Сборник задач по высшей математике 2 том

O"lt"lt'btit MOMe"ltrn Kxy

(HHa'le:1COSapUa'4"UJ1, COV(X, Y)), KOTOPbIit AIlH .n;HCKpeTHbIX

C. B. BbI'IHCJ1HeTCHrro <p0pMYJIe

n

m

Kxy = L

L(xi - ax)· (Yj - ay)· pij;

i=1 j=1

a AIlH HerrpepbIBHbIX -

rro <popMYJIe

HY.

~

Koa<p<PHIJ;HeHT KOpPeJlHIJ;HH XapaKTepH3yeT CTerreHb JIHHeiiHoit 3aBHCHMOCTH

CJ1Y'IaitHbIxBeJlH'IHHX

H Y.

1.

-1 ,,;;; TXY ,,;;; 1;

2.

ECJ1H X H Y - He3aBHCHMbIe C. B., TO TXY = 0;

3.

ECJ1H c. B. X H Y CBH3aHbI JIHHeitHoit 3aBHCHMOCTbIO Y = aX + b, a 1= 0, TO

ITxyl = 1;

4.

ECJ1H ITxyl = 1, TO C. B. X H Y CBH3aHbI JIHHeitHoit <PYHKIJ;HOHaJIbHOit 3aBH-

X\Y

1

2

3

1

0,16

0,12

0,08

2

0,28

0,11

0,25

6) B COOTBeTCTBHH C <POPMYJIOil: F(x,y) =

I: I: Pij rrOJIyqaeM:

eCJIH X ~ 1 H Y ~ 1, TO F(x,y) = P{X < X,

Xi<X Yj<Y

Y < y} = 0, TaK KaK C06bITH.H

{X < x} H {Y < y} B 9TOM CJIyqae .HBJI.HIOTC.H HeB03MO:lKHbIMH.

AHaJIOrHqHO rrOJIyqaeM:

F(x,y) = P{X = 1, Y = I} + P{X = 1,

Y = 2} = 0,16 + 0,12 = 0,28;

eCJIH 1 < x ~ 2 H 3 < y, TO

F(x,y) = P{X = 1, Y = I} + P{X = 1,

Y = 2} + P{X = 1, Y = 3} =

= 0,16 + 0,12 + 0,08 = 0,36;

eCJIH 2 < x H y ~ 1, TO F(x, y) = 0;

eCJIH 2 < x H 1 < y ~ 2, TO

F(x,y) = P{X = 1, Y = I} + P{X = 2,

Y = I} = 0,16 + 0,28 = 0,44;

eCJIH 2 < x H 2 < y ~ 3, TO

F(x,y) = P{X = 1, Y = I} + P{X = 2,

Y = I} + P{X = 1, Y = 2}+

rrpH

y::::;1 1 <y::::;2

2 <y::::;3

3<y

X::::; 1

°

0,16°

0,28°

0,36°

1 <x::::;2

2<x

°

0,44

0,67

1

6.12.2.

3aKoH pacrrpe,l.l;eJIeHH.H CHCTeMbI°

,l.l;HCKpeTHblx CJIyqail:Hblx BeJIHqHH

3a,l.l;aH Ta6JIHIJ;eil:

X\Y

1

2

3

4

1

0,10

0,15

0,04

0,06

2

0,12

0,08

0,05

0,04

3

0,03

0,02

0,11

D

Hail:TH:

a) 3HaqeHHe qHCJIa D;

6) 6e3YCJIOBHble 3aKOHbI pacrrpe,l.l;eJIeHH.H CJIyqail:Hblx BeJIHqHH X

HY;

B) BepO.HTHOCTH C06bITHil: {X = 1, Y ;;:: 2} H {X = Y}.

6.12.3.

,II;BYMepHa.H CJIyqail:HM BeJIHqHHa (X, Y) 3a,l.l;aHa 3aKOHOM pacrrpe-

,l.l;eJIeHH.H

X\Y

0,12°

1

°1

0,18

0,28

0,42

HaihH:

a) <PYHKIJ;HIO pacrrpe,l.l;eJIeHH.H ,l.l;. c. B. X;

6) <PYHKIJ;HIO pacrrpe.n;eJIeHH.H ,l.l;BYMepHoil: c. B. (X, Y);

B) BepO.HTHOCTb C06bITH.H {X ::::; Y}.

6.12.4.

IIo MHmeHH rrpOH3BO,l.l;HTC.H O,l.l;HH BbICTpeJI. BepO.HTHOCTb rrorra,l.l;a-

HH.H paBHa 0,75. IIYCTb

c. B. X

-

qHCJIO rrOrra,l.l;aHHil:; c. B. Y -

qHCJIO rrpOMaXOB. CocTaBHTb Ta6JIHIJ;y COBMeCTHoro pacrrpe,l.l;eJIe-

HH.H BepO.HTHOcTeil: CJIyqail:Hblx BeJIHqHH X H Y. OrrHcaTb <PYHKIJ;HIO

pacrrpe,l.l;eJIeHH.H F(x,y) CHCTeMbI C.B. (X,Y).

6.12.5.

HcrroJIb3Y.H YCJIOBHe 3a,l.l;aqH 6.12.1, YCTaHOBHTb, 3aBHCHMbI HJIH

HeT KOMrrOHeHTbI X H Y.

a YCJIOBHe He3aBHCHMOCTH c. B. X H Y B ,l.l;HCKpeTHOM CJIyqae HMeeT BH,l.l;:

P{X = Xi, Y = Yj} = P{X = xil' P{Y = Yj} M.H JII06bIX i

= 1,2, ... , n H

j

= 1,2, ... , m. IIpOBep.HeM: rrYCTb Xl = 1 H YI = 1. IIo yCJIOBHIO

P{X = 1,

Y = I} = 0,16,

a

P{X = I} = 0,36, P{Y = I} = 0,44 (9TH

BepO.HTHOCTH Hail:,l.l;eHbI B

XO,l.l;e

pemeHH.H 3a,l.l;aqH 6.12.1). IIOCKOJIbKY

P{X = 1, Y = I} = 0,16 1: 0,36'0,44 = P{X = I}· P{Y = I},

TO OTCIO,l.l;a 3aKJIIOqaeM: KOMrrOHeHTbI CHCTeMbI (X, Y) 3aBHCHMbI.

•

6.12.6. 3a,n;aHo pacJIpe,D;eJIeHHe ,D;BYMepHoii CJlyqaiiHOii BeJIHqHHbI (X, Y)

X\Y 1

1,5

2

1

1

1

1

12

24

24

2

1

1

1

12

24

24

2,5

1

1

1

"3

"6

"6

HaiiTH O,D;HOMepHble pacJIpe,D;eJIeHH.H KOMJIOHeHT CHCTeMbI. YCTaHo-

BHTb, 3aBHCHMbI JIH KOMJIOHeHTbI X 1'1 Y. HaiiTH P{X +Y ~ 3,5}.

6.12.7.

MCJIOJIb3Y.H YCJIOBHe

3a,n;aqH 6.12.2, YCTaHOBHTb, 3aBHCHMbI HJIH

HeT CJIyqaiiHble BeJIHqHHbI X 1'1 Y.

6.12.8.

3a,n;aHbI 3aKOHbI pacJIpe,D;eJIeHH.H ,D;ByX He3aBHCHMbIX ,D;pyr OT ,D;pyra

CJIyqaiiHbIX BeJIHqHH X 1'1

Y:

Pi

~P~.~·L-~~~~~~

OJIHCaTb <PYHKIJ;HIO paCJIpe,D;eJIeHH.H F(x, y) 1'1 BblqHCJIHTb ee 3Ha-

qeHHe B TOqKe (9,2; 8,5).

6.12.9.

3aKoH pacJIpe,D;eJIeHH.H CHCTeMbI ,D;HCKpeTHbIX CJIyqaiiHbIX BeJIHqHH

(X, Y) 3a,D;aH Ta6JIHu;eii

X\Y

-2

-1

0

1

-1

1

2

3

1

16

16

16

16

0

2

3

1

0

16

16

16

1

0

1

0

2

16

16

HaiiTH:

a) 6e3YCJIOBHble 3aKOHbI pacJIpe,D;eJIeHH.H CJIyqaiiHbIX BeJIHqHH X

HY;

6) YCJlOBHblii 3aKOH pacJIpe,D;eJIeHH.H c. B. Y JIPH X = 0;

a a)

B) JIpOBepHTb He3aBHCHMOCTb CJlyqaiiHbIX BeJIHqHH X 1'1 Y.

CJIyqaiiHM BeJIHqHHa X JIpHHHMaeT 3HaqeHH.H Xl = -1, X2 = 0,

X3 = 1,

BepO.HTHOCTH KOTOPbIX HaxO,D;HM CYMMHpoBaHHeM BepO.HTHOCTeii co-

OTBeTCTBeHHO B JIepBoii, BTOpoii 1'1 TpeTbeii CTPOKax Ta6JIHU;bI:

1231763

Pl = 16 + 16 + 16

+ 16 = 16'

P2 = 16' P3 = 16'

TaKHM o6pa30M,

6e3YCJIOBHble 3aKOHbI paCJIpe,D;eJIeHH.H X 1'1 Y HMeIOT

BH,D;:

1

Yi -2 -1 0

1

Xi -1 0

,

Pi

7

6

3

Pi

3

6

4

3

16

16

16

16

16

16

16

2

P{Y = -21 X = O} = 16 = ~ = ~,

6

16

3

P{Y = -11 X = O} = 16 = ~ = ~,

6

16

1

16

1

P {Y = 0 I X = 0} =""6 = 6'

16

P{Y = 1 IX = O} = ~

= O.

16

BH)J,

-2

-1

0 1

Yi

Px=o

1

1

1

0

3

2

6

B) TaK KaK 6e3YCJIOBHblfi H

YCJIOBHblfi

3aKOHbI paClIpe)J,eJIeHH5I C. B. Y

1

7

3

= PiX = -I}· P{Y = -2}.)

PiX = -1, Y = -2} = 16

-I 16 . 16

'IHCJIOBblexapaKTepHCTHKH, HCrrOJIb3Y5I «CTapble <POPMYJIbI»:

M(X) = 1· 0,36 + 2·0,64 = 1,64;

M(Y) = 1· 0,44 + 2·0,23 + 3·0,33 = 1,89.

6) Haxo)J,HM )J,HcrrepCHH D(X) H D(Y):

D(X) = [tt,(Xi - ax )2 . Pij ]

=

= (1 -

1,64)2 ·0,16 + (1 -

1,64)2 ·0,12 + (1 -

1,64)2 ·0,08+

+ (2 -

1,64)2 ·0,28 + (2 -

1,64)2 ·0,11 + (2 -

1,64)2 ·0,25 =

= 0,4096·0,36 + 0,1296 . 0,64 = 0,2304,

T. e. D(X) = 0,2304.

D(Y) = (1 -

1,98)2 ·0,16 + (1 -

1,89)2 ·0,28 + (2 -

1,89)2 ·0,12+

+ (2 -

1,89)2 . 0,11 + (3 -

1,89)2 . 0,08 + (3 -

1,89)2 ·0,25 =

= 0,7921 ·0,44 + 0,0121 . 0,23 + 1,2321 ·0,33 = 0,7579.

X: D(X) = [M(X2) - (M(X»2] = 1 . 0,36 + 22 ·0,64 -

(1,64)2

= 0,2304.

TO'IHOTaK )ICe

D(Y) = [M(y2) - (M(y»2] = 12 ·0,44 + 22 .0,23 + 32 .0,33 -

(1,89)2 =

= 4,33 -

3,5721 = 0,7579.

B) Terrepb Y)lCe JIerKO HaitTH a(X) = JD(X) = JQ,2304 = 0,48 H a(Y) =

::: VO;7579 ~ 0,87.

•

X\Y

-2

0,05°

2

0,2

0,03

0,12

0,6

0,15

0,30

0,35

HafiTH YCJIOBHOe MaTeMaTH'IeCKOeQ)KH,l.l;aHHe M(X IY = 2).

Q ,ILrr51 HaXQ)K,l.l;eHH5I HCKOMOfi BeJIH'IHHbIBOCIIOJIb3yeMC5I <popMYJIofi

n

M(X IY = Yj) = LXi' p(Xi IYj)'

r,l.l;e p(Xi IYj) = P{X = Xi IY = Yj}.

i=l

Hafi,l.l;eM CHa'laJIaYCJIOBHblfi 3aKOH pacIIpe,l.l;eJIeHH5I KOMIIOHeHTbI X IIPH

JIeHH5I C. B. X IIpH Y = 2 HMeeT BH,l.l;

Xi

0,2

0,6

PY=2

U,l~

U,J<> (CM. perneHHe 3a,l.l;a'lH6.12.9.)

0,47

0,47

CTaJIO 6bITb,

M(X IY = 2) = [X1P(Xl

IY3) + X2P(X2 IY3)] =

= 0,234 = 234 ,..., °498

=°2 . 0,12

°6 . 0,35

= 0,024 + 0,21

,

047 +,

047

047

047

470 "',

,

,

,

,

,

T. e. OKOH'IaTeJIbHOM(X IY = 2) ~ 0,498.

•

6.12.16.

I1CIIOJIb3Y5I YCJIOBHe 3a,l.l;a'lH6.12.15, HafiTH:

a) M(X);

6) M(X I Y = 0);

B) M(Y);

r) M(Y IX = 0,2).

6.12.17.

I1CIIOJIb3Y5I YCJIOBHe 3a,l.l;a'lH6.12.10, HafiTH

M(Y I X = -1), M(X IY = -2),

M(X IY = 0).

6.12.18.

I1CIIOJIb3Y5I YCJIOBHe 3a,l.l;a'lH 6.12.1, HafiTH KOppeJI5IIJ;HOHHblfi

M0-

MeHT Kxy (KoBapHaIJ;HIO) H K09<P<PHIJ;HeHT KOppeJI5IIJ;HH rXY.

Q MaTeMaTH'IeCKHeQ)KH,l.l;aHH5I

KOMIIOHeHT )')KeHafi,l.l;eHbI: ax = M(X) =

= 1,64; a y

= M(Y) = 1,89 (CM. 3a,l.l;a'lY6.12.12).

,n:HcIIepCH5I H Cpe,l.l;HeKBa,l.l;paTH'IeCKOeOTKJIOHeHHe KOMIIOHeHT TaK)Ke H3·

Kxy

0,0404

TXY = a(X) . a(Y) = 0,48. 0,87 ~ 0,096732,

T.e. TXY ~ 0,1.

•

6.12.19.

IICIIOJIb3Y5I YCJIOBMe 3a)J,a'fM6.12.3, HaiiTM Kxy M TXY.

6.12.20.

IICIIOJIb3Y5I YCJIOBMe 3a)J,a'fM 6.12.9, HaiiTM OCHOBHble XapaKTepM-

CTMKM M(X), M(Y), D(X), D(Y), Kxy, TXY )J,aHHoii CMCTeMbI

CJIY'faiiHbIXBJIM'fMH(X, Y).

6.12.21.

COBMecTHoe pacIIpe)J,eJIeHMe CJIY'faiiHbIXBeJIM'fMHX M Y 3a)J,aHO

IIJIOTHOCTblO pacIIpe)J,eJIeHM5I Bep05lTHOCTeii

f(x, y) = {c. (y -

xy), (x, y) ED,

0,

(x,y)¢D,

r)J,e 06JIaCTb D = {(x,y) : 0 ~ x ~ 1, 0 ~ y ~ I}. HaiiTM:

a) K09<P<PMlI,MeHT c;

6) IIJIOTHOCTM pacIIpe)J,eJIeHM5I OT)J,eJIbHbIX KOMIIOHeHT X M Y;

B) Bep05lTHOCTM IIOIIa)J,aHM5I CJIY'faiiHOiiTO'fKM (X, Y) B 06JIacTb

Dl = {(x,y) : 0,7 ~ x ~ 3, 0 ~ y ~ 0,3};

r) COBMeCTHYIO <PYHKlI,MIO paCIIpe)J,eJIeHM5I F(x, y).

y ________________ .A13

_____ ... A14

:

I

:

I-m~~~

I

I

I

Y

I

:

00

00

1

1

!

! f(x,y)dxdy= !dx !c. y (l-x)dy=

- 00 - 00

0

0

6) HaxO.n;HM nJIOTHOCTH pacnpe.n;eJIeHHH KOMnOHeHT X H Y:

00

1

2

1

h(x) = fx(x) = ! f(x,y) dy = !4y(1- X) dy = 4(1- x) . Y210 = 2(1- x),

- 00

0

T.e.

fx(x)

= {2(1- x), x E [0,1]'

0,

x tt [0,1];

00

1

(2) 1

h(Y)=fy(y)= ! f(x,y)dx= !4y(1-x)dx=4y X-~

10=2y,

- 00

0

T.e.

fy(y) = {2Y,

Y E [0,1],

0,

y tt [0,1].

1

0,3

3

0,3

Dl

0,7 0

1

0

1

Y

0,3

x 2

2

=2·0,09·

() 1

=4 !(l-x)dx·-1

x --

I =

0,7

2

0

2

o~

= 0,18 ( 1 -

1

0,49)

= 0,0981.

"2 - 0,7 +

-2-

B cJIY'Iae(a) MMeeM

x

y

x

y

F(x,y) =4 1 du

1(I-u)vdv=4/(I-u) du lvdv=

-00

-00

0

0

= 4 J(1 - u) du· y; = 2y2 (u - ~) I: = 2y2 (x -

x;) = xy2(2 - x);

o

B CJIY'Iae(6) IIOJIY'IaeM

y

1

~.

2

Y

= y2;

F(x,y) = 41vdv 1(1- u) du = 4·

v

10

o

2

0

1

1

2

1

F(x, y) = 41vdv j(1- u) du = 4·

~ . v

10

o

2

0

11, HaKOHeI.J;, B cJIY'Iae(1'):

o

0

TaKMM 06pa30M,

x < 0, y < °MJIM X < 0, y > °MJIM X > 0,

0,

y < 0,

F(x,y) =

xy2(2 - x),

(x,y) ED,

y2,

0::::;

y::::; 1,

x> 1,

x(2 - x),

0::::;

x::::; 1,

y> 1,

1,

x> 1, Y > 1.

6.12.22.

I1CIIOJIb3Y5I

pe3YJIbTaTbI,

IIOJIY'IeHHble IIpM peIIIeHMM

3a,IJ,a'lM

•