2015-20287

1 U S U

1.1 u(X) S E[u(X)] U

E[u(X)] =

Z 10

u(x)p(x)dx =

Z 1c

u(x)p(x)dx+

Z c0

u(x)p(x)dx

X Z 1c

u(x)p(x)dx+

Z c0

u(x)p(x)dx Z 1c

u(x)p(x)dx cZ 1c

u(p)(x)dx = cP [u(X) c]

; P [u(X) c] E[u(X)]c

1.2 u(X) = (X )2S c = k22 U

P (jX j2 k22) E[(X )2]

k22

2 U

P (jX j k) 1k2

1

U U S S U 2008 U

2 2.1 ABC U E[X] ABC U X U X U U

E[XjX x] =R x0yp(y)dyR x

0p(y)dy

;E[XjX x] =R1x

yp(y)dyR1x

p(y)dy

; E[X] = F (x)E[XjX x] + S(x)E[XjX x]:::(1)

F (x)E[XjX x]+S(x)E[XjX x] = F (x)E[XjX x]+S(x)E[Xx+xjX x]:::(2)

(2) E[X] S(x) U

S(x)E[X x+ xjX x] = E[X xjX x] + E[xjX x] = E[X xjX x] + x:::(3)

(3) (1)

E[X] = F (x)E[XjX x] + xS(x) + S(x)E[X xjX x]

ABC U A=F (x)E[XjX x], B=xS(x), C=S(x)E[X xjX x] UABC A,C E[X] = R1

0S(x)dx

2

U 0 x a X x F(a)/F(x) U 1-F(a)/F(x) E[XjX x]

E[XjX x] =Z x0

(1 F (a)F (x)

)da =

Z x0

[F (x) F (a)]F (x)

da =1

F (x)

Z x0

[S(a) S(x)]da

E[X xjX x]

E[X xjX x] = 1S(x)

Z 1x

S(a)da

U A, C

A = F (x)E[XjX x] =Z x0

[S(a) S(x)]da =Z x0

S(a)da xS(x)

C = S(x)E[X xjX x] =Z 1x

S(a)da

U

2.2 S S X ABC A,B,C U E[X] B = xS(x) for all x 0 x E[x]/x P [X x] U E[X] B +C U

E[X]

x+ E[X xjX x] P [X x]

E[X] A+B U

E[X] E[XjX x]x E[XjX x] P [X x]

3

x 1 U U

2.3 U Z S E[X] = S 2 = E[(X )2] X = (Z )2 E[X] = 2 U

: 2

x P [(Z )2 x]

: 2

x+ E[(Z )2 xj(Z )2 x] P [(Z )2 x]

: 2 E[(Z )2j(Z )2 x]x E[(Z )2j(Z )2 x] P [(Z )

2 geqx]

3 1.a 2008 U x 0.5 82 A=34, B=41, C=4 U 0.96, 0.88, 0.79 U x 82 U U U

4