . 2,6475 =10,59;
D(X2) = (0 - 2,95)2 ·0,25 + (1 - 2,95)2 ·0,35 + (4 - 2,95)2·0,2+
+ (9 - 2,95)2 ·0,2 = 11,0475.
360
nOCKOJIbKY PH,n: pacnpe,n:e.neHIUI ,n:. C. B. X 4 HMeeT BH,n:
TO D(X2) MO}l{HO HaAm npOrn;e:
D(X2) = M(X4) - (MX2)2 = 02·0,25+ 12 .0,35+42.0,2+92·0,2- (2,95)2 =
= 19,75 - 8,7025 = 11,0475. •
6.10.2. BpomeHbI 10 HrpaJIbHbIX KocTeA. HaATH M(X), D(X) U a(X), r,n:e c. B. X - cYMMa O'IKOB,BbmaBmHX Ha Bcex HrpaJIbHbIX KOCTHX.
a 0603Ha'lHM'1epe3Xi (i = 1,2, ... ,10) '1UCJIOO'IKOB,BbmaBmux Ha i-A KOCTH. Tor,n:a X = Xl + X 2 + X3 + ... + X lO . CJIY'IaAHbleBeJIH'IHHbIXl,
X2, ... , XlO HMeIOT O,n:HHaKOBble 3aKOHbI pacnpe,n:eJIeHUH, nOSTOMY M(Xl ) =
= M(X2) = ... = M(XlO) U D(Xl ) = |
D(X2) = |
... = D(XlO). II TaK KaK |
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C. B. Xi (i = 1,2, ... ,10) He3aBucuMbI, TO |
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M(X) = M(Xl + X 2 + ... + X lO ) = M(Xl ) + M(X2) + ... + M(XlO) = |
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= 10· M(Xl ), |
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D(X) = D(Xl +X2+ .. .+XlO ) = D(Xl )+D(X2)+ ...+D(XlO ) = lO·D(Xl ). |
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3aKOH pacnpe,n:eJIeHU11 C. B. Xl HMeeT BH,n: |
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Xl,i |
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IIoSTOMY, Y'IHTbIBaH,'ITO |
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D(X) = M(X)2 - |
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(M(X))2 = Lx~. Pi - (MX)2, |
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i=l |
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UMeeM |
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M(Xd = 1 . ~ + 2 . ~ + ... + 6 . ~ = ~, |
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D(Xl) = (12. ~ + 22. ~ + 32 . ~ + ... + 62 |
.~) _ (~)2 = 9i _ 4j = ~~. |
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CTaJIO 6bITb, M(X) = 10· M(Xl ) |
= 10· ~ |
= 35, D(X) = 10· D(Xl ) = |
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~ 10· ~~ = 1~5 = 29~, a(X) = ..jD(X) = Jl~5 ~ 5,4. |
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6.10.3. 3aKOH pacnpe,n:e.neHHH ,n:. c. B. X 3a,n;aH Ta6JIun;eA pacnpe,n:e.neHuH
Xi |
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Pi |
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HahH c, M(X), D(X), a(X), P{X < 3}.
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6.10.4. <l>YHKD;HH pacnpe.IJ:eJIeHHH .IJ:. c. B. X HMeeT BU.IJ:
0, |
x:::; 0, |
0,2, |
0< x:::; 1, |
F(x) = 0,6, |
1 < x :::; 2, |
0,9, |
2 < x:::; 3, |
1, |
3 < x. |
Hathu M(X), M(X2), D(X), a(X). |
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6.10.5. He3aBHcHMo ucnbiTbIBaIOTCH Ha Ha,n;e:lKHOCTb 3 npu6opa. BepoHT- |
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HOCTH BbIXO.IJ:a U3 CTPOH Ka:lK.IJ:OI'OnpH60pa 0.IJ:UHaKOBbi U paBHbi 0,6.
HaiiTH M(X) U a(X), r.IJ:e c. B. X - '1HCJIOBblille.IJ:illHX U3 CTPOH npu6opoB.
6.10.6. HaiiTU MaTeMaTH'IeCKOeO:lKH.IJ:aHHe CYMMbi '1UCJIaO'IKOB,KOTopble
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Bbma,n;aIOT npu 6pocaHuu .IJ:BYX UrpaJIbHbiX KocTeit. |
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6.10.7. |
CJIY'IaitHbieBeJIU'IHHbi X U |
Y He3aBUCHMbJ, npU'IeMD(X) = |
2 |
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U D(Y) = 6. HaiiTU D(Z), eCJIU Z = 12· X - 3Y + 2. |
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Q Ha OCHOBaHUH CBOitCTB .IJ:UCnepCHU (1-4) nOJIY'IaeM: |
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D(Z) = D(12X - 3Y + 2) = 144· D(X) + 9 . D(Y) = |
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= 144 . 2 + 9 . 6 = 342. |
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6.10.8. |
MaTeMaTH'IeCKOeO:lKH.IJ:aHHe H .IJ:HCnepCHH c. B. X COOTBeTCTBeHHO |
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paBHbi ~ H ~~. HaitTu MaTeMaTU'IeCKOeO:lKH.IJ:aHUe U .IJ:uCnepCuIO |
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CJIY'IaitHoitBeJIU'IHHbi4X - |
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Q COrJIacHo CBoitcTBaM MaTeMaTH'IeCKOrOO:lKU.IJ:aHUH U .IJ:Ucnepcuu, nOJIY- |
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'1aeM: |
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M(4X -1) = M(4X) +M(-l) = 4M(X) -1 = 4· ~ -1 = 13; |
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D(4X -1) = 16· D(X) = 16. ~~ = 1~0. |
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6.10.9. |
He3aBucuMbie CJIY'IaiiHbieBeJIU'IUHbiX U Y 3a,n;aHbi Ta6JIUD;aMU |
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pacnpe.IJ:eJIeHUHBepoHTHocTeit |
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U |
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Pi |
~P~'~'L-~~~~~~ |
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HaitTH D(X +Y) .IJ:BYMH cnoco6aMu: 1) COCTaBUB npe.IJ:BapHTeJIbHO |
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Ta6JIuD;y pacnpe.IJ:eJIeHUH c. B. Z = X + Y; 2) HCnOJIb3YH npaBUJIO |
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CJIO:lKeHUH .IJ:ucnepcuit. |
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6.10.10. |
HaitTu M(X) U a(X) .IJ:JIH CJIY'IaiiHOitBeJIH'IUHbI |
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i=l
r.IJ:e Xi (i = 1,2, ... , n) - He3aBHCUMbie .IJ:HCKpeTHbie CJIY'IaitHbJe BeJIU'IUHbI,UMeIOID;ue O.IJ:HO U TO :lKe MaTeMaTH'IecKoeO:lKU.IJ:aHUe a
U OMY H TY :lKe .IJ:UCnepCUIO a2 •
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6.10.11. BepoHTHoCTb nOHBJIeHHH C06bITHH A B KIDK,n;OM He3aBHCHMOM HCnbITaHHH o,n;HHaKOBa H paBHa p. HaitTH 9TY BepOHTHOCTb, eCJIH ,lI;J1H ,n;. C. B. X = {'1HCJIOnOHBJIeHHit C06bITHH A B 5 HcnbITaHHHx} ,n;HC-
nepCHH paBHa D(X) = 1,25.
6.10.12. BepoHTHoCTb TOro, 'ITO cTy,n;eHT C,n;acT 9K3aMeH Ha «5», paBHa
0,2, Ha «4» - 0,4. Onpe,n;eJIHTb BepOHTHOCTH nOJIY'leHHHHM on;e- HOK «3» H «2», eCJIH H3BeCTHO, 'ITOM(X) = 3,7, r,n;e,n;. c. B. X -
On;eHKa, nOJIY'leHHMcTy,n;eHTOM Ha 9K3aMeHe. 6.10.13. ,naH PH,n; pacnpe,n;eJIeHHH C.B. X:
HaitTH Ha'laJIbHbleH n;eHTpaJIbHble MOMeHTbI nepBblx '1eTblpexnoPH,n;KOB.
6.10.14. IIJIOTHOCTb BepoHTHoCTH c. B. X 3a,n;aeTCH <POPMYJIOit
f(x) = {236 (X - 3)2, |
X E [0,2], |
0, |
x 1. [0,2]. |
HaitTH ee '1HCJIOBblexapaKTepHcTHKH: MaTeMaTH'IeCKOeOJKH,n;aHHe, ,n;HcnepCHIO, cpe,n;HeKBa,n;paTH'IeCKOeOTKJIOHeHHe, aCHMMeTpHIO H 9KCn;ecc.
Q Hait.n;eM MaTeMaTH'IeCKOeOJKH,n;aHHe:
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M(X)= |
jx·f(x)dx= |
jx.Odx+jx· |
(x-3)2dx+ |
jx.Odx= |
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JL '" °692 |
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Tenepb OTbIIII;eM .n;HCnepCHIO: |
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D(X) = |
j (x - M(X))2 . f(x) dx = |
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- 00 |
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13 )2. 236 (x - |
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= j(x - |
3)2 dx = 236 j(x2 -1~x+ iD2 dx = |
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= ;6· j[(x - |
3)f dx = |
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13) (x - |
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o
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D(X) = ! |
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X . f(x) dx - (M(X))2 = !X |
. 26 (x - 3)2 dx - |
(13) |
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(JL)2 = 1- (X5 |
_ 6. X4 + 9. X3) 12 _R |
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= 1- f(x 4 _ 6x3 + 9x2) dx _ |
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169- |
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24 + 24) - 1869 = ~6~~ - 1i9 = :~ - 1869 = ~!~ ~ 0,259.) |
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= 26 (325 - |
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OTCIO.IJ:a Hait.IJ:eM Cpe.IJ:HeKBMpaTHqeCKOe OTKJIOHeHHe: |
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a(X) = JD(X) = JO,259 ~ 0,509. |
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HaKOHen;, BblqHCJIHM acHMMeTpHIO: |
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A = J1.3 |
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2 |
JL)3 .1- . (x - |
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1 |
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fI(x _ |
3)2 dx |
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(0,509)3 |
II |
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_ 105 X4 + 3870 x3 _ 60750 x2 + 32805 x _ 6561) dx = |
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'" _1_ . 1- fI(X5 |
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,... 0,132 |
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169 |
2197 |
2197 |
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2197 |
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_ 105x5 + 3870x4 _ 60750x3 + 32805x2 _ 6561X) 12 |
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= 1- . _1_ (X6 |
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26 |
0,132 |
6 |
13· 5 |
169·4 |
2197·3 |
2197· 2 |
2197 |
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~ 0,634. |
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H 3KCn;ecc: |
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E = ~: - 3 ~ 0,~67 |
2 |
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3)2 dx - 3 = |
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= 1- . _1_. fI(X6 _ 114 x5 + 4815 x4 _ 95580 X3+ |
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26 |
0,067 JI |
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169 |
2197 |
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380538 x + 59049) dx - |
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+973215 x 2 _ |
3'" -0545 |
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28561 |
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28561 |
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28561 |
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6.10.15. ,naHa nJIOTHOCTb pacnpe.IJ:eJIeHHg BepOgTHOcTeit CJIyqaitHoit BeJIHqHHbI X:
O, |
npH x < 0, |
f(x) = { ~. x, |
npH 0 ~ x < 4, |
0, |
npH 4 ~ x. |
HaitTH M(X), D(X) H a(X).
6.10.16. TIJIOTHOCTb pacnpe.IJ:eJIeHHg c. B. X
f(x) = {oX. e-AX , |
x ~ 0, |
(oX> 0), |
0, |
x < O. |
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HaitTH M(X), D(X), a(X). |
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364
6.10.17. CJIY'IaAHruIBeJIH'IHHaX "pHHHMaeT IIOJIo:lKHTeJIbHbIe 3Ha'leHHH, HMeeT IIJIOTHOCTb BepoHTHocTeA f(x) = -ax2 + 2ax. RaATH 3Ha- '1eHHenapaMeTpa a H MaTeMaTH'IecKoeo:lK:Jf,Il;aHHe C. B. X.
paBHa
1 |
, IIpH - 10 < x < 10, |
f(x) = { 11" V100 - |
x2 |
0, |
IIpH Ixl ~ 10. |
RaATH M(X), D(X), a(X).
6.10.19. llJIOTHOCTb pacIIpe,ll;eJIeHHH c. B. X 3a,ll;aHa B BH,Il;e
1 . |
IIpH 0 ~ X < 11", |
f(x) = { 2smx, |
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0, |
IIpH 0 < X H X ~ 11". |
HaATH M(X) H D(X).
6.10.20. HaATH MO,Il;y, Me,Il;HaHY, MaTeMaTH'IecKoeo:lKH,Il;aHHe H KBaHTHJIb
YPOBHH 0,75 CJIY'IaAHoABeJIH'IHHbIC IIJIOTHOCTblO BepOHTHOCTH
_4Z2 |
, IIpH X ~ |
0 |
f (x) = {8x . e |
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IIpH X < O. |
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a HaA,Il;eM TO'lKYMaKCHMYMa <PYHKII.UH f(x):
!,(x) = 8e-4z2 + 8x· e-4z2 (-8x) = 8e-4z2 (1 - 8x2 )j
OTCIO,Il;a f'(x) = 0 IIpH X = \" TO'lKax = |
\, HBJIHeTCH TO'lKOAMaKCH- |
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2v2 |
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2v2 |
MyMa <PYHKII,HH f(x) (TaK KaK, eCJIH x < |
1M |
, TO f'(x) > 0, a ecJIH x> 1M , |
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2v2 |
2v2 |
TO f'(x) < 0). CJIe.n;oBaTeJIbHO, MO,Il;a Mo(X) = 1M ~ 0,35. |
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2v 2 |
Me,Il;HaHa Me(X) = Xl OIIpe,Il;eJIHeTCH KaK 3Ha'leHHeCJIY'IaAHoABeJIH'IH- |
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HbI, KOTopoe ,Il;eJIHT IIJIOIlIa,ll;b <PHrYPbI, OrpaHH'IeHHOArpa<pHKOM <PYHKII,HH |
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f(x), Ha ,Il;Be paBHbIe '1acTH.l109TOMY |
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ZI |
+00 |
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!8x . e-4z2 dx = ~ (HJIH: |
! |
8x. e-4z2 dx = ~), |
o
ZI
ZI |
4z2 d(-4x2) |
1 T.e. |
_ e-4- |
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= _~, |
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1 |
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= 2' |
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o
", CJIe,Il;OBaTeJIbHO, e-4zi = ~. OTCIO,Il;a HaXO,Il;HM:
Xl = MeX = ~v'ln2 ~ 0,42.
365
Haxo,n:HM MaTeMaTH'IeCKOeO>ICH,n:aHHe C. B. X:
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M(X)= IX'f(x)dx= |
Ix.Odx+ |
Ix.8x2 .e-4X2 dx= |
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M |
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Jx . x2 . e-4x2 dx = |
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[no '1acTlIM:U = x2,du = 2xdx, dv = X· e-4x2 dx, v = _le-4X2 ] |
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= 8 lim |
(_1. x2 . _I_1M + ~ I:' e-4x2 dX) |
= |
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M --++00 |
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e4x2 0 |
8 |
° |
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_ ~ 1~-4X2d(_4X2)) |
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= lim |
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e4M |
8 o |
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= _1 |
lim (e- 4M2 - |
1) = -1(0 - |
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M--++oo |
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Hait,n:eM |
<PYHKII,HIO |
pacnpe,n:eJIeHHlI c. B. X. IIpe,n:BapHTeJIbHo 3aMeTHM, |
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'ITOeCJIH x < 0, TO |
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F(x) = 1 |
f(t)dt = 1 Odt = 0. |
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-00 |
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ECJIH :lKe x ~ 0, TO |
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F(x) = Jf(t) dt = J°dx + J8t· e-4t2 dt = _e-4t2 1: = _e- 4x2 + 1, |
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-00 |
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T.e. F(x) = l_e- 4x2 , x ~ 0.
KBaHTHJIb XO,75 HaxO,n:HM H3 paBeHCTBa F(XO,75) = 0,75. IIMeeM:
1 - |
e-4X~.75 = 0,75. |
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OTcIO,n:a Haxo,n:HM, 'ITOXO,75 = |
~v'ln4 ~ 0,59. |
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KPHBM pacnpe,n:eJIeHHlI (c. B. X pacnpe,n:eJIeHa no 3aKoHY PeJIell) npe,n:- |
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CTaBJIeHa Ha pHC. 88. |
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6.10.21. CJIY'IaitHMBeJIH'IHHaHMeeT nJIOTHOCTb pacnpe,n:eJIeHHlI BH,n:a |
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f(x) = |
*. x 2: a2' |
a > ° |
(pacnpe,n:eJIeHa no 3aKOHY KOlIIH). HaitTH MO.n:y, Me,n:HaHY H KBaH- |
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THJIH nOpll,n:Ka p = 0,25; 0,5; 0,75. |
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6.10.22. CJIY'IaitHMBeJIH'IHHaX 3MaHa <PYHKII,Heit pacnpe,n:eJIeHHlI |
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O, |
x ~ 2, |
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F(x) = { 19' (x 3 - 8), |
2 < x ~ 3, |
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1, |
x> 3. |
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HaitTH: M(X), Mo(X), Me(X).
366
f(x)
2 x
Puc. 88
6.10.23. IIJIOTHOCTb pacIIpe.n;eJIeHlUl c. B. X HMeeT BH.n;
f(x) = {lx2 , |
XE[0;2], |
0, |
x f/. [0;2]. |
HaitTH MO.u.y, Me.n;HaHY, MaTeMaTH'leCKOeO)lm.n;aHHe H KBaHTHJIb IIOpjf.n;Ka 0,25.
AononHIIITenbHble 3aACIHIIIH
6.10.24. HaitTH 3aKOH pacIIpe.n;eJIeHHjf .n;HcKpeTHoit c. B. X, 3HM, 'ITO:OHa "pHHHMaeT .n;Ba 3Ha'leHHjfXl H X2 (Xl < X2) H, KpOMe Toro, P(xd =
=0,4; M(X) = 2,6; D(X) = 8,64.
6.10.25.lICIIOJIb3Yjf YCJIOBHe 3a.n;a'lH6.8.3, HaitTH MaTeMaTH'IeCKOeO)l{H.n;a- HHe H .n;HCIIepCHIO c. B. Z.
6.10.26.lICIIOJIb3Yjf YCJIOBHe 3a.n;a'lH6.8.5, HaitTH cpe.n;Hee 3Ha'leHHe'1HCJIa oIIycKaHHit MOHeT B aBTOMaT.
6.10.27.lICIIOJIb3Yjf YCJIOBHe 3a.n;a'lH 6.8.9, HaitTH M(X), D(X) H a(X), r.n;e X - '1HCJIOIIepBOpa3pjf.n;HHKOB cpe.n;H .n;BYX BbI6paHHblx Hayra.n; CiiopTcMeHoB.
6.10.28.lICIIOJIb3Yjf YCJIOBHe 3a.n;a'lH6.8.10, HaitTH:
a)M(X) H M(X2) .n;JIjf <PYHKII,HH H3 YCJIOBHjf 3a.n;a'lH6.8.10 a;
6)D(X) H a(X) .n;JIjf <PYHKII,HH H3 YCJIOBHjf 3a.n;a'lH6.8.10 6.
6.10.29.lICIIOJIb3Yjf YCJIOBHe 3a.n;a'lH6.8.11, HaitTH M(X), M(X2), D(X),
D(X2).
6.10.30. lICIIOJIb3Yjf YCJIOBHe 3a.n;a'lH6.8.16, HaitTH M(X) H M(Y).
6.10.31. lICIIOJIb3Yjf YCJIOBHe 3a.n;a'lH6.8.17, HaitTH M(Z), M(W), D(X),
D(W).
367
6.10.32. IIPOH3BO,II.HTClI ,n:Ba He3aBHCHMbIX BbICTpeJIa. BepOllTHOCTb rro-
rra,n:aHHlI rrpH K~,n:OM BbICTpeJIe rrOCTOllHHa. RaihH ,n:HcrrepCHIO
c. B. X - '-IHCJIarrorra,n:aHHA, eCJIH M(X) = 1,6.
6.10.33."tJeJIOBeK HaxO,n:HTClI B Ha'laJIerrpllMoyroJIbHOA CHCTeMbI KOOp,n:H- HaT. OH rro,n:6pacbIBaeT MOHeTY. IIPH rrOllBJIeHHH rep6a ,n:eJIaeT mar
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HarrpaBO, rrpH rrOllBJIeHHH pemKH - mar HaJIeBO (.D:JIHHa mara paB- |
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Ha o,n:HoA e,n:HHHII,e MacmTa6a). IIYCTb X - a6CII,HCCa rrOJIO)KeHHH |
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'1eJIOBeKarrOCJIe Tpex 6pocaHHA. RaATH M(X) H D(X). |
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6.10.34. |
MCrrOJIb3YlI YCJIOBHe 3a,n:a'lH6.10.13, HaATH acHMMeTpHIO H 9KCII,eCC |
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C.B. X. |
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6.10.35. |
CJIY'IaAHMBeJIH'IHHaX 3a,n:aHa rrJIOTHOCTbIO pacrrpe,n:eJIeHHlI |
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f(x) = { |
10 - 2x |
rrpH 2 ~ x ~ 5, |
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9' |
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0, |
B OCTaJIbHbIX CJIY'IMX. |
BbI'IHCJIHTb:M(X), Ha'laJIbHbIeMOMeHTbI BToporo H TpeTbero rro-
pll,n:Ka, D(X).
6.10.36. IIJIOTHOCTb pacrrpe,n:eJIeHHlI BepOllTHOCTeA HerrpepbIBHoA cJIY'IaAHoA BeJIH'IHHbIX 3a,n:aHa <P0PMYJIoA
_1_ |
x E [-1'2] |
f(x) = { l'+ l' |
: ' |
0, |
x 1- [-1,2]. |
RaATH: rrapaMeTp 1', M(X) H D(X).
6.10.37. CJIY'IaAHMBeJIH'IHHaX 3a,n:aHa rrJIOTHOCTbIO pacrrpe,n:eJIeHHlI
f(x) = {* .sin2 x, |
x E [0; 7rJ, |
0, |
x 1- [0; 7r]. |
RaATH M(X) H D(X).
6.10.38. MCrrOJIb3YlI YCJIOBHe 3a,n:a'lH6.9.6, HaATH M(X), D(X) H a(X). 6.10.39. MCrrOJIb3YlI YCJIOBHe 3a,n:a'lH6.9.11, HaATH:
a) M(X); |
6) D(X); |
8)a(X).
6.10.40.MCrrOJIb3YlI YCJIOBHe 3a,n:a'lH6.9.14, HaATH:
a) M(X); |
6) D(X); |
B)MO.n:y C. B. X.
6.10.41.MCrrOJIb3YlI YCJIOBHe 3a,n:a'lH6.9.20, HaATH:
a) M(X); |
6) D(X); |
B)a(X).
6.10.42.MCrrOJIb3YlI YCJIOBHe 3a,n:a'lH6.9.27, HaATH:
a) M(X); |
6) Mo(X); |
B)Me(X);
r)XO,5 (KBaHTHJIb rrOpll,n:Ka p = 0,5).
6.10.43.MCrrOJIb3YlI YCJIOBHe 3a,n:a'lH6.9.30, HaATH MaTeMaTH'IecKoeO)KH- ,n:aHHe H MO.n:y C. B. X.
368
6.10.44. CJIY'iaitHaJIBeJIH'iHHaX 3a,rr.aHa nJIOTHOCTblO pacnpe,n:eJIeHHjJ
f(x) = {0,1, npH x E [OJ 10], 0, npH x ¢ [OJ 10].
RaitTH: M(X), D(X), Me(X), KBaHTHJIb nopjJ,n:Ka p = 0,25, p =
= 0,50 H P = 0,75.
KOHTponbHble Bonpocbl lit 60nee CnO)l(Hbie 3aACIHlltH
6.10.45. IIcnoJIb3YjJ YCJIOBHe 3a,rr.a'iH6.8.22, HaitTH cpe,n:Hee 'iHCJIOnonbITOK OTKpbITb ,n:Bepb.
6.10.46. IIcnoJIb3YjJ YCJIOBHe 3a,rr.a'IH6.8.24, HaitTH MaTeMaTH'IeCKOeO:>KH- ,n:aHHe, ,n:HcnepCHIO, MO.n:y H K09<P<PHn;HeHT aCHMMeTpHH CJIY'IaitHoit BeJIH'IHHbIX - 'IHCJIa,n:e<peKTHblx H3,n:eJIHit B BbI6opKe.
6.10.47. IIcnoJIb3YjJ YCJIOBHe 3a,rr.a'IH6.8.29, HaitTH MaTeMaTH'IeCKOeO:>KH,n:a- HHe, ,n:HcnepCHIO, n;eHTpaJIbHblit MOMeHT 'IeTBepToronopjJ,n:Ka, Ko- 9<P<PHn;HeHT 9KCn;ecca CJIY'IaitHoitBeJIH'IHHbIZ.
6.10.48. IIcnoJIb3YjJ YCJIOBHe 3a,rr.a'IH6.8.32, HaitTH MaTeMaTH'IecKoeO:>KH- ,n:aHHe 'IHCJIanpOBe,n:eHHbIX HcnbITaHHit.
npHHHMaeT 3Ha'IeHHem C BepojJTHoCTblO
Pn(m) = C::" . pm . qn-m (m = 0,1,2, ... , nj p + q = 1). RaitTH
M(X) H D(X).
6.10.50.X H Y - He3aBHCHMble CJIY'IaitHbleBeJIH'IHHbI. ,n:OKa3aTb, 'ITO
D(XY) = D(X) . D(Y) + (M(y))2 . D(X) + (M(X))2 . D(Y).
6.10.51. ,n:OKa3aTb, 'iTOm ~ M(X) ~ M, r,n:e X - .IJ:HCKpeTHaJI CJIY'IaitHaJI BeJIH'IHHa,m H M - COOTBeTCTBeHHO HaHMeHbwee H HaH60JIbWee 3Ha'IeHHjJC. B. X.
6.10.52. IIoKa3aTb, 'ITO,n:HcnepCHjJ 'IHCJIaycnexoB npH o,n:HOKpaTHOM npo-
Be,n:eHHH HcnbITaHHjJ He npeBocxo,n:HT 0,25.
6.10.53. Bblpa3HTb n;eHTpaJIbHble MOMeHTbI BToporo, TpeTbero H 'IeTBepToro
nopjJ,n:KoB 'Iepe3Ha'IaJIbHbleMOMeHTbI.
6.10.54. RenpepbIBHajJ c. B. X HMeeT nJIOTHOCTb pacnpe,n:eJIeHHjJ BepojJTHo-
cTeit BH,n:a
1 |
_Ix - |
al |
f(x) = 20" e |
U |
, r,n:e 0" > 0, a E lR. |
RaitTH M(X) H D(X).
6.10.55. IIJIOTHOCTb pacnpe,n:eJIeHHjJ BepojJTHocTeit c. B. X 3a,rr.aHa B BH,n:e
f(X)={A'(I-X2)~, |
Ixl<l, |
0, |
Ixl ~ 1. |
RaitTH A, M(X) H D(X).
6.10.56. IIcnoJIb3YjJ yCJIOBHe 3a,rr.a'iH6.9.28, HaitTH MaTeMaTH'IeCKOeO:>KH- ,n:aHHe, Ha'IaJIbHbleH n;eHTpaJIbHble MOMeHTbI nepBoro H BToporo nopjJ,n:KOB c. B. X.
369