強化第七天概率專題三

Cov(X,Y)9

5

EXEY99P{X1,Y1}9P{X1,Y0}P{X0,Y1} P{XY1} P{X0,Y0} 9f(x,y) (y|x) (x) ,0yx 例3（1F(zP{XY當z0P{XYz當0z2P{X

z z}

1

1 當2z4P{XYz}1P{XY21z2

dxz

1242

zz2當z當z4P{XYz0,zz2,0

zF(z)P{X

z} z

1,2z1z,0

zf(z)F/(z)

z,24

z（2F(z)P{XY當z2時{XYz當2z0P{X

z}

1zz 18

xz當0z2P{XYz}1P{XY12dxxz1dy11(z 當z2P{XYzF(z)P{XYz}

0,11

2)2,2z 1

2)2,0z 1

2),2zf(z)

F/(z)

14

2),0z（3）F(z)P{XYz}P{YzX當z 時{Yz}當0z4P{XYz}1P{XY

zdx

141z1zln21zln 當z4P{XYz0,z F(z)P{XYz}

z zln24 4

z ,02

z1ln2 zf(z)F/(z) 4ln2,（4F(z)P{YX

z}P{YzX當z 時YzX}當0z1

zX}

20

z

14

12當 時， 2z z

dxzx41 F(z)X

z}P{YzX}2

z,01

z ,z 練習（1F(z)P{XYz}當z0 P{XYz}0當0z1時，P{XYz}dxz ezz當z1時PXYzdxzxeeze1z（2F(z)P{(XY)2 XY zz當z0F(z)zz當0z4時F(z) 4z16F(z)1P{XY z1(4 zz1 1(4z當z16時F(z)

（4解F(z)P{2XYzP{Y2X當z0F(z)當0z2時F(z)1P{Y2X1dx2xz z當z2F(z)f(z)F(z)

1z,0zX0X01PP{XY0}P{XY0,X0}P{XY0,XP{00,X0}P{Y0,XP{X0}P{Y0}P{X1[1(0)] ：X01：X01Pf( ey,y F( 1ey,y 0,y 0,yF(2,1)P{Y2,Z1}P{Y2,(2X1)YP{Y2,(2X1)Y1,XP{Y2,(2X1)Y1,XP{Y2,Y1,X0}12

2,Y1,XP{Y2,Y1}P{X0}P{Y2,Y1}P{X1P{Y2,Y1}1P{Y2,Y 1P{1Y2}1P{Y [F(2)F(1)] 例3F(z)P{XYP{XYz,Y1}P{XYz,YP{X1z,Y1}P{X2z,Y1[P{X1z}P{X2z}]1[P{Xz1}P{Xz2}]1[F(z1)F(z F(z)P{XYz}1[F(z1)F(z f(z)F/(z)1[ (z1) (z 0,z,z,2z

4,2z,3z 1,3z,4z,z 4,4z練習（1FZ(zP{XYP{XYz,Y0}P{XYz,YP{0z,Y0}P{Xz,Y1[P{0z}P{Xz}]1P{Xz},z 1(z),z1[1P{Xz}],z

1[1(z)],zZ(z)P{XY1313

1z}P{Xz}P{X1z1}P{Xz}P{Xz1[F(z1)F(z)F(z f(z)F/(z)1[ (z1) (z) (z f(z)1[ (z1) (z) (z 0,z,z

,1z,1z ,0z 1,0

z,1z

1,z

,1z例1.解：FZ(zP{max{X,YP{Xz,Yz}P{Xz}P{YFX(z)FY(z)例2.FZ(zP{min{X,Y1P{min{X,Y}z}1P{Xz,Y1P{Xz}P{Y1[1P{Xz}][1P{Y1[1FX(z)][1FY(z)]P{U0}P{X0,Y0}9P{U1}P{X0,Y1}P{X1,YP{X1,Y1}3P{U2}P{X0,Y2}P{X1,YP{X2,Y P{X2,YP{X2,Y2}9P{V0}P{X0,Y0}P{X0,YP{X0,Y2}P{X1,Y例4.P{maxX,Y)0}1P{max(X,Y)AX0,BY

1P{X0,YP{X0,Y0}P( P(A 1P(AP{max(X,Y)0}P(A P(A)P(B)P(AB)7

dxe2x22xyy2

ex2

ex22xyy2 exdxe(y A

A

(x)

f

y)dy

e2x22xyy2

1e

e(yx)2 例2.解： （1）X(x) f(x,y)dyxex,x0,xfY(y)f(x,y)dxy dx,y0,y y,yey,yy0,y xdy,0x (x)f(x,y)dy2xdy,1x x,0x2x,1xYf(y)Y

f

2yy)dxy

dx,0y

y（2f

y) 1

x (x)

f(x,y)dy

dy,0x

221,1