Сборник задач по высшей математике 2 том
.pdf6.10.57. 3MaHa IIJIOTHOCTb paCIIpe.IJ:eJIemUI c. B. X:
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HaATH A, M(X) H D(X).
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0,1,2,.... , n C COOTBeTCTBYIOIIJ;lfMlf BepOHTHOCTHMlf:
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npe,n;eJIeHlfe, HaxO,n;HTCH no q,0pMYJIaM:
M(X) = np, D(X) = npq.
1'13q,OPMYJIbI BepHYJIJIlf CJIe.n;yeT, 'ITOC. B. X - '1lfCJIOnOHBJIeHlfi!: C06bITlfH A
B ceplflf lf3 n He3aBlfClfMbIX lfcnbITaHlfi!: (P(A) = p) - pacnpe,n;eJIeHa no 6lfHOMlfaJIbHOMY 3aKOHY.
~ 2. )J,lfcKpeTHaH CJIY'Iai!:HaHBeJIlf'llfHalfMeeT pacnpeiJe.l!e'Hue IIyacco'Ha (lfJIlf pacnpe,n;eJIeHa no 3aKOHY IIyaccoHa), eCJIlf OHa nplfHlfMaeT C'IeTHOe '1lfCJIO 3Ha- '1eHlfi!::0,1,2, ... , m ... , C cooTBeTcTBYIOIIJ;lfMlf BepOHTHOCTHMlf
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IIyaccoHa, HaxO,n;HTCH no q,0pMYJIaM: |
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M(X) = a, |
D(X) = a. |
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Pacnpe,n;eJIeHlfe IIyaccoHa HBJIHeTCH npe,n;eJIbHbIM MH 6lfHOMlfaJIbHOI'O, eCJIlf '1lfCJIOonbITOB n YCTpeMJIHeTCH K 6eCKOHe'lHOCTlf,a BepOHTHOCTb C06bITlfH p CTpeMlfTCH K HYJIIO, nplf'leMlfX npolf3Be,n;eHlfe np = a OCTaeTCH nOCTOHHHbIM. IIplf 3TlfX
YCJIOBlfHX (T.e. n-+oo, p-+O, np=a=const) BepOHTHOCTb P{X=m}=C;::pmqn-m,
= 1 - p, Haxo,n;lfMaH no q,0pMYJIe BepHYJIJIlf, CTpeMlfTCH K BepoHTHoCTlf
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~ 3. )l;HcKpeTHaH eJIY'lail:HaHBeJlH'IHHaHMeeT
eCJIH OHa npHHHMaeT C'IeTHOe'1HCJIO3Ha'leHHiI::1,2, ... , m, ... C cooTBeTcTBYJOIII;HMH BepoHTHOCTHMH:
pm = PiX = m} = qm-l . p, r,ll;e m = 1,2, ... , °< p < 1, q = 1- p. ~
)l;.nH c. B. X, HMeJOIII;eil: reOMeTpH'IeCKOepacnpe,ll;eJleHHe
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x f. [a, b]. |
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X ~ R[a,b].
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~(x) = 0,5 + ~o(x).
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c, a + c), CHMMe- |
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P{ a - |
c < X < a + c} = P{IX - al < c} = 2~o (~) = 2~ (~) - 1. |
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B qacTHOCTH, |
P{IX - al < 3u} |
~ 0,9973, T. e. npaxTHqeCKH |
,ll;OCTOBepHO, qTO |
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c. B. X ~ N(a,u) npHHHMaeT CBOH 3HaqeHHH B npOMe)KYTKe (a - |
30', a + 30'). 3To |
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<) B ,Il;aHHOM cJIyqae MbI HMeeM ,Il;eJIO co cxeMoil: HcnbITaHHiI: BepHYJIJIH, no9TOMY c. B. X HMeeT 6HHOMHaJIbHOe pacnpe,Il;eJIeHHe. IIcnoJIb3YH <P0PMYJIbI
M(X) = np H D(X) = npq, HaXO,Il;HM (npH n = 150, p = 0,2, q = 0,8) |
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M(X) = 150·0,2 = 30, D(X) = 150·0,2·0,8 = 24. |
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6.11.2. Hail:TH Cpe,Il;Hee qHCJIO JIOTepeil:Hblx 6HJIeTOB, Ha KOTopble Bbma- )1;JT BbIHrpbIlliH, eCJIH npH06peTeHo 20 6HJIeTOB, a BepOHTHOCTb BblHrpbIllia O,Il;HOrO 6HJIeTa paBHa 0,1. Hail:TH ,Il;HCnepCHIO qHCJIa
ycnexoB B ,Il;aHHOM onbITe.
6.11.3. llpoBo)1;HTCH 3 He3aBHCHMbIX HCnbITaHHH, B K~,Il;OM H3 KOTOPbIX BepOHTHOCTb HacTynJIeHHH HeKOToporo C06bITHH nOCTOHHHa H paB-
Ha p. qHCJIO nOHBJIeHHiI: C06bITHH A B 9TOM onbITe. Hail:TH D(X), eCJIH H3BeCTHO, qTO M(X) = 2,1.
6.11.4. BepoHTHoCTb nop~eHHH u;eJIH npH O,Il;HOM BbICTpeJIe paBHa 0,4.
CKOJIbKO Ha)1;O npOH3BeCTH BbICTpeJIOB, qT06bI MO)KHO 6bIJIO O)KH-
,Il;aTb B Cpe,Il;HeM 80 nOna)1;aHHiI: B u;eJIb?
6.11.5. llpoBepHeTcH napTHH H3 10000 H3,Il;eJIHiI:. BepoHTHoCTb TOro, qTO H3,Il;eJIHe OK~eTCH 6paKOBaHHbIM, paBHa 0,002. Hail:TH MaTeMaTHqeCKOe O)KH,Il;aHHe H ,Il;HCnepCHIO qHCJIa 6paKoBaHHblx H3,Il;eJIHiI: B 9TOil: napTHH. Hail:TH BepoHTHoCTb Toro, qTO B napTHH eCTb XOTH
6bI O,Il;HO 6PaKOBaHHOe H3,Il;eJIHe.
<) l..JHCJIO onbITOB |
(n = |
10000) ,Il;OCTaTOqHO BeJIHKO, a BepoHTHoCTb (p = |
= 0,002) «ycnexa» |
B K~,Il;OM H3 HHX MaJIa, n09ToMY MO)KHO CqHTaTb, qTO |
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CJIyqail:HM BeJIHqHHa X - |
qHCJIO 6paKOBaHHblx H3,Il;eJIHiI: - pacnpe,Il;eJIeHa |
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no 3aKOHY llyaccoHa: ee B03MO)KHble 3HaqeHHH 0,1,2, ... , 10000, a COOTBeTCTBYlOllJHe BepoHTHoCTH BblqHCJIHIOTCH no <popMYJIe
ame- a
Pm = P{X = m} = --,-.
m.
llo <popMYJIe a = np onpe,Il;eJIHeM napaMeTp a H MaTeMaTHqeCKOe O)KH-
,Il;aHHe c. B. X:
M(X) = a = 10000·0,002 = 20.
,1l;HcnepcHH qHCJIa 6paKoBaHHblx H3,Il;eJIHiI: paBHa
D(X) = npq = 10000·0,002·0,998 = 19,96,
T.e. M(X) ~ D(X). lloJIaI'MM(X) = D(X) = 20, HaxO,Il;HM npH6JIH)KeHHO HCKOMYIO BepoHTHoCTb C06bITHH A = {B napTHH CO,Il;ep)KHTCH XOTH 6bI O,Il;HO 6paKoBaHHoe H3,Il;eJIHe}:
P(A) = 1 - |
- |
PlOOOO(O) = 1- |
200 . |
e-20 |
= |
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P(A) = 1 - |
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= 1 - |
e- 20 |
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= 1 - 2,06 . 10-9 ~ 1. |
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374
6.11.6.,I:LHcKpeTHM CJIY'IafiHMBeJIH'IHHaX pacnpe.n;eJIeHa no 3aKoHY IIyaccoHa C napaMeTpoM a = 0,324. HafiTH MaTeMaTH'IeCKOeO)KH-
.n;aHHe H cpe.n;HeKBa.n;paTH'IeCKOeOTKJIOHeHHe 9TOfi CJIY'IafiHOfiBeJIH'IHHbI.
6.11.7.B Mara3HH OTnpaBJIeHbI 1000 6YTbIJIOK MHHepaJIbHOfi BO.n;bI. Bepo-
.HTHOCTb Toro, 'ITOnpH nepeB03Ke 6YTbIJIKa OKa)KeTC.H pa36HTOfi, paBHa 0,002. HafiTH:
a) cpe.n;Hee 'IHCJIOPa36HTbIX 6YTbIJIOK;
6) BepO.HTHOCTb Toro, 'ITOMara3HH nOJIy'IHT60JIee .n;ByX pa36HTbIX
6YTbIJIOK.
6.11.8.Co06meHHe co.n;ep)KHT 1000 CHMBOJIOB. BepO.HTHOCTb HCKa)KeHH.H o.n;Horo CHMBOJIa paBHa 0,004. HafiTH cpe.n;Hee 'IHCJIOHCKa)KeHHbIX CHMBOJIOB; HafiTH BepO.HTHOCTb Toro, 'ITO6y.n;eT HCKa)KeHO He 60JIee 3-x CHMBOJIOB.
6.11.9.IIpoH3Bo.n;HTC.H CTpeJIb6a no u;eJIH .n;o nepBoro nona.n;aHH.H. BepO.HT-
HOCTb nona.n;aHH.H npH Ka)K.n;OM BbICTpeJIe paBHa 0,2. HafiTH MaTeMaTH'IeCKOeO)KH.n;aHHe H .n;HcnepcHIO CJIY'IafiHOfiBeJIH'IHHbIX -
'IHCJIanpoH3Be.n;eHHblx BbICTpeJIOB, C'IHTM,'ITO:
a) CTpeJI.HTb MO)KHO HeOrpaHH'IeHHOe'IHCJIOpa3;
B) B HaJIH'IHHeCTb Bcero 5 naTpOHOB.
Q a) CJIY'IafiHMBeJIH'IHHaX HMeeT reOMeTpH'IeCKOepacnpe.n;eJIeHHe, ee p.H.n; pacnpe.n;eJIeHH.H HMeeT BH.n;
Xi |
1 |
2 |
3 |
... |
Pi |
P |
qp |
q2p ... |
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l.fI1CJIOBble xapaKTepHCTI1KH 9Toro pacnpe.n;eJIeHH.H: M(X) = ~, D(X) = q2'
P
CJIe.n;OBaTeJIbHO, 3Ha.H, 'ITOP = 0,2 H q = 0,8, HMeeM:
M(X) = 012 = 5; |
D(X) = |
~084 = 20. |
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6) P.H.n; pacnpe.n;eJIeHH.H c. B. X HMeeT BH.n; |
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Xi |
1 |
2 |
3 |
4 |
5 |
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Pi |
P |
qp |
q2p |
q3 p q4 |
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n09TOMY, M(X) = 1 . P + |
2 . qp + |
3 . q2p + |
4 . q3p + |
5q4; npH p = 0,2 H |
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q = 0,8 HMeeM M(X) = 0,2 + |
0,32 + 0,384 + |
0,4096 + |
2,048 = 3,3616, T. e. |
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M(X) = 3,3616; ,I:LaJIee |
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D(X) = M(X)2 - (M(X))2 = l·p+4·qp+9.q2p + 16.q3p +25q4- (M(X))2;
npl1 p = 0,2 H q = 0,8 nOJIy'IHM
D(X) = 0,2+0,64+1,152+1,6384+10,24-(3,3616)2 ~ 13,8704-11,3 = 2,57,
T. e. D(X) = 2,57. •
375
6.11.10. IIrpoK nOKynaer JIOTepeitHbIe 6HJIeTbI ,n:o nepBoro BbIHrpbIma. Be-
pmlTHOCTb BbIHrpbIma no O,n:HOMY 6HJIeTY paBHa 0,1. HaitTH M(X),
r,n:e c. B. X - qHCJIO KynJIeHHbIX 6HJIeTOB, eCJIH HrpOK MO:>KeT Ky-
nHTb:
a)TOJIbKO qeTbIpe 6HJIeTa;
6)HeOrpaHHqeHHOe (nycTb TeOpeTHqeCKH) qHCJIO 6HJIeToB.
6.11.11.BepoHTHocTb npoH3Bo,n:cTBa HecTaH,n:apTHoit ,n:eTaJIH paBHa 0,05.
KOHTpOJIep npOBepHeT napTHIO ,n:eTaJIeit, 6epH no O,n:HOit ,n:o nepB0ro nOHBJIeHHH HecTaH,n:apTHoit ,n:eTaJIH, HO He 60JIee 3 mTYK. HaitTH MaTeMaTHqeCKOe O:>KH,n:aHHe H ,n:HcnepCHIO qHCJIa npOBepeHHbIX CTaH,n:apTHbIX ,n:eTaJIeit.
6.11.12.IIrpaJIbHM KOCTb no,n:6pacbIBaeTcH ,n:o nepBoro nOHBJIeHHH nHTH OqKOB. KaKOBa BepOHTHOCTb Toro, qTO nepBoe BblDa,n:eHHe nHTepKH npoH30it,n:eT npH nHTOM no,n:6pacbIBaHHH HrpaJIbHoit KOCTH?
6.11.13.CJIyqaitHM BeJIHqHHa X pacnpe,n:eJIeHa paBHoMepHo Ha oTpe3Ke
[a, b), T. e. X "" R[a, b]. HaitTH BepoHTHoCTb nona,n:aHHH c. B. X Ha OTpe30K [a,.8), II;eJIHKOM co,n:ep:>KamHitCH BHyTpH oTpe3Ka [a, b].
Q BocnoJIb3yeMcH H3BecTHoit q,0PMYJIOit
{3
Pia ~ X ~.8} = Jf(x)dx,
a
r,n:e nJIOTHOCTb BepOHTHOCTH c. B. x"" R[a, b] HMeeT BH,n:
-Ib , X E [a,b), f(x) = { - a
0, xi [a,b].
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Pia ~ X ~.8} = PiX E [a,.8]} = J-b- dx = -b- ·x |
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T.e. OKOHqaTeJIbHO PiX E [a,.8]} |
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6.11.14. TIJIOTHOCTb BepoHTHoCTH HenpepbIBHoit CJIyqaitHoit BeJIHqHHbI X
HMeeT BH,n:
f(x) = {0,25 . A, |
x E [0,4], |
0, |
xi [0,4]. |
HaitTH A, F(x), M(X), D(X), u(X), PiX E [0; I,l]}.
Q K09q,q,HII;HeHT A
BepOHTHOCTH
J00 f(x)dx = 1.
- 00
376
Ih1eeM |
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j I(x)dx= |
jOdx+ jO,25Adx+ jOdx=l, |
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T. e. 0,25Axl~ = 1. OTCIO,ll;a CJIe.rr.yeT, 'ITOA = 1. IlTaK,
I(X)={~' XE[0,4),
0, xi [0,4),
II, 3Ha'IHT,CJIY'IaitHruIBeJIH'IHHaX paBHOMepHO pacnpe,ll;eJIeHa Ha OTpe3Ke
[0,4),
<I>YHKIJ;H8 pacnpe,ll;eJIeHH8 ,ll;JI8 c. B. |
X""" R[0,4) HMeeT BH,ll; |
O, |
x ~ 0, |
F(x) = { i, |
< x ~ 4, |
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1, |
4 < x. |
qHCJIOBbIe xapaKTepHCTHKH 9Toro pacnpe,ll;eJIeHH8 TaKOBbI:
M(X) = [a~b] = 0~4 =2,
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D(X) = [(b-a)2] |
_42 -1 |
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f'TV"V\] |
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a(X) = [V D(X) |
va |
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Bepo8THoCTb nOna,IJ;aHH8 c. B. X |
B |
npOMe:>KYTOK [OJ 1,1) HaxO,ll;HM, HC- |
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IIOJIb3Y8 <P0PMYJIY P{X E |
[0'., (3l) = |
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(CM. 3a,IJ;a'IY 6.11.13). OTCIO,ll;a |
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P{X E [OJ I,ll} = 141_-0° = |
141 = 0,275. |
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6.11.15. |
HeKTo O:>KH,ll;aeT TeJIe<poHHbIit 3BOHOK Me:>K.rr.y 19.00 H 20.00. BpeM8 |
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O:>KH,ll;aHH8 3BOHKa eCTb HenpepbIBHruI c. B. X, HMelOIIJ;ruI paBHOMep- |
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Hoe pacnpe,ll;eJIeHHe Ha OTpe3Ke [19,20). HaitTH Bep08THOCTb Toro, |
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'ITO3BOHOK nocTynHT B npOMe:>KYTKe OT 19 'lac22 MHHYT,ll;O 19 'lac |
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46 MHHYT. |
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6.11.16. |
CJIY'IaitHruI BeJIH'IHHa X, |
pacnpe,ll;eJIeHHruI paBHoMepHo, HMeeT |
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CJIe.rr.yIOIIJ;He 'IHCJIOBblexapaKTepHcTHKH |
M(X) = 2, D(X) = 3. |
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HaitTH F(x). |
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6.11.17. |
IIpo C.B. X H3BeCTHO, 'ITOX """ |
R[4, 7). HaitTH: |
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B) P{X E (6j6,81)}.
6.11.18.«PYHKIJ;H8 pacnpe,ll;eJIeHH8 HenpepbIBHoit CJIY'IaitHoitBeJIH'IHHbIX
HMeeT BH,ll;, YKa3aHHbIit Ha pHC. 90. HaitTH aHaJIHTH'IeCKHeBbIpa-
:>KeHH8,ll;JI8 F(x), I(x), M(X) H D(X).
6.11.19.HenpepblBHruI CJIY'IaitHruIBeJIH'IHHaX HMeeT nOKa3aTeJIbHOe pacnpe,ll;eJIeHHe. HaitTH Bepo8THoCTb nona,IJ;aHH8 c. B. X B HHTepBaJI
(a, b), r,ll;e a ~ 0.
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F(x) |
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1 -------------------------- |
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Puc. |
90 |
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a BOCrrOJIb30BaBIIIHcb <POPMYJIOit |
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P{a ~ X ~,B} = |
j I(x)dx, |
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0< |
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P{a < X < b} = j-Xe- Ax dx = -e-AX!a = _e- Ab + e- Aa , |
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T.e. P{X E (a,b)} = e- Aa - e- Ab . |
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3TOT JKe pe3YJIbTaT MOJKHO rrOJIY'IHTb,HCrrOJIb3YH <POPMYJIY |
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P{a < X < b} = F(b) - |
F(a). |
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TaK KaK F(x) = 1 - |
e- AX , X ;:: 0, TO |
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P{a < X < b} = (1- e- Ab ) - |
(1- e- Aa ) = e- Aa _ e- Ab . |
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6.11.20. |
BpeMH T BbIxo,n:a H3 CTPOH pa,n:HOCTaHII;HH rro,n:'IHHeHOrrOKa3aTeJIb- |
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HOMY 3aKOHY pacrrpe,n:eJIeHHH C rrJIOTHOCTbIO |
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FT(t) = {0,2 . e-O,2t, |
rrpH |
t;:: 0, |
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t < O. |
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HaitTH: <PYHKII;HIO pacrrpe,n:eJIeHHH FT(t); MaTeMaTH'IeCKOeOJKH,n:a- |
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HHe H ,n:HCrrepCHIO CJIY'IaitHoitBeJIH'IHHbIT; BepOHTHOCTb TOrO, 'ITO |
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pa,IJ,HOCTaHII;HH COXpaHHT pa60TOCrroco6HOCTb OT 1 ,n:o 5 'lac.pa6o- |
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6.11.21. |
C. B. X pacrrpe,n:eJIeHa rro rroKa3aTeJIbHoMY 3aKoHY C rrapaMeTpoM |
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HaitTH ,n:H<p<pepeHII;HaJIbHYIO H HHTerpaJIbHYIO <PYHKII;HH |
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pacrrpe,n:eJIeHHH (T. e. I(x) H |
F(x)), |
a(X), a TaKJKe BepOHTHOCTb |
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rrOrra,IJ,aHHH 3Ha'IeHHitc. B. X B HHTepBaJI (0,25; 5). |
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6.11.22. |
C.B. X, KOTOPM paBHa ,n:JIHTeJIbHOCTH pa60TbI 9JIeMeHTa, HMeeT |
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rrJIOTHOCTb pacrrpe,n:eJIeHHH I(t) |
= O,003e-O,003t, t |
;:: 0. |
HaitTH: |
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cpe,n:Hee BpeMH pa60TbI 9JIeMeHTa; BepoHTHoCTb Toro, 'ITO9JIeMeHT |
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rrpopa6oTaeT He MeHee 400 'IacOB. |
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378
6.11.23. Cpe,lJ;mul npO,lJ;OJDKHTeJIbHOCTb TeJIe<poHHoro pa3rOBOpa paBHa
3 MHH. HalI:TH BepOHTHOCTb Toro, qTO npOH3BOJIbHbIit TeJIe<poH-
Hblit pa3rOBOp 6Y,lJ;eT npO,lJ;OJDKaTbCH He 60JIee 9 MHHyT, CqHTM, qTO BpeMH pa3rOBOpa HBJIHeTCH c. B. X, pacnpe,lJ;eJIeHHoit no nOKa3aTeJIbHOMY 3aKoHY.
6.11.24. Onpe,lJ;eJIHTb 3aKOH pacnpe,lJ;eJIeHHH CJIyqaitHoit BeJIHqHHbI X, eCJIH
ee nJIOTHOCTb BepoHTHoCTH HMeeT BH,lJ;
I(x) = A . e- x2 + 2x+1.
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HaitTH: |
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6) u(X); |
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a) M(X); |
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B) 3HaqeHHe K09<P<PHII;HeHTa A; |
r) M(X2); |
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.u.) P{1 < X < 3}. |
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(x _ 1)2 |
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CPaBHHB ,lJ;aHHYlO <PYHKII;HlO |
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2 |
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I(x) = A· e- x2 + 2x+1 = A· e-(x-1)2+2 = A· e 2 . e-(x-1)2 = A . e2 . e |
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C nJIOTHOCTblO |
1 |
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20'2 |
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u·.j2;
HOpMaJIbHOrO pacnpe,lJ;eJIeHHH, 3aKJIlOqaeM, |
qTO c. B. X HMeeT HOpMaJIbHOe |
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pacnpe,lJ;eJIeHHe. |
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a) OqeBH,lJ;HO, M(X) = 1. |
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1 |
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6) u(X) = y'2' |
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B) 3HaqeHHe K09<P<PHII;HeHTa A Hait,lJ;eM H3 paBeHCTBa A . e2 = |
~, |
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r,lJ;e u = ~ ~ 0,71. OTClO,lJ;a |
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u· |
211' |
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A = |
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e2 ._1 . y'2. y'7i e2 ."y'7i· |
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V2 |
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CJIe,lJ;OBaTeJIbHO, nJIOTHOCTb BepoHTHoCTH c. B. X HMeeT BH,lJ; |
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(x - 1)2 |
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I(x) = |
1 |
.e2.e-2·(~r |
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e2 |
. y'7i |
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T.e. |
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(x - 1)2 |
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I(x) = |
1 |
.e |
2·(~r |
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_1 . .j2; |
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V2
aCHO, qTO X'" N (1; ~).
r) M(X2) Hait,lJ;eM, HCnOJIb3YH <POPMYJIY D(X) ,= M(X2) - (M(X»2. B
Ra1IIeM cJIyqae M(X) = 1, D(X) = (U(X»2 = ~. II09ToMY
M(X2) = D(X) + (M(X»2 = ~ + 1 = ~.
379