Сборник задач по высшей математике 2 том
.pdfr.D:e 06JIacTb D z - |
'1aCTb9TOrO npHMoyrOJIbHHKa, JIe)KaIII,M HH)Ke npHMoti |
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x + y = Z, T. e. y = |
-x + Z; |
f(x, y) - |
nJIOTHOCTb pacnpe.D:e.rreHHH .D:BYMepHoti |
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C. B. (X, Y); JI(X) |
= |
fx(x) |
H h(y) = |
Jy(y) - |
nJIOTHOCTH paCnpe.D:eJIeHHH |
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BepOHTHocTeti CJIY'latiHblxBeJIH'!HHX H Y COOTBeTCTBeHHO. ITo YCJIOBHIO |
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JI(x) = {1' |
x E [0;1], |
H |
h(y)={l, |
YE[-1:2], |
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0, |
x ft |
[0; 1], |
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0, |
y ft [-1,2]. |
TaK KaK C. B. X H |
Y He3aBHCHMbI, TO |
f(x, y) = |
JI (x)h(Y) = 1 ·1 3 |
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=lB |
npHMoyrOJIbHHKe ABeD (BHe era f(x, y) = 0).
y
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Puc. |
95 |
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CJIe.D:OBaTeJIbHO, |
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Fz(z) = III dxdy = |
1IIdxdy = |
lSDz |
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Dz |
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Dz |
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(3.D:eCb SDz - nJIow:a.n:b 06JIacTH Dz). OTCIO.D:a HMeeM: |
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1) eCJIH z ~ -1, TO F(z) = 0; |
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2) eCJIH -1 < z ~ 0, TO F(z) |
1 |
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+ l)(z + 1) = |
(z+1)2 |
= ['3SDz |
= |
'3' |
2(z |
6 ' |
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nOCKOJIbKY B 9TOM CJIy'lae06JIaCTb Dz - |
npHMoyraJIbHblti TpeyrOJIbHHK C |
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KaTeTaMH z + 1 H Z + 1 (Ha pHC. 95 06JIacTb Dz 3aIlITpHXOBaHa).
420
IIJIoru;a,rr.b 06JIacTH D z MO)KHO, KOHel-IHO, HatiTH C nOMOru;bIO HHTerpa.na
(XOTH 9TOT cnoco6 60JIee rpOM03.D:KHti):
z+1 |
z-z |
z+1 |
z+1 |
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SDz = ! dx |
! |
dy = |
! dx· y[~z = |
! (z - x + 1) dx = |
o |
-1 |
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0 |
0 |
2 |
) IZ+1 |
= z2 + z - |
(z + 1)2 |
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= ( zx - ~ + x |
0 |
2 |
+ z + 1 |
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2z2 + 4z + 2 - |
z2 - |
2z - |
1 Z2 + 2z + 1 |
(z + 1)2 |
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2 |
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= -'---~2---'- |
3) eCJIH 0 < z ~ 2, TO F(z) |
1 |
(z + 1) + z |
·1 = |
2z + 1 |
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= 3 . |
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- 6 - ' TaK KaK |
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06JIacTb D z B 9TOM CJIY'-Iaenpe,n:CTaBJIHeT co6oti npHMoyrOJIbHyIO TpaneIJ.HIO c OCHOBaHHHMH z + 1 HZ (npOBepbTe!); ee BbICOTa paBHa 1.
4) eCJIH 2 < z ~ 3, TO |
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1 |
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1 ( |
1· 3 - |
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2)) |
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F(z) = 3(SOABCD-St.CEF) = |
3 |
2(1- (z - 2)) . (1 - (z - |
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= ~ (3 - ~(3 - |
z)2) = 1- ~(3 - |
z2). |
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liIJIH, HHaqe, HCnOJIb3YH HHTerpa.nbI: |
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F(z) = ~( 72dX |
j dy+ |
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j dx 7~Y) = |
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o |
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-1 |
z-2 |
-1 |
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1 |
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= ~(3XI:-2+ !(Z-X+l)dX) = ~ (3Z-6+ (ZX- ~2 +X)I:_2) = |
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z-2 |
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1 ( |
3z - |
6 + z - z |
2 |
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2z - |
1 |
(z - 2)2 |
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= |
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= 3 |
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2 + |
2 |
+ 1 - z + 2 |
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= 1 (_Z2 + 5z _ 'I+ (z - |
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2)2) = 1(_2Z2 + lOz _ 7 + Z2 - |
4z + 4) = |
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3 |
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2 |
2 |
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6 |
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= 1(_Z2 + 6z - |
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3) = _1(z2 - 6z + 9 - 6) = 1 _l(z - |
3)2; |
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6· |
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5) eCJIH 3 < Z, TO F(z) = 3SDz = |
3S0ABCD = |
3.3 = 1. TaKHM 06Pa30M |
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0, |
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z ~ -1, |
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(z + 1)2 |
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-1 < z ~ 0, |
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6 |
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F(z) = -2z6+ -1 , |
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0< z ~ 2, |
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1- (z - |
3)2 , |
2 < z ~ 3, |
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1, |
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6 |
3 < z. |
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421
OTCIO.Ll:a
0, |
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z ~ -1, z > 3, |
z+1 |
-1 < z ~ 0, |
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Jz(z) = F'(z) = ~ |
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3' |
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0< z ~ 2, |
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2 < z ~ 3. |
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1- |
~ |
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3' |
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Cnoco6 2. Hati.Ll:eM nJIOTHOCTb pacnpe,n,eJIeHlUI c. B. Z = X +Y, HCnOJIb3Y.H <P0PMYJIY CBepTKH:
00
Jz(z) = Jh(x)h(z ,- x) dx.
- 00
C1>YHKIJ;HH no.Ll: 3HaKOM HHTerpaJIa OTJIH'IHbIOT HYJI.H JIHilib B CJIY'Iae
{o~ x ~ 1, |
T.e. {o ~ x ~ 1, |
(13.1) . |
-1 ~ z - x ~ 2, |
z - 2 ~ x ~ z + 1. |
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PeIIIeHHe CHCTeMbI 3aBHCHT OT 3Ha'leHH.Hz. |
2; z + 1] |
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ECJIH z < -1, CHCTeMa (13.1) HeCOBMeCTHa; OTpe3KH [0; 1] H [z - |
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He nepeceKaIOTC.H (CM. reOMeTpH'IeCKYIOHJIJIIOCTpaIJ;HIO Ha pHC. 96). CJIe,n,o-
BaTeJIbHO, B 9TOM CJIY'Iaeh(z - |
x) = 0 H, 3Ha'lHT,J(z) = O. |
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fie""", 'J |
x |
1""'C"'1 |
x |
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z-2"""i'+1 |
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;;;, |
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'" 'I |
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0""1 |
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0"" |
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Puc. 96 |
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Puc. 97 |
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ECJIH -1 < z ~ 0, CHCTeMa (13.1) 9KBHBaJIeHTHa HepaseHCTBY 0 ~ X ~ |
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~ Z + 1 (pHC. 97). IIo9TOMY |
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z+1 |
z+1 |
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1 |
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J(z) = J3 dx = -3-· o
ECJIH 0 < Z ~ 2, CHCTeMa (13.1) 9KBHBaJIeHTHa HepaBeHCTBY 0 ~ X ~ 1
(pHC. 98). IIo9TOMY
I', , , , , , , 'I |
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z-'i"" ';+1 |
x |
;;;; |
x |
0"'1 |
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Puc. 98 |
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422
ECJIH 2 < z :::; 3, CHCTeMa (13.1) 9KBHBaJIeHTHa HepaBeHCTBY z - 2 :::; x :::; 1
(pac. 99). IIo9ToMY |
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J(Z) = !1 |
1 dx = 1(1 - z + 2) = 1 - |
~. |
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z-2 |
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.',, , , , , , 'J |
x |
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1""('("1 • |
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z-2"'" ''';+1 |
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;,;, |
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;;;; |
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0"" 1 |
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0" |
"1 |
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Puc. 99 |
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Puc. 100 |
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ECJIH 3 < z, CHCTeMa (13.1) HeCOBMeCTHa, TaK KaK OTpe3KH [z - |
2; z + 1] |
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a [0; 1] He nepeceKalOTca (pHC. 100); T.e. h(z - x) = °H J(z) = 0. |
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I1TaK, KaK H B nepBOM CJIY'lae, |
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0, |
x:::; -1, x> 3, |
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z+l , |
-1 < z :::; 0, |
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Jz(z) = |
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< z:::; 2, |
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3' |
° |
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1- ~ |
2 < z:::; 3. |
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3' |
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6.13.19. Hail:TH nJIOTHOCTb pacnpe.n;eJIeHHa BepoaTHocTeil: CYMMbI .n;BYX He-
3aBHCHMbIX CJIY'lail:Hblx BeJIH'lHH, paBHOMepHO pacnpe.n;eJIeHHbIX Ha OTpe3Ke.[0; 2].
6.13.20. Hail:TH 3aKOH pacnpe.n;eJIeHHa CYMMbI Z .n;BYX He3aBHCHMbIX CJIy-
'lail:HblxBeJIH'lHHX H Y, pacnpe.n;eJIeHHbIX no HOPMaJIbHOMY 3a-
KOHY: X '" N(O; 1), Y '" N(O; 1). HaiI:TH M(Z) H D(Z).
H Y He3aBHCHMbI H pacnpe.n;eJIeHbI no
I |
_!x |
X ~ |
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h(y) = {0,2e- |
O,2Y |
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y ~ 0, |
fr(x) = { 4e |
4 , |
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0, |
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x < 0, |
0, |
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y < 0. |
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Hail:TH KOMn03HIJ;HlO 9THX 3aKOHOB.
Q IIJIOTHOCTb pacnpe.n;eJIeHHa C. B. Z = X +Y Hail:.n;eM, HCnOJIb3ya <POPMYJIY
g(z) = !z fr(x) . h(z - x) dx. o
g(z) = !z0,25e-O,25X • 0,2e-O,2(Z-X) dx = 0,05e-O,2z !ze-O,25x+O,20x dx =
o |
0 |
:= 0,05e-O,2Z !Z e-O,05x dx = _e-O,2z • e-O,05x I: = e-O,2z • (1 _ e-O,05Z). •
o
423
6.13.22. |
Hatl:TH 3aKOHbI |
pacnpe,n;eJIeHHH c. B. Z = X + Y, r,Il;e X H Y - |
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He3aBHCHMble CJIY'iatl:HbleBeJIH'iHHbI,HMelOIIIHe paBHoMepHoe pac- |
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npe,Il;eJIeHHe Ha oTpe3Ke [0; 1) |
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( fX(X) = {I, |
x E [0; 1), |
fy(y) = {I, |
y E [0; 1), ). |
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0, |
x ~ [0; 1), |
0, |
y ~ [0; 1) |
6.13.23. |
113BecTHo, 'iTOC. B. X pacnpe,Il;eJIeHa no nOKa3aTeJIbHOMY 3aKOHY |
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fx(x) = {0,3e-O,3X, x ~ 0, |
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0, |
x < 0, |
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a c. B. Y '"R[O; 2) H He 3aBHCHT OT X. Hatl:TH nJIOTHOCTb pacnpe- ,Il;eJIeHHH CJIY'iatl:Hotl:BeJIH'iHHbIZ = X + Y.
6.13.24. |
CJIY'iatl:Hbltl: BeKTop (X, Y) pacnpe,Il;eJIeH no 3aKoHY, |
3a,IJ;aHHoMY |
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Ta6JIHu;etl: |
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X\Y |
-2 |
-1 |
0,24° |
1 |
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-1 |
0,02 |
0,16 |
0,03 |
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°2 |
0,05 |
0,15 |
0,10 |
0,06 |
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0,03 0,09 0,06 0,01 |
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Hatl:TH: |
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a) 3aKOHbI pacnpe,Il;eJIeHHH c. B. X H Y; |
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6) 3aKOHbI pacnpe,Il;eJIeHHH c. B. Z = -5X + 2 H W = ly3 -11; |
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B) MaTeMaTH'ieCKHeOlKH,Il;aHHH H ,Il;HCnepCHH CJIY'iatl:HblxBeJIH'iHH |
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ZHW. |
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6.13.25. |
,1l;HcKpeTHM c. B. X 3a,IJ;aHa CBOHM PH,Il;OM pacnpe,Il;eJIeHHH |
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6.13.26. |
Hatl:TH ,Il;HCnepCHIO c. B. Y = JX + 2. |
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HenpepbIBHM c. B. X |
pacnpe,Il;eJIeHa |
paBHOMepHO B |
HHTepBarre |
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(-i; i)· Hatl:TH: |
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a) nJIOTHOCTb pacnpe,n;eJIeHHH c. B. Y = sin x; |
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6) 'iHCJIOBbleXapaKTepHCTHKH M(Y) H D(Y). |
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6.13.27. |
CJIY'iatl:HMBeJIH'iHHaX pacnpe,Il;eJIeHa no 3aKOHY KOIIIH |
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x E III
HaHTH <PYHKU;HIO pacnpe,n;eJIeHHH H nJIOTHOCTb BepOHTHOCTH CJIy- 'iatl:Hotl:BeJIH'iHHbI:
a)Y=I-X; |
6)Y=X 3 • |
424
6.13.28. CJIyqatiHM BeJIHqHHa X HMeeT rrJIOTHOCTh pacrrpe,n;eJIeHHH
f(x) = {~' |
XE(-i;i), |
0, |
xi (-i; i) . |
HatiTH rrJIOTHOCTh pacrrpe.n;eJIeHHH BepOHTHocTeti g(y) CJIyqatiHoti
BeJIHqHHhI: |
6) Y = IXI; |
a) Y = 6X + 3; |
B) Y = e- x2 .
6.13.29.IIycTh f(x) - rrJIOTHOCTh pacrrpe,n;eJIeHHH H. c. B. X, x E (-00,00).
HatiTH rrJIOTHOCTh pacrrpe.n;eJIeHHH g(y) CJIyqatiHoti BeJIHqHHhI Y, eCJIH:
a) Y = -X; |
6) Y = X + 10; |
B) Y = X2; |
r) Y = arctgX. |
6.13.30. IIycTh F(x) - <PyHKIJ;HH pacrrpe,n;eJIeHHH H.C.B. X. BhIpa3HTh qe-
pe3 Hee <PYHKIJ;HIO pacrrpe.n;eJIeHHH F(y) CJIyqatiHoti BeJIHqHHhI Y,
eCJIH: |
+ 3; |
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a) Y = 2X |
6) Y = e- X ; |
B) Y = X3.
6.13.31.CJIyqatiHaH BeJIHqHHa X HMeeT rrJIOTHOCTh pacrrpe,n;eJIeHHH
1 _",2
f(x) = - e 2 .
~
HatiTH rrJIOTHOCTh pacrrpe,n;eJIeHHH BepOHTHocTeti g(y) CJIyqatiHoti
BeJIHqHHhI Y, eCJIH: |
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a) Y = 1.; |
6) Y = IX - 11; |
X |
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B) Y = 5X + 1.
6.13.32.113BecTHo, qTO H. C. B. X '" R[O, 2]. HatiTH <PYHKIJ;HIO pacrrpe,n;eJIeHHH H rrJIOTHOCTh pacrrpe,n;eJIeHHH CJIyqatiHhIx BeJIHqHH
Y = -2X + 4, |
Z = IX-1I- |
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HatiTH M(Y) H D(Y). |
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6.13.33. 3aKOHhI pacrrpe,n;eJIeHHH qHCJIa OqKOB, |
BhI6HBaeMhIx K8JK,n;hIM H3 |
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,n;ByX CTpeJIKOB (X H Y cooTBeTcTBeHHo), TaKOBhI: |
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8 |
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10 |
8 |
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OrrHcaTh 3aKOH pacnpe,n;eJIeHHH CyMMhI OqKOB, Bh16HBaeMhIx 9THMH CTpeJIKaMH.
6.13.34. I1crroJIh3YH YCJIOBHe 3a,n;aqH 6.13.15, HaftTH FT(t) H h(t), r,n;e
T= ~.
6.13.35. COBMecTHoe pacrrpe,n;eJIeHHe c. B. X H Y 3a,n;aHO rrJIOTHOCThIO pac-
rrpe,n;eJIeHHHBepoHTHocTeti
f(X'Y)={~' eCJIHO~x~I,O~y~7r,
0, B rrpOTHBHOM CJIyqae.
HatiTH rrJIOTHOCTh pacrrpe,n;eJIeHHH BepoHTHocTeti c. B. Z = X + Y.
425
6.13.36. CJIyqail:Hble BeJIHqHHbI X H Y He3aBHCHMbI H O,Il;HHaKOBO pacrrpe,Il;e- JIeHbI rro 3aKOHY N(O, 1). Hail:TH pacrrpe,Il;eJIeHHe c. B. Z = X 2 +y2.
6.13.37. MCrrOJIb3YH YCJIOBHe 3a,IJ;aqH 6.13.18, Hail:TH <PYHKII.HIO H rrJIOTHOCTh pacrrpe,n;eJIeHHH BepoHTHocTeil: c. B.
y
Puc. 101
6.13.38. CJIyqail:HaH TOqKa |
(X, Y) pacrrpe,Il;eJIeHa paBHOMepHO B KBa,IJ;pa- |
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Te co CTOPOHOil: 1 (pHC. |
101). Hail:TH 3aKOH |
pacrrpe,Il;eJIeHHH c. B. |
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S=XY. |
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°::::; x ::::; 1, °::::; y ::::; |
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(fXY(X, Y) = |
{ ~: |
1; ) |
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B rrpoTHBHOM CJIyqae. |
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6.13.39. Hail:TH <PYHKII.HIO pacrrpe,Il;eJIeHHH H rrJIOTHOCTb pacrrpe,Il;eJIeHHH BePOHTHocTeil: c. B. Z = max{X, Y}, r,Il;e X H Y - HerrpepbIBHble, He3aBHCHMble, paBHOMepHO pacrrpe,Il;eJIeHHble Ha OTpe3Ke [0,1] CJIyqail:Hble BeJIHqHHbI.
6.13.40. 3a,IJ;aHbI rrJIOTHOCTH pacrrpe,n;eJIeHHH BepoHTHocTeil: ,Il;ByX He3aBHCH-
MbIX C.B. X H Y:
fx(x) = {0,25X, |
x |
E [1,3], |
H Jy(y) = {01" Y E [1,2], |
0, |
x |
f. [1,3] |
Y f. [1,2]. |
Hail:TH rrJIOTHOCTb pacrrpe,Il;eJIeHHH CYMMbI Z
6.13.41. BpeMH T1 , B TeqeHHe KOTOPOro KJIHeHT OlKH,Il;aeT 06cJIY:lKHBaHHH, H caMO BpeMH 06CJIJ:lI(HBaHHH T2 - ,Il;Be He3aBHCHMble HerrpepbIBHble CJIyqail:Hble BeJIHqHHbI, HMeIOW;He rrOKa3aTeJIbHOe pacrrpe,Il;eJIeHHe C rrapaMeTpaMH >'1 H >'2 ~ >'1 COOTBeTCTBeHHO. Hail:TH rrJIOTHOCTb pacrrpe,n;eJIeHHH BepOHTHocTeil: o6w;ero BpeMeHH T = T1 + T2 , npoBe,Il;eHHOrO KJIHeHTOM B CHCTeMe MaccoBoro 06CJIY:lKHBaHHH.
6.13.42. He3aBHcHMble CJIyqail:Hble BeJIHqHHbI X H Y pacrrpe,n;eJIeHbI rro O,Il;- HOMY H TOMY :lKe:
a) rroKa3aTeJIbHoMY 3aKOHY C rrapaMeTpoM >.
6) paBHOMepHOMY 3aKOHY B HHTepBarre Hail:TH M(Z), r,Il;e Z = X + Y.
426
KOHTponbHble BOnpOCbl lit 60nee CnmKHble 3aACIHlltR
6.13.43. IhBecTHo, 'ITOC. B. X '" N(a, a). IloKa3aTb, 'ITOc. B. Y, CBH3aHHaH CO c. B. X JUfHeitHoit <PYHKII.HOHa.J1bHOit 3aBHCHMOCTbIO Y = kX + b
(k, b E 1R), TaIOKe paCIIpe,n,e.JIeHa no HOPMa.J1bHOMY 3aKOHY. l..{eMY
paBHbI M(Y) H a(Y)?
6.13.44. IhBecTHo, 'ITOC.B. X", R[-1,2]. HaitTH IIJIOTHOCTb pacIIpe,n;eJIeHHH c. B. Y = IXI. (HaitTH fx(x) B HHTepBa.J1e (-1; 1), a 3aTeM B
HHTepBa.J1e (1; 2).)
6.13.45. IlycTb F(x) - <PYHKII.HH pacIIpe,n;eJIeHHH HeilpepbIBHoit cJIY'laitHOit Be.JIH'lHHbI X. HaiiTH <PYHKII.HH pacIIpe,n;eJIeHHH CJIY'laiiHbIX
BeJIH'lHHY = 16X2 - 9, |
Z = e-3X , BbIpa3HB HX 'lepe3<PYHKII.HIO |
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paCIIpe,n,eJIeHHH c. B. X. |
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6.13.46. CJIY'laiiHaHBe.JIH'lHHaX |
HMeeT nOKa3aTeJIbHOe pacnpe,n;eJIeHHe C |
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nJIOTHOCTbIO |
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f(x) = {e-X , |
x ~ 0, |
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0, |
x < o. |
By,n;eT JIH HMeTb IIOKa3aTeJIbHOe pacIIpe,n;eJIeHHe c. B. Y = X + I?
HaitTH <PYHKIJ.HIO IIJIOTHOCTH paCIIpe,n,eJIeHHH CJIY'laiiHbIXBeJIH'lHH
Y = 2ex + 6, Z = X2.
6.13.47.By.n;yT JIH He3aBHCHMbI c. B. X H Y, eCJIH HX COBMeCTHoe pacIIpe- ,n;e.JIeHHe f (x, Y) HBJIHeTCH paBHOMepHbIM:
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a) B 06JIacTH D = {(x, y): |
0::::; |
x ::::; |
1, |
0 ::::; y ::::; 1 - |
x}; |
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6) B 06JIaCTH F = {(x,y): |
0::::; |
x::::; |
1, |
0::::; y::::; 2}? |
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6.13.48. |
lICilOJIb3YH YCJIOBHe 3a,n;a'lH 6.13.35, HaiiTH Fz(z) |
H |
fz(z), r,n;e |
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Y |
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Z= X. |
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6.13.49. |
1I3BecTHO, 'ITOX H Y - He3aBHCHMbIe CJIY'laiiHbIeBeJIH'lHHbI,HMe- |
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IOW;H~ HOpMa.J1bHOe pacnpe,n;eJIeHHe: X |
'" N(O; 1), |
Y |
'" N(O; 1). |
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HaitTH pacIIpe,n,eJIeHHe c. B. Z = JX2 + y2. |
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6.13.50. HaitTH IIJIOTHOCTb paCIIpe,n;eJIeHHH CYMMbI ,n;BYX paBHOMepHO pacnpe,n;e.JIeHHbIX Ha OTpe3Ke [-1; 1] He3aBHCHMbIX CJIY'laiiHbIXBe.JIH- 'lHHX H Y. l..{eMY paBHa Fx+y(z)?
6.13.51. CTy,n;eHT IIpH IIoe3,n;Ke B HHCTHTYT nOJIb3yeTcH MeTpo H TpoJIJIeii6YCOM. Iloe3,n; MeTpo npHxo,n;HTCH O:ll{H,n;aTb He 60JIee 3 MHHYT, a O:>KH,n;aHHe TpoJIJIeit6yca He 60JIee 8 MHHYT. C'lHTaeTCHBpeMH O:>KH- ,n;aHHH noe3,n;a B MeTpo H TpoJIJIeii6yca HeilpepbIBHbIMH c. B. X H
Y, paCIIpe,n,eJIeHHbIMH paBHOMepHO COOTBeTCTBeHHO B npOMe:>KYTKax [0; 3] H [0; 8], HaitTH nJIOTHOCTb pacnpe,n;eJIeHHH BepoHTHocTeit o6w;ero BpeMeHH O:>KH,n;aHHH Z = X + Y.
427
§ 14. nPEAEl1bHblE TEOPEMbl TEOPlt1lt1
BEPOflTHOCTEt1
Bo MHorHX 3a,n;a'laxTeopHH BepOHTHocTeft H3Y'IaIOTCHCJIY'IaftHbIeBeJIH'IHHbI,
HBJIHIOIIJ;HeCH CYMMaMH 60JIbillOro '1HCJIa,lI;PyrHx CJIY'IaftHbIxBeJIH'IHH,T. e. 3aBHCHIIJ;He OT 60JIbillOrO '1HCJIaCJIY'IaftHbIX<l>aKTopOB. Orrpe,ll;eJIeHHbIe CBoftcTBa TaKHX CJIY'IaftHbIxBeJIH'IHHorrHCbIBaIOTCH cOBoKyrrHocTbIO TaK Ha3bIBaeMbIX npeiJeJl'b'lt'bl.X meopeM, KOTopbIe, B CBOIO O'lepe,ll;b,pa36HBaIOTCH Ha ,lI;Be rpyrrrrbI TeopeM - «3aKOH
60JIbIIIHX '1HCeJI»H «rreHTpaJIbHaH rrpe,ll;eJIbHaH TeopeMa».
rpyrrrra TeopeM, Ha3bIBaeMbIX «3aKOHOM 60JIbIIIHX '1HCeJI», YCTaHaBJIHBaeT ycTOft'lHBOCTb CPe,ll;HHX 3Ha'leHHft:rrpH 60JIbiliOM '1HCJIeHcrrbITaHHft HX Cpe,ll;HHft pe3YJIbTaT MOlKeT 6bITb rrpe,ll;CKa3aH C ,lI;OCTaTO'lHoftTO'lHOCTbIO.,I4>yraH rpyrrrra TeopeM, Ha3bIBaeMaH «rreHTpaJIbHoft rrpe,ll;eJIbHoft TeopeMoft», YCTaHaBJIHBaeT, 'ITOrrpH ,lI;OCTaTO'lHO06IIJ;HX H eCTeCTBeHHbIX YCJIOBHHX 3aKOH pacrrpe,ll;eJIeHHH CYMMbI 60JIbIIIoro '1HCJIaCJIY'laftHbIxBeJIH'IHH6JIH30K K HOPMaJIbHOMY.
HepaBeHCTBO Ye6blWeBa ~ 3aKOH 60nbW~X ... ~cen
TeopeMa 6.9 (HepaBeHcTBo MapKoBa). Ecn~ c. B. X np~H~MaeT HeOTp~Lla
TenbHble aHa'leH~ft ~ ~MeeT MaTeMaT~'leCKOe O)K~AaH~e M(X), TO Anft mo6oro c > 0 ~MeeT MeCTO HepaBeHCTBO:
M(X)
P{X ~ c} ~ - c - '
2ho HepaBeHCTBO, O'leBH,lI;HO,paBHOCHJIbHO CJIe,ll;yIOIIJ;eMY
M(X)
P{X < c} ~ 1 - - c - '
TeopeMa 6.10 (HepaBeHcTBo "Ie6blweBa). Ecn~ c. B. X ~MeeT MaTeMan'leCKOe
O)K~AaH~e M(X) ~ A~cnepc~1O D(X), TO Anft nlO6oro c > 0 ~MeeT MeCTO HepaBeHCTBO
P{IX - M(X)I ~ c} ~ DC;). c
HepaBeHCTBO qe6bIIIIeBa MOlKHO 3aMeHHTb PaBHOCHJIbHbIM |
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P{IX - M(X)I < c} ~ 1 _ D(.;). |
(14.1) |
c
HepaBeHCTBa MapKoBa H qe6bIIIIeBa MOlKHO HCrrOJIb30BaTb ,!J,J1H OrreHKH BepoHTHocTeft C06bITHft, CBH3aHHbIX co CJIY'IaftHoftBeJIH'IHHOft,pacrrpe,ll;eJIeHHe KOTOPOit HeH3BeCTHO.
428
ECJIH C. B. X = m HMeeT 6UHOMUaJlMWe paCnpeOeJleHUe C MaTeMaTH'IecKHM O»m,LlaHHeM M(X} = a = np H ,LlHCrrepcHeil: D(X} = npq, HepaBeHCTBO qe6hllleBa
P{lm - npl < c} ~ 1 _ n~q.
c
~H omHOCUmeJlbHOil "tacmom'bl. r;: C06hlTHH A B n He3aBHCHMhlX HCrrhlTaHH-
jlX, B K8JK,!lOM H3 KOTOPhlX OHO MOlKeT rrpoH30il:TH C BepOHTHOCTbIO p, HepaBeHCTBO qe6hllleBa rrpHHHMaeT BH,!l
P {I r;: - pi < c} ~ 1 - pq2· |
(14.2) |
nc
OCHOBHOil: <P0PMOil: «3aKOHa 60JIbllHX '1HCeJI» C'IHTaeTCH
TeopeMa 6.11 ('"Ie6blwea). ECI1II1 clly'la~Hble BeI1 111'1111 Hbl Xl, X2, ... , X n , ... He3aBIIICIIIMbi III CYUlecTByeT TaKOe 'IIIICIlOC> 0, 'ITOD(Xi) ~ c (i = 1,2,3, ... ), TO Alll'l11I060ro c > 0 BblnOIlHl'IeTCl'IHepaBeHCTBO
(14.3)
1:13 HepaBeHCTBa (14.3) CJIe.n;yeT rrpe,!leJIbHOe paBeHCTBO
TeopeMa qe6hllleBa rrOKa3b1BaeT, 'ITOCpeOHee apU(/jMemU"teCICOe 60JlbUJ,OZO "tU- c.!!a cJly"tailH'bI.X 6eJlU"tUH ICaIC YZOOHO MaJlO omJlu"taemCJJ (C BepOHTHOCTbIO 6JIH3KOil:
K 1) om cpeoHezo apu(/jMemu"tecICozo ux MameMamu"tecICUX O'JICuoaHuil.
CnElACTBlile 6.1. ECIlIII c. B. Xl, X 2 , ••• , X n , ... He3aBIIICIIIMbi III OAIIIHaKOBO pacnpeAelleHbl, C MaTeMaTIII'IeCKIIIM0)f(IIIAaHllleMaIllAlIIcnepCllle~q2.TOAlll.lIll060ro
13:>0
CnElACTBlile 6.2 (TeopeMa 6epHynnlll). ECIlIII B yCIlOBlIIl'IX cxeMbl 6epHYllllIII Be-
POl'lTHOCTbHacTynlleHlIIl'IC06blTIIIl'IA B OAHOM onblTe paBHa p, 'IIIICIlOHacTynlleHIII~
3Toro C06blTIIIl'I nplll n He3aBIIICIIIMbiX IIIcnblTaHlIIl'IXpaBHO m, TO Alll'll1Io60ro c > 0
lim P {I r;: - pi < c} = 1.
n-+oo
429