6.12.66. cI>YHKIJ;HjI pacnpe,n:eJIeH~jI CHCTeMbI (X, Y) HenpepbIBHbIX c. B. 3a-
,n:aHa B BH,n:e |
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x < °HJIH Y < 0, |
a, |
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F(x, y) = { 0,5(sin x + sin y - sin(x + y», |
°::;; x ::;; ~, |
0::;; y ::;; ~, |
1, |
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7r |
7r |
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x> 2 H Y |
= 2· |
HaihH: |
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a) P{(X, Y) ED}, r,n:e |
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6" |
::;;x::;; 3' |
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7r |
::;;y::;; |
7r) |
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D= (x,y): |
[{3 |
2' |
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{~ |
::;; x::;; |
~ |
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1 |
3 |
2' |
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0::;; y::;;~.
6)fxy(x,y).
6.12.67.,IJ,BYMepHblit CJIY'faitHblitBeKTop (X, Y) paBHoMepHo pacnpe,n:eJIeH
(f(x, y) = c) B 06JIaCTH D = {(x, y): Ixl + Iyl ::;; I} (BHe 06JIacTH f(x,y) = 0). HaitTH:
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a) fxy(x, y); |
6) fx(x) H fy(y). |
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3aBHcHMbI JIH CJIY'faitHbleBeJIH'fHHbIX H Y? |
6.12.68. |
3a,n:aaa <PYHKIJ;HjI f(x, y) |
= K . e-(ax2+bx+cy2). KaKHM YCJIOBHjlM |
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,n:OJDKHbI y,n:oBJIeTBOpjlTb 'fHCJIaa, b H C ,n:JIjI TOro, 'fTo6bI9Ta <PYHK- |
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IJ;HjI MOrJIa 6bI 6bITb nJIOTHOCTblO pacnpe,n:eJIeHHjI BepOjlTHOcTeit? |
6.12.69. |
nYCTb CJIY'faitHbleBeJIH'fHHbI X H Y He3aBHCHMbI H HOPMaJIbHO |
pacnpe,n:eJIeHbI: X '" N(O; 1) H Y '" N(O; 1). HaitTH: a) COBMeCTHYIO nJIOTHOCTb pacnpe,n:eJIeHHjI fxy(x,y);
6)P{(X,Y) ED}, r,n:e D = {(x,y): 2::;; Jx2 +y2 < 3}.
6.12.70.HenpepbIBHajl c. B. X""R[-2; 4], a HenpepbIBHM c. B. Y ""N( -1; 2). 1I3BecTHo, 'ITOrxy=-0,5. HaitTH M(XY).
6.12.71.3a,n:aHa HenpepbIBHM c. B. X C nJIOTHOCTblO pacnpe,n:eJIeHHjI Bepo-
jlTHOcTeit fx(x) = A . e- X 2 • 1I3BecTHo, 'ITO,n:pyrM C. B. Y CBjI3aHa CO C. B. X paBeHCTBOM Y = X2. qeMY paBeH K09<P<PHIJ;HeHT KOppeJIjlIJ;HH C. B. X H Y? KaKoit BbIBO,n: CJIe.n:yeT H3 nOJIY'feHHOrO pe3YJIbTaTa?
§ 13. (J)YHKU.LiILiI Cl1Y....A~HbIX BEl1Li1 .... LilH
(J)YHK4111111 OAHOM CI1Y'-laMHOMBellIII'-III Hbl
IIYCTb paCCMaTpHBaIOTCS: ,ll;Be CJIY'IafiHbleBeJIH'IHHblX H Y, CBS:3aHHble <PYHK-
IJ;HOHaJIbHOfi 3aBHCHMOCTbIO
Y = rp(X).
ECJIH X - .n:HCKpeTHaJI C. B., 3aKOH pacnpe.n:eJIeHHH KOTOPOi!: Onpe.n:eJIHeTCH <POp- r.lYJIoi!: Pi = P{X = Xi}, i = 1,2,3, ... , TO C. B. Y TaKlKe .n:HCKpeTHa, a ee 3aKOH pacnpe.n:eJIeHHH BbIpalKaeTCH <P0PMYJIOi!: Pi = P{Y = Y;}, i = 1,2,3, ... , r.n:e Yi = ip(x;),
p{Y = y;} = P{X = Xi}.
MaTeMaTH'IeCKOeOlKH.n:aHHe H .n:HcnepCHH C. B. Y onpe.n:eJIHIOTCH COOTBeTCTBeH-
HO paaeHCTBaMH
M(Y) = M(ip(X)) = LYiPi = L ip(Xi)Pi
i i
D(Y) = D(ip(X)) = L(Yi - ay)2pi = L(ip(X;) - ay)2pi'
i
me ay = M(Y).
ECJ1H X |
- HenpepblBHaJI C. B. C nJIOTHOCTbIO pacnpe.n:eJIeHHH f (X) H eCJIH |
y = ip(X) - |
.n:H<p<pepeHD:HPyeMaJI H MOHOTOHHaJI <PYHKD:HH, TO llJIOTHOCTb pacnpe- |
p;eJIeHHH g(y) C. B. Y = ip(X) BblpalKaeTCH <P0PMYJIOi!: |
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g(y) = f(1f;(y)) ·11f;'(y)l, |
rp;e 1f;(y) = ip-1(y) = X - <PYHKD:HH, 06paTHaH <PYHKD:HH Y = ip(X) (::na <PYHKD:HH cY11l,eCTByeT B CHJIY MOHOTOHHOCTH ip(X)).
ECJ1H <PYHKD:HH Y = ip(X) HeMOHOTOHHaJI, TO 'IHCJ10BaHnpHMaJI pa36HBaeTCH
Ha n npOMelKYTKOB MOHOTOHHOCTH H 06paTHaJI <PYHKD:HH 1f;i(Y) HaxO.n:HTCH Ha KalK.n:OM H3 HHXj llJIOTHOCTb pacnpe.n:eJIeHHH g(y) C. B. Y = ip(X) onpe.n:eJIHeTCH B aTOM
CJ1Y'Iaeno <popMYJIe
n
g(y) = L f(1f;i(Y)) ·11f;;(y)l· i=1
,I1;.rrH HaxolK.n:eHHH MaTeMaTH'IeCKOrOOlKH.n:aHHH H .n:HcnepCHH C. B. Y = ip(X)
Heo6H3aTeJIbHO HaxO.n:HTb 3aKOH ee pacnpe.n:eJIeHHHj MOlKHO BOCnOJIb30BaTbCH <popMYJIaMH
M(Y) = M(ip(x)) = J00ip(X)· f(x)dx,
, = |
= |
- 00 |
J |
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00 |
D(Y) D(ip(x)) |
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(ip(X) - ay)2 f(x) dx. |
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- 00 |
::? IIycTb paccMaTpHBaeTCH CHCTeMa .n:BYX CJ1Y'Iai!:HblxBeJIH'IHH(X, Y). ECJ1H KalK.n:oi!: nape (X, y) B03MOlKHblX 3Ha'leHHi!:C. B. X H Y COOTBeTcTByeT O.n:HO B03MOlKHoe 3Ha'leHHez = ip(X, y) (Haxo.n:HMoe no onpe.n:eJIeHHOMY 3aKOHY) C. B. Z, TO Z Ha3b1BalOT ljjyH,7C'Il,Ueit iJayx c.n,y"l,aitH,'btx apeYMeH,mOa X H Y:
Z = ip(X, Y).
)JjrH <PYHKD;HH .n:BYX (H 6oJIee) aprYMeHTOB y.n:06Hee CHaqaJIa HaxO.n:HTb ee
<PYHKD;HIO pacnpe.n:eJIeHHH G(z), a 3aTeM - |
nJIOTHOCTb pacnpe.n:eJIeHHH g(z): |
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g(z) = G'(z). |
ECJIH (X, Y) - CHCTeMa .n:HCKpeTHbIX C. B., TO |
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G(z) = P{Z < z} = |
Pij; |
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i,j:<P(Xi ,Yj )<z |
eCJIH (X, Y) - |
CHCTeMa HenpepbIBHblX C. B., TO |
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G(z)=P{Z<z}= !!!(x,y)dXdy, |
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D. |
r.n:e !(x, y) - |
nJIOTHOCTb pacnpe.n:eJIeHHH CHCTeMbI (X, Y); |
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Dz = ((x,y): cp(x,y) < z}. |
BalKHOe ,l.l;JIH npaKTHKH 3HaqeHHe HMeeT 3a.n:aqa onpe.n:eJIeHHH 3aKOHa pacnpe-
.n:eJIeHHH CYMMbI .D:BYX CJIyqaitHblX BeJIHqHH: Z = X + Y.
ctlYHKD;HH pacnpe.n:eJIeHHH C. B. Z MOlKeT 6b1Tb Hait.n:eHa no <popMYJIe
Gz(z) = 1(J~(X'y) dY) dx,
- 00 - 00
r.n:e !(x, y) - nJIOTHOCTb pacnpe.n:eJIeHHH CHCTeMbI (X, Y). IlJIOTHOCTb pacnpe.n:e- JIeHHH CYMMbI .n:BYX CJIyqaitHblx BeJIHqHH BbIpalKaeTCH <P0PMYJIOit
g(z)= !00!(x,z-x)dx |
(HJIH: g(z)= !00 |
!(z-y,y)dy). |
- 00 |
- 00 |
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Oco6eHHO BalKeH CJIyqait, Kor.n:a CJIyqaitHble BeJIHqHHbl X H Y He3aBHCHMbI. |
Tor.n:a !(x,y) = /1 (x)h(Y) H |
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00 |
00 |
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g(z)= !/1(x)h(z-x)dx |
(HJIH: g(z) = ! /1(z - Y)h(Y) dy), |
- 00 |
- 00 |
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r.n:e /1 (x) H h (y) - nJIOTHOCTH pacnpe.n:eJIeHHH C. B. X H Y COOTBeTCTBeHHO. ECJIH B03MOlKHble 3HaqeHHH aprYMeHTOB HeOTpHD;aTeJIbHbI, TO g(x) Haxo.n:HM no <popMYJIe
z
g(z) = !/1(x)h(z-X)dX. o
IIoCJIe.n:HIOIO <P0PMYJIY Ha3b1BaIOT ifjoP.My/IOit CBepmICU HJIH ifjoP.MyJ!oit II:O.Mno3U'4UU iJBYX pacnpeiJeJ!e'ltuit, a CPYHKD;HIO g(z) - CBepmll:oit ifjy'ltll:'4uit /1(x) H h(Y);
3aKOH pacnpe.n:eJIeHHH CYMMbI Z = X +Y .n:BYX He3aBHCHMblX C. B. Ha3b1BaIOT II:O.M- n03U'4Ueit (CBepTKoit) 3aKOHOB pacnpe.n:eJIeHHH CJIaraeMblX.
6.13.1.,il;HcKpeTHruI c. B. X 3a,rr,aHa CBOHM P.H)WM pacIIpe,n;eJIeHH.H
HaitTH:
a) pacIIpe,n;eJIeHHe c. B. Y = -3X2 + 1;
6) 3aKOH pacIIpe,n;eJIeHH.H c. B. T = cos (~X) - 1, a TaIOKe M(T)
H D(T). |
CJIe.n;yIOID;He 3Ha'IeHH.H: Yl = |
o a) CJIY'Iail:HruI BeJIH'IHHa Y "pHHHMaeT |
= -3(-2)2+1 = -11, Y2 = -3(-1)2+ 1 = -2, Y3 = I, Y4 = -2·1+1 = -2, |
Y5 = -3·4 + 1 = -11, Y6 = -3.32 + 1 = -26, Y7 = |
-3.42 + 1 = |
-47. |
BepO.HTHOCTH, 9THX 3Ha'IeHHil: TaKHe )Ke, KaK |
H y c. B. |
X, T. e. PI = |
0,05, |
P2 = 0,10 H T.,n;. 3aKOH paCIIpe,n;eJIeHH.H c. B. Y MO)KHO 3aIIHCaTb B BH,n;e
llJIll (y'IHTblBruI,'ITOP{Y=-11}=Pl +P5=O,05+0,15=O,20, P{Y=-2}=
=0,10+0,25=0,35) B 60JIee KOMilaKTHOM BH,n;e '
6) AHaJIOrH'IHOIIOJIY'IaeM3aKOH pacIIpe,n;eJIeHH.H c. B. T = cos 1'(;- - 1:
("POBePKa: t Pi = 1).
,=1
M(T) = -2·0,20 + (-1) ·0,55 + 0·0,25 = -0,95;
D(T) = [M(T2)_(M(T))2) = (-2)2.0,20+(-1)2.0,55+02.0,25-(-0,95)2 = = 1,35 - 0,9025 = 0,4475. •
6.13.2.,il;HcKpeTHruI c. B. X 3a,rr,aHa 3aKOHOM pacIIpe,n;eJIeHH.H
Xi |
0 |
1'( |
1'( |
1'( |
1'( |
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6 |
'4 |
3 |
2' |
1'( |
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Pi |
0,15 |
0,15 |
0,25 |
0,30 |
0,10 |
0,05 |
HaitTH:
a) 3aKOH pacIIpe,n;eJIeHH.H c. B. Y = 4 sin2 X;
6)M(Y), D(Y), a(Y).
6.13.3.,il;HcKpeTHruI c. B. X 3a,rr,aHa Ta6JIHlJ;eil: pacIIpe,n;eJIeHH.H
= X2.
HaitTH:
a) 3aKOHbI pacnpe)l,eJIeHH.H C. B. Y = ~IXI, Z = x - M(X);
6)M(Y), D(Y), a(Y), M(Z), D(Z).
6.13.4.IIJIOTHOCTb pacnpe)l,eJIeHH.H BepO.HTHOcTeit HenpepbIBHoit c. B. X
HMeeT BH)l,
1 |
x2 |
< x < 00. |
f(x) = . m=e- s |
, -00 |
2· y |
21T |
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HaitTH nJIOTHOCTb pacnpe)l,eJIeHH.H c. B. Y
<) OTMeTHM, qTO 3a,n,aHHa.H H. c. B. X pacnpe)l,eJIeHa no HOPMaJIbHOMY 3aKOHY: X '" N(O; 2). PeIIIHM 3a,n,aqy )l,BYM.H cnoc06aMH:
1) npe)l,BapHTeJIbHO Hait)l,.H <PYHKIIHIO pacnpe)l,eJIeHH.H G(y) C.B. Y, a 3a-
TeM, BOCnOJIb30BaBIIIHCb paBeHcTBoM g(y) = G'(y), H HCKOMYIO nJIOTHOCTb pacnpe)l,eJIeHH.H g(y) c. B. Y;
2) HCnOJIb3Y.H <P0PMYJIY g(y) = f("p(y)) ·1"p'(y)l· |
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Cnoco6 1. B03MO>KHble 3HaqeHH.H CJIyqaitHblx BeJIHqHH X |
H Y CB.H3aHbI |
3aBHCHMOCTbIO y = x 2 • |
TaK KaK c. B. Y He npHHHMaeT OTpHIIaTeJIbHblX 3Ha- |
qeHHit, TO G(y) = P{Y < y} = 0)l,JI.H Y :::;; O. IIycTb Y > |
O. Tor)l,a |
G(y) = P{Y < y} = p{X2 < y} = P{IXI < y'y}= |
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y'Y |
2 |
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y'Y 2 |
= P{-y'y < X < y'y}= J |
_1_e- s dx = ~. Je- Xs dx, |
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X |
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-y'Y2~ |
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2~ 0 |
T.e. |
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y'Y |
2 |
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G(y) = { k·!e- Xs dx, |
npH y > |
0, |
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0, |
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npH y:::;; O. |
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1 |
_ (y'Y)2 |
, |
1 |
_1l. |
1 |
g(y) = G'(y) = { |
~e |
s |
. (.jY) |
, = { ~e s· 2.jY' |
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0, |
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0, |
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T.e. |
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.~e-~, npHY>O, |
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g(y) = { 2y 21TY |
npH y:::;; O. |
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0, |
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3a.Me"ta'Hue. Bblp8JKeHHe )l,JI.H <PYHKIIHH |
pacnpe)l,eJIeHH.H |
c. B. Y MO:>KHO |
3anHcaTb HHaqe: |
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G(y) = P{Y < y} = ... = P{-y'y < X < y'y}=
= P{X < y'y}- P{X:::;; -y'y}= P{X < y'y}- P{X < -y'y}=
= Fx(y'y) - Fx(-y'y).
f('l/J(y)) . I'l/J'(y) I, lIOJIY'IaeM:
)J,l1cPcPepeHIJ;HPYH lIOJIY'IeHHOepaBeHCTBO lIO y, lIOJIY'IaeM |
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g(y) = G~(y) = |
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=Fx(..;y) . 2~ + Fx( -..;y) . 2~ = |
2~(fX(..;y) + fx( -..;y)) = |
1 |
(1 |
_(v'Y)2 |
1 |
_(-v'Y)2) |
1_1l. |
= 2JY· |
2J27f e |
8 + 2J27f e |
8 |
= 2J2Y7f e 8, |
T. e. TaKoit )Ke pe3YJIbTaT |
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g(y) = {_2..lftifY_e-~, |
y > 0, |
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0, |
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y ~ o. |
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Cnoco62. B HHTepBaJIe (-00; 00) cPYHKII,HH y = |
x2 He MOHOTOHHa. Pa30- |
6beM 9TOT HHTepBaJI Ha ,lJ;Ba HHTepBaJIa (-00; 0) H (0; 00), B KOTOPbIX cPYHKIJ;I1H Y = x 2 MOHOTOHHa. Ha HHTepBaJIe (-00; 0) 06paTHaH cPYHKII,HH K cPYHK-
IJ;I1H Y = |
x 2 |
eCTb |
Xl = '1/h(Y) |
= -JY, Ha HHTepBaJIe |
(0;00) HMeeM X2 = |
= 'l/J2(y) |
= |
JY. MCKOMYIO lIJIOTHOCTb |
paclIpe,lJ;eJIeHHH |
Hait,lJ;eM, HClIOJIb3YH |
paBeHCTBO |
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g(y) |
= fX('l/JI(Y)) |
. l'l/JUy) 1 + fx('l/J~(y)) ·1'l/J~(y)l· |
TaK KaK |
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1'l/J~(y)1 = |
1(-..;yY 1 = 1-2~1 = |
2~' |
H 1'l/J~(y)1 = |
1(..;yY 1 = |
12~1 = 2~' |
TO |
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1 |
_(v'Y)2 |
1 |
1 |
_(-v'Y)2 |
1 |
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1_1l. |
g(y)=--e |
8 · -- + -- e |
8 |
· -- = -- e 8. |
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2J27f |
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2JY |
2J27f |
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2JY |
2J27fY |
TaK KaK y = |
x2 , X |
E lR = |
(-00; 00), TO y > 0, lI09TOMY g(y) = 0 lIpH Y ~ o. |
lfTaK, |
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~e-~, |
y > 0, |
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• |
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g(y) = { 2y 27fY |
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y ~ o. |
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0, |
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6.13.5. |
HelIpepbIBHaH CJIY'IaitHaHBeJIH'IHHaX HMeeT paBHoMepHoe pac- |
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1 |
lIpH X E [1; 3],) |
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lIpe,lJ;eJIeHHe Ha OTpe3Ke [ 1; 3) |
( T. e. f(x) |
= { |
-2 , |
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. |
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0, |
lIpH X ~ [1; 3) |
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HaitTH lIJIOTHOCTb paClIpe,lJ;eJIeHHH cPYHKII,HH: |
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a)Y=2X; |
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6)Y=X2. |
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Q a) <l>YHKII,HH y = 2x Ha OTpe3Ke B03MO)KHbIX 3Ha'leHHitc. B. X MOHOTOHHa. n09TOMY 06paTHaH eit cPYHKII,HH X = 'l/J(y) = ~y CYIIJ;eCTByeT H TaK)Ke MOHO-
TOHHa Ha OTpe3Ke [2; 6) (TaK KaK 1 ~ x ~ 3, TO 2 ~ y = 2x ~ 6). MClIOJIb3YH CPOPMYJIY g(y) =
g(y) ~{t1Gy)'1 |
lIpH 2 |
~ Y ~ 6, |
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lIpH Y f/. [2; 6).
Kax BU,n;UM, H. C. B. Y UMeeT TaJOKe pasHoMepHOe paclIpe,n;eJIeHUe, T. e.
Y.....,R[2;6].
6) Cl>YHKIJ;UH y = x 2 TO:>Ke MOHOTOHHa Ha OTpe3Ke [1; 3] U II09TOMY UMeeT o6paTHYIO <PYHKIJ;UIO x = 'I/J(y) = Vfj, KOTOPaH TaJOKe MOHOTOHHa Ha OTpe3Ke
[1;9]. OTcIO,n;a x' = 'I/J'(y) = 2~' I'I/J'(y) I= 2~ U, CJIe,n;OBaTeJIbHO,
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g(y) = {~. 2~' |
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1 ~ y ~9, |
• |
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0, |
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y~[1;9]. |
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6.13.6. |
IhBecTHo, 'ITOIIJIOTHOCTb paclIpe,n;eJIeHUH C. B. X UMeeT BU,n; |
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cosx, |
x E (O;~), |
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f(x) = { |
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X ~ (O;~). |
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0, |
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Hathu: |
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a) IIJIOTHOCTb paclIpe,n;eJIeHUH C. B. Y = X2; |
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6) '1UCJIOBbIexapaxTepucTUKU M(Y) U D(Y). |
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6.13.7. |
HelIpepbIBHaH C.B. X (0 < x < 00) UMeeT IIJIOTHOCTb paclIpe,n;eJIe- |
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HUH BepOHTHocTei;!: f(x) U <PYHKIJ;UIO paClIpe,n;eJIeHUH BepOHTHocTei;!: |
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F(x). ,LLrrH C. B. Y = InX Hai;!:Tu IIJIOTHOCTb paClIpe,n;eJIeHUH Bepo- |
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HTHocTei;!: g(y) U <PYHKIJ;UIO paclIpe,n;eJIeHUH BepOHTHocTei;!: G(y). |
6.13.8. |
CJIY'Iai;!:HaHBeJIU'IUHaX UMeeT IIJIOTHOCTb paclIpe,n;eJIeHUH |
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f(x) = {3X2 , |
x E [0; 1], |
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0, |
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x ~ [0; 1]. |
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Hai;!:Tu IIJIOTHOCTb paclIpe,n;eJIeHUH c. B. Y = IX - |
21. |
6.13.9. |
3a.n;aHa IIJIOTHOCTb paclIpe,n;eJIeHUH H. c. B. X: |
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fx(x) = {e- x , |
x ~ 0, |
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0, |
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x < 0. |
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Hai;!:Tu Fy(y) U Jy(y), eCJIU Y |
= e- x . |
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6.13.10. CJIY'Iai;!:HaHBeJIU'IUHaUMeeT IIJIOTHOCTb paclIpe,n;eJIeHUH |
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X-2 |
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x E [2; 4], |
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fx(x) = { - 2 - ' |
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0, |
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x ~ [2;4]. |
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Hai;!:TU: |
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a) IIJIOTHOCTb paclIpe,n;eJIeHUH gy(y); |
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6) MaTeMaTU'IeCKOeO:>Ku,n;aHue M(Y) U ,n;UCIIepcuIO D(Y) c. B. Y, |
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KOTOPaH IIpe,n;CTaBJIHeT co6oi;!: IIJIO~a.n;b Kpyra pa.n;uyca X. |
6.13.11. |
CJIY'Iai;!:HaHBeJIU'IUHaX UMeeT IIJIOTHOCTb paclIpe,n;eJIeHUH |
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f(x) = {1' IIpU X E [1; 2], |
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0, |
IIpU x ~ [1; 2]. |
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,I:4>yrruI C. B. Y CBH3aHa C X <PYHKIUIOHaJIbHOit 3aBHCHMOCTbIO Y =
=2X3 +1. RaitTH MaTeMaTH'.JeCKoeO)KH,ll;aHHe H ,ll;HCnepCHIO c. B. Y: a) He HaxO,IVI nJIOTHOCTH gy(y);
6) Hait,IVI npe,ll;BapHTeJIbHO nJIOTHOCTb gy (y ).
6.13.12. COBMecTHoe pacnpe,ll;eJIeHHe,ll;. c. B. X H Y 3a,ll;aHO Ta6JIHn;eit
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X\Y |
0 |
4 |
9 |
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1 |
0,20 |
0,15 |
0,10 |
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4 |
0,30 |
0,20 |
0,05 |
o |
OnHcaTb 3aKOH pacnpe,ll;eJIeHHH c. B. Z = X - /Y. |
3anHweM 3aKOHbI pacnpe,ll;eJIeHHH COCTaBJIHIOIIJ;HX X H Y: |
3aKOH pacnpe,ll;eJIeHHH c. B. /Y HMeeT BH,ll;
CJIyqaitHruI BeJIHqHHa Z = X - /Y npHHHMaeT 3HaqeHHH Zl = 1 - 0 = 1,
Z2 = 1 - 2 = -1, Z3 = 1 - 3 = -2, Z4 = 4 - 0 = 4, Zs = 4 - 2 = 2,
Z6 = 4 - 3 = 1. BepoHTHoCTH 9THX 3HaqeHHit TaKOBbI:
P{Z = I} = P{Z = zd + P{Z = Z6} =
=PiX = 1, JY = O} + PiX = 4, JY = 3} =
= PiX = 1, Y = O} + PiX = 4, Y = 9} = 0,20 + 0,05 = 0,25;
P{Z = -I} = PiX = 1, JY = 2} = PiX = 1, Y =:= 4} = 0,15;
P{Z=-2}=P{X=I, Y=9}=0,1O;
P{Z = 4} = PiX = 4, Y = O} = 0,30;
P{Z=2}=P{X=4, Y=4}=0,20.
TaKHM o6pa30M, 3aKOH pacnpe,ll;eJIeHHH c. B. Z = X - /Y HMeeT BH,ll;
•
6.13.13. IIcnoJIb3YH YCJIOBHe 3a,n:aqH 6.13.12, onHcaTb 3aKOH pacnpe,ll;eJIeHHH C. B.:
a) Zl = X + Y; |
6) Z2 = IX - YI; |
B)Z3 = viX2 + y2.
6.13.14.X H Y - He3aBHCHMbIe CJIyqaitHbIe BeJIHqHHbI, pacnpe,ll;eJIeHHbIe
no O,ll;HOMY H TOMY )Ke reOMeTpHqeCKOMY 3aKOHY C napaMeTpOM p = 0,7 (p - BepOHTHOCTb ycnexa B O,ll;HOM HcnbITaHHH). QnHcaTb 3aKOH pacnpe,ll;eJIeHHH c. B. Z = X + Y.
14 C60PHHK 3IIJUI~ no ...eweR MareMOTH". 2 KYPC |
417 |
6.13.15. COBMecTHoe pacIIpe,r:t;eJIeHHe c. B. X H Y 3a.,Il;aHO IIJIOTHOCThlO pac-:
I
IIpe,r:t;eJIeHHfl BepoflTHocTeft
f(x,y) = {X + y, |
IIpH X E [0; 1], y E [0; 1], |
0, |
B "POTHBHOM cJIyqae. |
HaftTH:
a) cPYHKII;HIO paCIIpe,r:t;eJIeHHfl BepoflTHocTeft c. B. Z = X + Y;
6) IIJIOTHOCTh paCIIpe,r:t;eJIeHHfl f z (z ).
a a) Fz(z) = Fx+y(z) = P{X+Y < z} = jj(x+y)dxdy, r,r:t;e06JIacThDz
D%
eCTh MHO)KeCTBO TOqeK IIJIOCKOCTH Oxy, Koop,r:t;HHaThI KOTOPhIX y,r:t;OBJIeTBOpfl-
lOT HepaBeHCTBY x+y < z, r,r:t;e z - |
"POH3BOJIhHOe qHCJIO (Ha pHC. 9306JIacTh |
D z eCTh qaCTh KBa.,Il;paTa (0 ~ X |
~ 1, 0 ~ y ~ 1), JIe)Karn,a» |
HH)Ke "PflMOft |
y = -x + z). IIPH Z |
~ 0, |
OqeBH,r:t;HO, F(z) = 0 |
(BHe KBa.,Il;paTa |
f(x,y) = 0). |
ECJIH 0 < Z ~ 1 (06JIacTh Dz 3aIlITpHXOBaHa Ha pHC. 93), TO |
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2) |
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z |
-x+z |
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z |
( |
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+ |
F(z)=jj(x+y)dxdy=jdx |
j(x+y)dy=jdx |
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xy+~ I~xz= |
D% |
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0 |
0 |
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0 |
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~(2 |
(z - |
X)2) |
(X3 |
x2 |
1 |
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(z - |
3 |
X)3) IZ |
= JI -x + xz + |
2 |
dx = |
-3 + z . 2" - "2 . |
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0 = |
o |
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Z3 |
Z3 |
1 |
3 |
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z3 |
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Z3 |
z3 |
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= -- |
+ - + |
-(z - 0) |
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= - |
+ - |
= - |
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3 |
2 |
6 |
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6 |
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6 |
3· |
Puc. 93 |
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Puc. 94 |
ECJIH 1 < z ~ 2 (CM. pHC. 94), TO |
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F(z) = jj(x+y)dxdy= |
z-1 |
1 |
1 |
-x+z |
j dx j(x+y)dy+ |
j |
dx j (x+y)dy= |
D% |
0 |
0 |
z-1 |
0 |
HaKOHeIJ;, eCJUI Z > 2, TO |
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1 |
1 |
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F(z) = / /(X + y) dxdy = / dx |
/(X + y) dy = ... = 1. |
Dz |
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0 |
0 |
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TaKHM 06pa30M, |
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0, |
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npH z:::;; |
0, |
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Z3 |
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npH °< z:::;; 1, |
Fz(z)= |
- |
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1 |
3;3 |
2 |
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-"3 + z |
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- 3' npH 1 < z:::;; 2, |
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1, |
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npH z > 2. |
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6) HaxO,D,HM fz(z), HCnOJlb3Yjf paBeHCTBO Jz(z) = F~(z): |
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a, |
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npH |
z :::;; ° |
Z |
> 2, |
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HJlH |
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fz(z) = { z2, |
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npH |
°< z:::;; 1, |
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-z2 + 2z, |
npH 1 < z:::;; 2. |
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00 |
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• |
MO)KHO y6e,D,HTbCjf, 'ITO |
/ Jz(z) dz = 1. |
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- 00 |
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6.13.16. l1cnoJlb3Yjf YCJloBHe 3a,D,a'lH 6.13.15, HaitTH
Z=X-Y.
6.13.17. McnOJlb3Yjf YCJloBHe 3a,D,a'lH 6.13.15, HaitTH
Z=X·Y.
6.13.18. CJlY'laitHbleBeJIH'IHHbIX H Y He3aBHcHMI?I H HMeIOT paBHOMepHoe pacnpe,D,eJleHHe: X '"R[O; 1], Y '" R[-l; 2]. HaitTH nJlOTHOCTb pacnpe,D,eJleHHjf CJlY'laitHOitBeJlH'IHHbIZ = X + Y.
Q Hait,D,eM 3aKOH pacnpe,D,eJleHHjf CYMMbI He3aBHCHMblX c. B. ,D,BYMjf cnoco-
6aMH.
Cnoco6 1. CHa'laJIaHait,D,eM <PYHKIJ;HIO pacnpe,D,eJleHHjf c. B. Z = X + Y.
CHcTeMa ,D,BYX c. B. (X, Y) paBHoMepHo pacnpe,D,eJleHa B npjfMoyroJlbHHKe
ABeD (CM. pHC. 95), n09TOMY
Fz(z) = P{Z < z} = P{X + Y < z} = //f(x, y) dxdy = //h(x)h(Y) dxdy,
Dz Dz
