Сборник задач по высшей математике 2 том
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~)l;HcKpeTHhIe C. B. X H Y Ha3hIBaIOTCH HeSaSUC'U.M'btMU, eCJIH He3aBHCHMbi CO-
6hITHH {X = Xi} H {Y = Yi} llpH JIlO6hIX i = 1,2,3, ... , n, j = 1,2, ... , m. |
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6.S.1. |
B ypHe 4 6eJIbIX H 3 qepHbIX mapa. 11:3 Hee nOCJIe,L1;OBaTeJIbHO BhI- |
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HHMaIOT maphI .D.O nepBoro nOjfBJIeHHjf 6eJIoro mapa. llocTpoHTh |
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Pjf.D. H MHoroyrOJIhHHK pacnpe.D.eJIeHHjf .D.. c. B. X - qHCJIa H3BJIe- |
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qeHHhIX mapOB. |
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Q B03MO)J(HhIMH 3HaqeHHjfMH c. B. X jfBJIjfIOTCjf qHCJIa Xl = 1, X2 = 2, X3 =
= 3, X4 = 4. 3HaqeHHe X3 = 3, HanpHMep, 03HaqaeT, qTO nepBhdl: H BTOPOit maphI 6hIJIH qepHhIMH, a TpeTHit - 6eJIhlit.
CooTBeTcTBYIOIlIHe HM BepOjfTHOCTH PI , P2, P3, P4 Hait.D.eM, BOCnOJIh30BaB-
mHCh npaBHJIOM YMHO)J(eHHjf BepOjfTHocTeit:
PI = P{X = 1} = P{1-it map 6eJIhlit} = ~,
P2 |
= P{X = 2} = P{1-it map qepHhlit, 2-it - 6eJIhlit} = ¥.~ = ~, |
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P3 |
= P {X = 3} = '7 |
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35' |
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P4 |
= P {X = 4} = '7 |
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. 5 . 4 = |
35' |
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Pi |
( KOHTPOJIh: i~Pi = '7+ '7+ 35 + 35 |
= 1. |
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'7 |
'7 |
35 |
35 |
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MHoroyrOJIhHHK pacnpe.D.eJIeHHjf c. B. X npe.D.CTaBJIeH Ha pHC. 78. |
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'7 |
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Puc. 78
6.S.2. B ypHe 4 6eJIhIX H 3 qepHhIX mapa. 11:3 Hee HaY.D.aqy H3BJIeKJIH TpH mapa. RaitTH:
a) Pjf.D. pacnpe.D.eJIeHHjf .D.. c. B. Y - qHCJIa H3BJIeqeHHhIX 6eJIhIX mapOB;
340
6) BepOfiTHOCTb C06bITHfi A = {H3BJIe'"leHO He MeHee 2-x 6eJIbIX
IIIapOB}.
o a) CJIY'"IaitHMBeJIH'"IHHaY MO)l{eT npHHflTb CJIe,llj'IOIIJ;He3Ha'"leHHfI:Yo =
== 0, Yl = 1, Y2 = 2, Y3 = 3. CooTBeTcTBYIOIIJ;He HM 3Ha'"leHHfIPi Hait,Ll;eM, HCXO,Ll;fl H3 KJIaccH'"IeCKOrOonpe,Ll;eJIeHHfI BepOfiTHOCTH:
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C~· cl |
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Po = P{Y = O} = C; |
= 35' Pl=P{Y=I}= |
C? |
=35' |
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P |
= P{Y = 2} = CJ .Cl = 18 |
P3 = P{Y = 3} = |
cg· Cl |
2 |
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C? |
4
= 35·
OTCIO,LI;a pfl,Ll; pacnpe,Ll;eJIeHHfI c. B. Y HMeeT BH,LI; |
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Pi |
( KOHTPOJIb: i~Pi = 35 + 35 + |
35 + |
35 = 1. |
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35 |
35 |
35 |
35 |
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6) Hait,Ll;eM HCKOMYIO BepOfiTHOCTb, HCnOJIb3YfI pfl,Ll; pacnpe,Ll;eJIeHHfI c. B. |
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Y: P(A) = P{Y ~ 2} = P{Y = 2} + P{Y = 3} = |
18 |
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35 |
+ 35 = 35· |
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6.8.3. |
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MOHeTa nO,Ll;6paCbIBaeTcfI 5 pa3. IIOCTPOHTb MHoroyrOJIbHHK pac- |
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npe,Ll;eJIeHHfI ,LI;. c. B. Z - '"IHCJIaBbIIIa,D;eHHit rep6a. |
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6.8.4. |
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CTpeJIY. BepofiTHoCTH HX nOna,Ll;aHHfI B u;eJIb COOTBeTCTBeHHO paB- |
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HbI 0,5, 0,6, |
0,8. IIOCTPOHTb pfl,Ll; pacnpe,Ll;eJIeHHfI c. B. X - |
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nOna,D;aHHit B u;eJIb. |
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6.8.5. |
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BepofiTHoCTb TOI'O, '"ITO aBTOMaT npH |
onYCKaHHH |
O,Ll;HOit |
MOHeTbI |
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cpa6aTbIBaeT npaBHJIbHO, paBHa 0,98. IIOCTPOHTb pfl,Ll; pacnpe,Ll;e- |
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JIeHHfI c. B. - |
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BHJIbHOrO cpa6aTbIBaHHfI aBTOMaTa. HaitTH BepOfiTHOCTb Toro, '"ITO 6Y,LI;eT onYIIJ;eHO 5 MOHeT. PeIIIHTb Ty )l{e 3a,D;a'"lYnpH YCJIOBHH, '"ITO
BHaJIH'"IHHBcero 3 MOHeTbI.
6.8.6.IIOCTPOHTb pfl,Ll; pacnpe,Ll;eJIeHHfI '"IHCJIanOna,Ll;aHHit B BopOTa npH ,LI;ByX O,LI;HHHa,D;u;aTHMeTpOBbIX y,Ll;apax, eCJIH BepOfiTHOCTb nona,D;a- HHfI npH O,Ll;HOM y,Ll;ape paBHa 0,7.
6.8.7.,I1;HcKpeTHM c. B. X 3a,Ll;aHa Pfl,Ll;OM pacnpe,Ll;eJIeHHfI
HaitTH:
a)<PYHKU;HIO pacnpe,Ll;eJIeHHfI F(x);
6) BepOfiTHOCTH C06bITHit A = {X < 2}, B |
{I:::;; X < 3}, |
C = {I < X:::;; 3}; |
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B) nOCTpOHTb rpa<PHK <PYHKU;HH F(x).
Q a) IIo onpe,Ll;eJIeHHIO <PYHKU;HH pacnpe,Ll;eJIeHHfI HaXO,Ll;HM:
eCJIH x:::;; -2, TO F(x) = PiX < x} = 0;
eCJIH -2 < x:::;; 1, TO F(x) = PiX < x} = PiX = -2} = 0,08;
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eCJUI 1 < x :::; 2, TO F(x) = P{X = -2} + P{X = I} = 0,08 + 0,40 = 0,48; eCJUI 2 < x :::; 3, TO F(x) = P{X = -2} + P{X = I} + P{X = 2} =
= 0,08 + 0,40 + 0,32 = 0,80;
eCJUI 3 < X, TO F(x) = P{X = -2} +P{X = I} +P{X = 2} +P{X = 3} =
= 0,08 + 0,40 + 0,32 + 0,2 = 1.
0, |
eCJUI x :::; -2, |
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0,08, |
eCJUI |
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2 < x :::; 1, |
lITaK, F(x) = 0,48, |
eCJIH |
1 |
< x:::; 2, |
0,80, |
eCJIH 2 |
< x :::; 3, |
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1, |
eCJIH 3 < x. |
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6) CHaqaJIa Hail:,!l;eM HCKOMbIe BepOfiTHOCTH HenOCpe,!l;CTBeHHO:
P(A) = P{X < 2} = P{X = -2} + P{X = I} = 0,08 + 0,40 = 0,48;
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P(B) = P{1 :::; |
x < 3} = 0,40 + 0,32 = 0,72; |
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P(C) = P{1 < x :::; |
3} = 0,32 + 0,2 = 0,52. |
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9TH :>Ke BepOfiTHOCTH MO:>KHO Hail:TH, HCnOJIb3YfI <P0PMYJIbI |
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F(x) = P{X < X} |
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P{a:::; X |
< b} = F(b) - F(a). |
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Tor,!l;a |
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P(A) = P{X < 2} = F(2) = 0,48; |
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P(B) = P{1 :::; X |
< 3} = F(3) - |
F(I) = 0,80 - 0,08 = 0,72; |
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P(C) = P{1 < X |
:::; 3} = P{1 :::; |
X < 3} - |
P{X = I} + P{X = 3} = |
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= F(3) - F(I) - |
0,40 + 0,2 = 0,72 - |
0,2 = 0,52. |
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B) rpa<PHK <PYHKIJ;HH F(x) H306pa:>KeH Ha pHC. 79. |
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F(x) |
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1 ------"1",. _ - |
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Puc. 79 |
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6.S.S. |
HathH <PYHKIJ;HIO pacnpe,!l;eJIeHHfI CJIyqail:Hoil: BeJIHqHHbI X, 3aKOH |
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pacnpe,!l;eJIeHHfI KOTOPOil: nOJIyqeH npH pemeHHH 3a,n;aqH 6.8.1. |
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a Rait,n.eM F(x), HCnOJIb3Y51 cPOpMyJIy F(x) = 2: Pi ('ITO6b1CTpee npHBO-
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Xi<X |
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,Il,HT K II.eJIH, qeM HCnOJIb30BaHHe onpe,n:eJIeHH5I F(x)). Tor,n:a |
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0, |
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x ~ 1, |
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1 < x ~ 2, |
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7' |
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2 < x ~ 3, |
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F(x) = 7 |
+ 7 |
= 7' |
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4 |
2 |
4 |
34 |
3 < x ~ 4, |
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7 |
+ 7 |
+ 35 = |
35' |
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2 |
4 |
1 |
x> 4.' |
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+ 7 |
+ 35 + 35 = 1, |
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6.8.9.B KOMaH,n:e 16 cnopTCMeHOB, H3 KOTOPbIX 6 nepBOpa3p5l.D:HHKOB. Ra- y,n:aqy Bbl6HpaIOT ,n:ByX cnopTCMeHOB. IIocTpoHTb P5l,n: pacnpe,n:eJIeHH5I H cPYHKII.HIO pacnpe,n:eJIeHH5I qHCJIa nepBOpa3p5l,n:HHKOB cpe,n:H BbI6paHHblx.
6.8.10.3a,n;aHa cPYHKII.H5I pacnpe,n:eJIeHH5I c. B. X. RaitTH P5l,n: pacnpe,n:eJIe-
HH5I, a TaK)l{e Bep05lTHOCTH: PiX = I}, P{1 < X ~ 8}.
O, |
npH x ~ 0, |
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a) F(x) = { 0,3, |
npH °< x ~ 1, |
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1, |
npH 1 < x; |
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0, |
x ~ 1, |
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0,2, |
1 < x ~ 3, |
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6) F(x) = 0,35, |
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< x ~ 6, |
0,8, |
6 |
< x ~ 8, |
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8 < x. |
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6.8.11.
IIoCTpoHTb MHoroyrOJIbHHK pacnpe,n:eJIeHH5I, rpacPHK cPYHKII.HH
pacnpe,n:eJIeHH5I, HanTH Bep05lTHOCTH
P{X> 1,4}, P{1,4 ~ x ~ 2,3}.
6.8.12.IIo,n:6pacblBaIOT ,n:Be MOHeTbI. RanTH cPYHKII.HIO pacnpe,n:eJIeHH5I c. B.
X - qHCJIa BbIIIa,n;eHHn rep6a.
6.8.1S. 3a,n;aHo pacnpe,n:eJIeHHe ,n:. c. B. X
IIoCTpoHTb P5l,n: pacnpe,n:eJIeHH5I CJIyqanHbIX BeJIHqHH:
a)Y=2X; |
6)Z=X2. |
Q a) B03MO)l{Hble 3HaqeHH5I c. B. Y TaKOBbI: |
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Yl=2·(-2)=-4, |
Y2=2·(-I)=-2, Y3=2, Y4=4, Y5=6. |
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BepmlTHOCTH 3THX 3HaqeHHi\ paBHbI BepOfiTHOCTfiM COOTBeTCTBYIOID;HX 3HaqeHHi\ C. B. X (HanpHMep, P{Y = -4} = P{X = -2} = 0,20 H T.,ll;.). TaKHM o6pa30M
6) 3HaqeHHfI C.B. Z TaKOBbI: ZI = (_2)2 = 4, Z2 = (_1)2 = 1, Z3 = 12 =
= 1, Z4 = 22 = 4, Z5 = 32 = 9. TIPH 3TOM
P{Z = 4} = p{X2 = 4} = P{X = -2} + P{X = 2} = 0,20 + 0,15 = 0,35
H T.,ll;. TI03TOMY PM paCnpe,ll;eJIeHHfI C. B. Z HMeeT BH,ll;
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6.8.14. ,1l,aHbI 3aKOHbI pacnpe,ll;eJIeHHfI ,ll;ByX He3aBHCHMbIX CJIyqai\HbIx BeJIHqHH X H Y:
Hai\TH 3aKOH pacnpe,ll;eJIeHHfI CJIyqai\HbIx BeJIHqHH
a) Z=X+Y; |
6) W=X·Y. |
°= |
Q a) Hai\,ll;eM B03MO)KHbIe 3HaqeHHfI |
Zij = Xi + Yj: -1 = 1 + (-2), |
= 1+(-1),0 = 2+ (-2),1 = 2+ (-1),1 = 3+ (-2),2 = 3+(-1), T.e.
CJIyqai\HM BeJIHqHHa Z npHHHMaeT 3HaqeHHfI ZI = -1, Z2 = 0, Z3 = 1 H
Z4 = 2. HaXO,ll;HM BepOfiTHOCTH 3THX 3HaqeHHi\:
PI = P{Z = -I} = P{X = 1, Y = -2} = P{X = I} . P{Y = -2} =
= 0,3·0,4 = 0,12;
P2 = P{Z = o} = P{X = 1, Y = -I} + P{X = 2, Y = -2} =
= 0,3 . 0,6 + 0,5 . 0,4 = 0,38;
P3 = P{Z = I} = P{X = 2, Y = -I} + P{X = 3, Y = -2} =
= 0,5 . 0,6 + 0,2 . 0,4 = 0,38;
P4 = P{Z = 2} = P{X = 3, Y = -I} = 0,2·0,6 = 0,12.
HanOMHHM, qTO 3anHCb BH,ll;a P{X = 3, Y = -I} 03HaqaeT BepOfiTHOCTb HacTynJIeHHfI ,ll;ByX He3aBHCHMbIX C06bITHi\ {X = 3} H {Y = -I}, T. e.
P{X = 3,Y = -I} = P{{X = 3}· {Y = -I}} = P{X = 3}· P{Y = -I}.
TIPH HaxO)K,ll;eHHH BepofiTHoCTH P3 = P{ Z = I} H P2 MbI BOCnOJIb30BaJIHCb npaBHJIOM CJIO)KeHHfI HeCOBMeCTHbIX C06bITHi\.
B HTore nOJIyqaeM 3aKOH pacnpe,ll;eJIeHHfI c. B. Z = X + Y:
( KOHTPOJIb: t Pi = 1.)
t=1
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6) AHaJIOrJPIHO |
HaxO,Ll;HM |
(npoBepbTe!) |
p5l,Ll; |
pacnpe,Ll;eJIeHH5I |
c. B. W = |
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== X· Y: |
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( KOHTPOJIb: t Pi = 1.) |
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r--'--~'---~r-~-r--~.-~~ |
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Pi |
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~=1 |
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6.8.15. |
3a,L1;aHO pacnpe,Ll;eJIeHHe ,LI;HCKpeTHoit c. B. X |
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RaitTH pacnpe,Ll;eJIeHHe C. B.: |
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a) Y = IXI; |
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6) Z = X 3 + 1. |
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6.8.16. |
,nHcKpeTHM c. B. X HMeeT p5l,Ll; pacnpe,Ll;eJIeHH5I |
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Xi |
0 |
7r |
7r |
37r |
7r |
57r |
37r |
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2" |
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2" |
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Pi |
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llOCTPOHTb: |
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a) p5l,Ll; pacnpe,Ll;eJIeHH5I c. B. Y = sin ( X - i); |
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6) rpacPHK cPYHKn:HH pacnpe,Ll;eJIeHH5I c. B. Y. |
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6.8.17. llOCTPOHTb p5l,Ll; pacnpe,Ll;eJIeHH5I ,LI;JI5I CJIyqaitHblx BeJIHqHH |
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Z=X+Y |
H W=X·Y, |
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eCJIH X H Y - |
He3aBHCHMble CJIyqaitHble |
BeJIHqHHbI, |
3a,Ll;aHHble |
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p5l,Ll;aMH pacnpe,Ll;eJIeHH5I |
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RaitTH YCJIOBHYIO Bep05lTHOCTb C06bITH5I {Z < 4} npH YCJIOBHH, |
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'ITO{Z > 2}. |
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cP0PMYJIOit P {X = k} = C . k, |
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6.8.18. |
Pacnpe,Ll;eJIeHHe,Ll;. c. B. X |
3a,L1;aHO |
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r,Ll;e k = 2,3,4,5,6. RaitTH: |
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41 < I}. |
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a) 3HaqeHHe C; |
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6) P{IX - |
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6.8.19. llo,Ll;6pOIIIeHbI 2 HrpaJIbHble KOCTH. llOCTPOHTb p5l,Ll; pacnpe,Ll;eJIeHH5I:
a)CYMMbI BbIIIaBIIIHX OqKOB; 6) pa3HOCTH BbIIIaBIIIHX OqKOB.
6.8.20.Bep05lTHOCTb TOro, 'ITOCTY,LI;eHT Hait,Ll;eT B 6H6JIHOTeKe HY)l{HYIO eMY KHHry, paBHa 0,4. llOCTPOHTb p5l,Ll; pacnpe,Ll;eJIeHH5I qHCJIa 6~6-
JIHOTeK, KOTopble OH MO)l{eT noceTHTb, eCJIH eMY ,LI;OcTynHbI qeTblpe 6H6JIHOTeKH.
6.8.21.ABTOM06HJIb Ha nYTH K MecTY Ha3HaqeHH5I BCTpeTHT 5 cBeTocPopoB, KaJK,LI;blit H3 KOTOPbIX nponYCTHT ero C Bep05lTHOCTblO ~. llOCTPo-
HTb p5l,Ll; pacnpe,Ll;eJIeHH5I qHCJIa cBeTocPopoB, npoit,Ll;eHHbIX MaIIIHHOit ,LI;O nepBoit OCTaHOBKH HJIH ,LI;O npH6bITH5I K MeCTY Ha3HaqeHH5I.
345
6.8.22.Y .D:e)KypHoro llMeeTCH 7 pa3HbIX KJIlOqei!: OT pa3HbIX KOMHaT. BbI-
HYB HaY.D:aqy KJIlOq, OH npo6yeT OTKpbITb .D:Bepb O.D:HOi!: 113 KOMHaT. IIOCTPOllTb pH.D: pacnpe.D:eJIeHllH qllCJIa nOnbITOK OTKpbITb .D:Bepb (npOBepeHHbli!: KJIlOq BTOPOi!: pa3 He llCnOJIb3yeTCH). IIOCTPOllTb MHoroyrOJIbHllK <noro pacnpe.D:eJIeHllH.
6.8.23.ATe 06CJIY)KllBaeT 1500 a6oHeHToB. BepoHTHoCTb Toro, 'ITOB Te-
qeHlle 3 MllHyT Ha ATe nocTynllT BbI30B, paBHa 0,002. IIOCTPOllTb pH.D: pacnpe.D:eJIeHllH c. B. X, paBHoi!: qllCJIY BbI30BOB, nocTynllBIIIllx Ha ATe B TeqeHlle 3 MllHyT. Hai!:Tll BepOHTHOCTb Toro, 'ITO3a :'lTO BpeMH nocTynllT 60JIee Tpex BbI30BOB.
6.8.24. B napTllll, cO.D:ep)KaIII;ei!: 20 113.D:eJIlli!:, llMeeTCH qeTblpe 113.D:eJIllH C
.D:e<peKTaMll. HaY.D:aqy OTo6paJIll Tpll 113.D:eJIllH .D:JIH npoBepKll llX KaqeCTBa. IIoCTpOllTb pH.D: pacnpe.D:eJIeHllH qllCJIa .D:e<peKTHbIX 113-
.D:eJIlli!:, CO.D:ep)KaIII;IlXCH B YKa3aHHoi!: Bbl6opKe.
6.8.25. I1cnOJIb3YH YCJIOBlle 3a.D:aqll 6.8.4, Hai!:Tll <PYHKIJ;1l1O pacnpe.D:eJIeHllH c. B. II nOCTpOllTb ee rpa<PllK.
6.8.26. I1cnOJIb3YH YCJIOBlle 3a.D:aqll 6.8.6, Hai!:Tll <PYHKIJ;1l1O pacnpe.D:eJIeHllH c. B. II nOCTpOllTb ee rpa<pllK.
6.8.27. I1cnOJIb3YH YCJIOBlle 3a.D:aqll 6.8.22, Hai!:Tll <PYHKIJ;1l1O pacnpe.D:eJIeHllH c. B. II nOCTpOllTb ee rpa<PllK.
6.8.28. IIo.D:6pOIIIeHbI2 llrpaJIbHble KOCTII. IIoCTpOllTb pH.D: pacnpe.D:eJIeHllH II <PYHKIJ;1l1O pacnpe.D:eJIeHllH .D:. c. B. X - qllCJIa Bbllla.D:eHlli!: qeTHOro qllCJIa OqKOB.
6.8.29.X II Y - He3aBllCllMble .D:llCKpeTHble CJIyqai!:Hble BeJIllqllHbI, 3a-
.D:aHHble Ta6JIllIJ;aMll pacnpe.D:eJIeHllH
II
a) pH.D: pacnpe.D:eJIeHllH c. B. Z = X . Y;
6)P{X + Y > 5};
B) P{(X + Y > 5) I (X = 2)}.
6.8.30.3a.D:aHbI pacnpe.D:eJIeHllH .D:ByX He3aBllCllMbIX CJIyqai!:Hblx BeJIllqllH
XllY:
|
II |
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a) |
<PYHKIJ;1l1O pacnpe.D:eJIeHllH c. B. X; |
X + Y, W = |
6) |
pH.D: pacnpe.D:eJIeHllH CJIyqai!:Hblx BeJIllqllH Z |
=X-Y;
B)P{IX - YI ~ 2};
r)nOCTpOllTb MHoroyrOJIbHllKll pacnpe.D:eJIeHllH c. B. Z II W.
6.8.31. Hai!:Tll <PYHKIJ;1l1O pacnpe.D:eJIeHllH c. B. Y = sin ~X, r.D:e c. B. X -
qllCJIO OqKOB, Bbllla.D:alOIII;ee npll 6pocaHllll llrpaJIbHoi!: KOCTII.
346
6.8.32.I1cnbITaHlul no cxeMe BepHYJIJUI C BepofiTHoCTbIO ycnexa p B O,n;HOM lICnblTaHIUI nOBTopflIOTCfI ,n;o ,n;ByX ycnexoB. IIocTpollTb PfI,n; pacnpe,n;eJIeHlIfi 1.IlICJIa npoBe,n;eHHblx lICnbITaHlIt!:. Hat!:TlI BepofiTHoCTb Toro, 'iTO B nepBbIX N lICnbITaHlIfiX 'illCJIO ycnexoB MeHbille 2.
6.8.33.IIOJIb3YflCb YCJIOBlIeM 3a,n;a'i1l 6.8.29, nocTpollTb PM pacnpe,n;eJIeHlIfi C.B. Z = min{X, Y}.
6.8.34.KaKM 113 HIDKenpllBe,n;eHHbIX nOCJIe,n;OBaTeJIbHOCTet!: f1BJIfleTCfI pacnpe,n;eJIeHlIeM BepOfiTHOCTet!: HeKOTopot!: ,n;lICKpeTHot!: CJIY'iat!:Hoi!: BeJIlI'illHbI?
an = n(n1+ 1);
n |
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d |
2! |
dx |
nEN. |
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4n - l |
-4 |
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n |
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n=:rr |
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C = (n _ l)!e |
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1+x2 |
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n-l
6.8.35. .II:lICKpeTHM c. B. X npllHlIMaeT IIeJIO'illCJIeHHble 3Ha'ieHlIfi Xl = 1,
X2 = 2, X3 = 3, .... I13BecTHO, 'iTO Pn = P{X = n} = 2 C |
. |
n +3n+2 |
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Hat!:TlI: • |
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a) 3Ha'ieHlIe napaMeTpa c; |
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6) BepOfiTHOCTb C06bITlIfi D = {X = 5}. |
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6.8.36.MO:>KeT JIll <PYHKIIlIfi F(x) 6bITb <PYHKIIlIet!: pacnpe,n;eJIeHlIfi HeKO-
TOpOt!: c. B., eCJIlI: |
|
a) F(x) = e- x ; |
6) F(x) = eX; |
B) F(x) = 1 - eX; |
r) F(x) = 1 - e- x ; |
.u.) F(x) = 0,5 + #.arctg(x); |
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e) F(x) = { 0,4,O' |
x:::; 0, |
°< x:::; 1, |
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0,35, |
1 < x:::; 5, |
1, |
5 < x? |
6.8.37.MO:>KHO JIll YTBep:>K,n;aTb, 'iTO C06bITlIe C f1BJIfleTCfI HeB03MO:>KHbIM, eCJIlI P(C) = O?
6.8.38.COBna,n;aIOT JIll 3aKOHbI pacnpe,n;eJIeHlIfi ,n;lICKpeTHbIX CJIY'iat!:HbIX BeJIlI'illH X + X II 2 . X?
6.8.39. .II:lICKpeTHM C. B. X npllHlIMaeT HaTypaJIbHble 3Ha'ieHlIfI, npll'ieM
3Ha'ieHlIe n C BepOfiTHOCTbIO 1 IIOCTPOllTb Pfl,n; pacnpe,n;eJIeHlIfi
.
n
2
BepOfiTHocTei!: ,lI;JIfi C. B. Y = 1sin CfX ) I·
§ 9. HEnPEPblBHblE CllYYA~HbIE BElllllYIIIHbl
B rrpe,n;bI,IU'm;eMrraparpa¢e 6bIJIO BBe,n;eHO rrOHXTHe HerrpepblBHo:li CJIY'Ia:liHo:li
BeJIH'IHHbI(H. c. B.). MO:lKHO ,n;aTb ,n;pyroe, 60JIee cTporoe, orrpe,n;eneHHe H. C. B., HC-
nOJIb3yX rrOHXTHe <PYHKIIHH pacrrpe,n;eneHHX.
347
~C.J1Y'IaiiHaJIBeJIH'IHHaX Ha3hIBaerCH Henpep'bl.8HOil, eC.J1H ee <PYHKn;HH pacrrpe-
p;eJIeHHH F(x) HerrpephIBHa Ha Bceii '1HC.J10BOiiOCH. |
$ |
B OT.J1H'IHeOT p;HCKpeTHhIX C.J1Y'IaiiHbIXBe.J1H'IHHBepOHTHOCTb OTp;e.J1bHOrO 3Ha- |
|
'1eHHH ,n;.J1H HerrpepbIBHOii C.J1Y'IaiiHOii Be.J1H'IHHhI paBHa HY.J1IO: P{X = c} |
= 0, |
'ric E JR. IImlToMY ,n;.J1H H. C. B. X HMeeM: |
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P{a ~ X < b} = P{a < X < b} = P{a < X ~ b} = P{a ~ X ~ b} = F(b) - |
F(a). |
IIoMHMO <PYHKn;HH pacrrpep;eJIeHHH ,n;.J1H HerrpepbIBHbIX C.J1Y'IaiiHhIXBeJIH'IHH,CYIIIecTByeT eIIIe OP;HH YP;06HbIii crroco6 3ap;aHHH 3aKOHa pacrrpep;eJIeHHH - rr.J10THOCTb BepOHTHOCTH.
~IIycTb <PYHKn;HH pacrrpep;eJIeHHH F(x) p;aHHoii H. C. B. X HerrpepbIBHa H P;H<p<pe-
peHn;HpyeMa BCIO,D;y, KpOMe, MO:lKeT 6bITb, OTP;eJIbHbIX TO'leK. Torp;a rrpOH3Bop;HaH I (x) ee <PYHKn;HH pacrrpep;eJIeHHH Ha3hIBaeTCH n.n.omHOCm'b'lO pacnpeiJe.n.ewuSI Herrpe-
PhIBHOii c. B. X (H.J1H «rr.J10THOCTbIO BepOHTHOCTH», H.J1H rrpOCTO «rr.J10THOCTbIO»):
I(x) = F'(x).
HapH,D;y C 0603Ha'leHHeM I(x) ,n;.J1H rr.J10THOCTH pacrrpep;eJIeHHH HCrr0.J1b3yeTcH TaK:lKe 0603Ha'leHHep(x) (T.e. p(x) = F'(x)).
CBOMCTBa nnOTHOCTIo1 pacnp~eneHIo1H:
1.I (x) ~ 0 (cBoiicTBO HeOTpHn;aTeJIbHOCTH);
2.j00I (x) dx = 1 (cBoiicTBO HOpMHpoBaHHocTH);
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- 00 |
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b |
3.P{a~X~b}= jl(X)dX; |
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a |
4. |
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'" |
F(x) = j I(t) dt; |
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- 00 |
5. |
lim |
I(x) = O. |
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"'..... ±oo |
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~ rpa<pHK |
rr.J10THOCTH pacrrpep;e.J1eHHH I (x) Ha3hIBaeTCH IGpu80il pacnpeiJe.n.e- |
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H~. |
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$ |
6.9.1.3a.rr,aHa <PYHKIl;fUI paclIpe,n:e.neHlul H. C. B. X
O' |
IIpH X < 3, |
F(x) = { C(x - 3)2, |
IIpH 3 ~ x ~ 5, |
1, |
IIpH 5 < x. |
RaitTH: |
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a) K09<P<PHIJ;HeHT C; |
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348
6) IIJIOTHOCTb pacIIpe,n:eJIeHHH f(x) c. B. X H IIOCTPOHTb rpa<PHKH q,YHKIJ.Ht!: F(x) H f(x);
B) PiX E [3,4)}.
o a) TaK KaK c. B. X - HeIIpepbIBHa, TO F(x) ,n:OJDKHa 6bITb HeIIpepbIBHot!: q,YHKIJ.Het!: B JIlo6ot!: TO'"lKe,B '"IaCTHOCTH,H IIpH X = 5. TaK KaK F(5) = 1, TO
C· (5 - |
3)2 = 1, oTKy,n:a C = ~. TaKHM o6pa30M, |
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O' |
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IIpH x < 3, |
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F(x) = { ~(x - 3)2, |
IIpH 3:::; x:::; 5, |
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1, |
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IIpH 5 < x. |
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6) llJIOTHOCTb paCIIpe,n:eJIeHHH f(x) |
= F'(x) BblpaJKaeTCH <POPMYJIOt!:: |
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O' |
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IIpH X < 3, |
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f(x) = { ~(x - 3), |
IIpH 3 :::; x :::; 5, |
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0, |
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IIpH 5 < X. |
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fPa<PHKH q,YHKIJ.Ht!: F(x) H |
f(x) |
IIpe,n:cTaBJIeHbI Ha pHC. 80 H pHC. 81. |
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F(x) |
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I(x) |
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1 |
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o |
3 |
5 |
x |
o |
3 |
5 |
x |
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Puc. 80 |
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Puc. 81 |
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b |
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B) lICIIOJIb3YH q,OPMYJIY P{a:::; X |
:::; b} = |
Jf(x) dx., HaxO,n:HM, '"ITO |
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a |
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4 |
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PiX E [3,4)]} = P{3:::; X < 4} = J~(x - 3)dx =~. |
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3 |
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• |
ilJIH, HHa'"leP{3 :::; X < 4} = F(4) - F(3) = |
~ - °= ~. |
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6.9.2. |
llPH KaKHX 3Ha'"leHHHIIapaMeTpOB k H b <PYHKIJ.HH |
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O' |
x:::; |
-1, |
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F(x) = { kx + b, |
-1 < x :::; 2, |
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1, 2<x
MO)KeT 6bITb <PYHKIJ.Het!: paCIIpe,n:eJIeHHH HeKoTopot!: HeIIpepbIBHot!: c. B. X? Hat!:TH BepoHTHoCTb Toro, '"ITOC. B. X IIpHMeT 3Ha'"leHHe, 3aKJIIO'"IeHHOeB IIpOMe)KYTKe (-2,3; 1,5). llOCTPOHTb rpa<PHK IIJIOTHOCTH pacIIpe,n:eJIeHHH 9TOt!: CJIY'"Iat!:Hot!:BeJIH'"IHHbI.
349