概率论课后习题答案应坚刚、何萍复旦大学出版社.pdf
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概率论课后习题答案应坚刚、何萍复旦大学出版社.pdf
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2009?
12?
10?
1.1?
1.?
DeMorgan?
1.1.1.2.?
f?
X?
Y?
B?
Y?
X?
f1(B):
=f1(B):
BB.?
B?
(?
)?
f1(B)?
(?
)?
3.?
f?
X?
Y?
A?
X?
Y?
B:
=BY:
f1(B)A.?
A?
(?
)?
B?
(?
)?
4.?
0-1?
0?
1?
0?
0-1?
5.?
52?
(1)?
(2)?
(3)?
(1)945;
(2)36;(3)4?
132?
1336.?
x1+xr=n?
(x1,xr)?
A=(x1,xr):
x1+xr=n,x10,xr0;B=(x1,xr):
x1+xr=n+r,x10,xr0.?
(x1,xr)=(x1+1,xr+1).?
A,B?
|A|=|B|.7.?
Tn?
1,2,n?
T1=1,T2=2.?
Tn+1=1+nXk=1?
nk?
Tk.1?
2?
1,2,n+1?
n+1?
A,?
A?
n+1?
k?
Ak,?
0kn.?
k?
nk?
A?
nk?
Tnk?
Tn+1=nXk=0|Ak|=|An|+X0k1(35/36)24.6.?
6n?
n?
6,?
6(n+1)?
n+1?
6,?
7.?
26?
2?
226?
5226?
/?
10426?
.8.?
12.?
(1)?
(2)?
a?
b?
12=a(a1)(a+b)(a+b1).
(1)a=3,b=1;
(2)a=15,b=6.9.?
(1)?
(2)?
(1)?
(2)?
10.?
1?
2?
411.?
1,2,n?
k?
nk+1k?
/?
nk?
.1j1j21,?
lu=ju(u1),?
1l1l20.12.?
n?
1,2,2n.?
1?
Ai:
i?
i+1?
(2n+1?
1).P(2ni=1Aci)=1P(2ni=1Ai)=12nXk=1
(1)k1Xi11,0,ji=1.P(Ai1Aik)=1(n)k,iuiu11,1uk1,0,otherwise.?
2n?
k?
Sk,n.?
m?
k?
Nk,m,?
11?
Nk,m=?
mkk?
.?
Sk,n?
1,2n?
Sk,n=Nk1,2n2+Nk,2n=?
2nkk?
2n2nk.?
nXk=0
(1)k?
2nkk?
2n2nk1(n)k.13.?
m?
n?
r?
k?
5P(Ak)=?
k1r1?
(m)kr(n)r(m+n)k14.
(1)?
n?
2k?
s?
(2)?
n?
2k?
s?
(1)?
ns?
ns2k2s?
22ks?
2n2k?
(2)2?
ns?
ns2k2s?
2n2k?
15.?
(a)?
(b)?
(1)?
63?
/63;
(2)?
83?
/63,n?
k?
1j1j2jkn,?
lu=ju+(u1),?
n+k1?
k?
16.(a)?
(b)?
17.?
n?
2n?
2n?
(2n)!
?
2n?
n1?
2n2?
2n(2n2)(2n2)(2n4)64422=(2n)!
(2n2)!
?
(2n)!
(2n2)!
(2n)!
=(2n2)!
(2n1)!
.18.?
A?
a?
k?
19.?
A,B,C?
(1)A?
6?
B?
C?
(2)?
6?
A?
6?
B?
6?
C?
6(?
1.4)?
A1,A2?
(1),
(2)?
(1)?
A,?
B?
C?
6?
p=1/6+(5/6)21/6+(5/6)41/6+=6/11.?
P(A1)=1/6p+(5/6)31/6p+(5/6)61/6p+=6/1136/91.
(2)?
P(A2)=P(ABC.,A?
6)P(BCA.,B?
6)P(CAB.,C?
6)=(P(ABC.,A?
6)3=Xn0?
56?
3n163=?
3691?
320.?
(1)?
1.
(2)?
1.
(1)2/10;
(2)4/1021.?
A,B,C?
P(ABC)=12(P(A)+P(B)+P(C).?
ABC=,?
ABcCc=AcBCc=AcBcC=.?
ABC,BCA,CAB,?
P(A)=P(ABAC)=P(AB)+P(AC)P(ABAC)=P(AB)+P(AC).?
P(B)=P(BA)+P(BC),P(C)=P(CA)+P(CB),?
P(ABC)=P(A)+P(B)+P(C)P(AB)P(BC)P(CA)=12(P(A)+P(B)+P(C).22.?
n?
23.?
(?
3cm)?
2?
4cm?
724.?
a,b,c?
l,?
a,b,cl,?
21?
Buffon?
25.?
(?
),?
Bk?
k?
k3?
|Bk|=(k1)(|Bk1|+|Bk2|),?
|B1|=0,|B2|=1.?
|Bk|.1.3?
1.?
1AB=1A1B,1AB=1A1B,1Ac=11A,1AB=|1A1B|.2.?
1AB=1A+1B?
A,B?
3.?
-?
-?
4.?
-?
2?
5.?
(,F)?
P?
F?
P()=1,?
(a)P?
(b)P?
(c)P?
6.?
A,B?
P(A)=1,?
P(AB)=P(B).7.?
P?
P(A)=34,P(B)=13,?
112P(AB)13.8.?
(1)?
(2)?
F?
F?
9.?
m?
50?
n?
100?
mn,?
50?
Si?
i?
50?
100?
(0,0)?
(m+n,mn)?
x?
(?
)?
(i,Si).?
810.?
P(ni=1Ai)=XiP(Ai)+
(1)XijP(AiAj)+XijkP(AiAjAk)+
(1)n1P(A1An).DeMorgan?
1.2.111.?
P(ni=1Ai)nXi=1P(Ai)XijP(AiAj).?
1.2.112.?
P(ni=1Ai)minknXi=1P(Ai)Xi:
i6=kP(AiAk).?
13.?
A1,An,Nk?
k?
Ai?
P(Nk)=nkXi=0
(1)i?
k+ik?
Sk+i,?
Sj=Pi1i2ijP(Ai1Ai2Aij).n:
=1,2,n.Nk?
A1,An?
k?
Nk=In,|I|=k(iIAi)(i6IAi)c.?
AI:
=TiIAi,?
Nk=SI(AI(i6IAi)c),?
1.2.1.14.?
F,?
xR,P(x)=F(x),x?
F?
xR,P(=x)=0,?
(diffused).15.?
A1,An,?
P(Ai)=p,P(AiAj)=q,i6=j,?
p1n,q2n.?
=SiAi,?
1=P(SiAi)np,?
p1/n.?
1=npn(n1)/2q,?
q=2(np1)n(n1)2/n.?
916.?
m?
P(nAk?
m0,A1,An,An+m?
A1,An,Sn+mk=n+1Ak?
(2)?
n,A=m1kmAk=mnkmAk.?
10?
A,A1,An?
n?
A,A1,An,?
A?
Dynkin?
07?
Bn=TknAck,?
P(A)=limnP(knAk)=1limnP(Bn),Bn?
P(Bn)=QknP(Ack).?
a=limnP(Bn),?
0a1,?
m,?
nm?
a(1+a)/2P(Bn)a,Ymka,?
(1.4.1)?
(?
m?
)?
n,?
P(Bm)=aP(Bm),?
a=1?
m,P(Bm)=0,?
a=0.5.?
7?
6.?
A?
P(A)?
0?
1.7.?
?
x?
2x,x=1,2,.?
P(min(,)x),P(),P(=),P(k),(?
k?
)P(?
),P(=r),?
r?
8.?
X1,X2,X3?
P(Xi=x)=(1pi)px1i,x=1,2,i=1,2,3.(a)?
P(X1X2X3).(b)?
P(X1X2X3).9.?
X?
n,p?
Y?
m,p?
X,Y?
X+Y?
10.?
X?
k0?
h(k):
=P(X=k|Xk).?
Ui:
i0?
0,1?
Z:
=minn:
Unh(n)?
X?
k0,Z=k?
ih(i)?
Ukh(k).?
11.?
A1,An?
1A1,1An?
12.?
Bernoulli?
n?
m?
11?
n?
Pascal?
P(n)=qn;
(2)?
m,nZ+,?
P(m+n|m)=P(n).(3)?
Z+?
14.?
?
15.?
un?
n?
4?
n4,?
un=12un1+14un2+18un3+116un4,?
u8.?
u8=26/32.16.?
1,2,m?
m?
k?
k?
m?
17.?
A,B?
n?
pn=19(1pn1),p0=1,p1=0.18.?
A?
B?
n?
B?
pn=pn134+(1pn1)13,p0=1.19.?
A,B?
B?
A?
?
B,A?
P(k).?
=k,k?
k1?
k?
B?
P(=k,k)=(1P(AB)k1P(B),?
P()=P(B)/(P(A)+P(B).20.?
7?
11,?
2,3?
12,?
7?
12?
P(A)=XkP(A|=k)P(=k)=6/36+2/36+Xk4,5,6,8,9,10P(A|=k)P(=k),?
P(A|=k)=P(?
7?
k?
).?
18?
P(A|=4)=P(A|=10)=3/9,P(A|=5)=P(A|=9)=4/10,P(A|=6)=P(A|=8)=5/11.?
P(A)=244/495=127990.?
50%,?
21.?
X,Y,?
X,Y?
F,G,?
p?
Bernoulli?
X+
(1)Y?
pF+(1p)G22.?
1,n?
k?
k?
=infk:
k?
.?
:
=Pk1k1=k.23.?
An:
n1?
Pn1An=Yn1P(An).24.?
5?
3?
2?
A?
B?
P(A|B)?
P(B|A).25.?
X,?
X?
Y.?
Y?
26.?
1,2?
1,2?
Poisson?
nk0,?
P(1=k|1+2=n).27.?
1,r?
1+r=n?
(1,r)?
P(1=n1,r=nr|1+r=n)=1/?
n1nr?
.?
1328.?
n?
Bernoulli?
i?
i?
(0in).?
1+n=r,rn?
i?
29.?
A,B?
AB,?
P(A)=0?
P(B)=1.30.?
?
Bernoulli?
p,?
+?
p=0,1,1/2.31.?
?
1,1?
:
=.?
?
1.5?
1.?
A?
E(;A)=PxR()xP(=x,A).2.?
:
?
2.?
P(=2)=1/2,P(=3)=1/4,P(=k)=2/2k.?
E=3.3.?
E=Pn=1P(n).E=Xn1nP(=n)=Xn1nXk=1P(=n)=Xk1XnkP(=n)=Xk1P(k).4.?
n,p?
E11+=1(1p)n+1(n+1)p.E(1/(1+)=(1qn+1)/pn+1.5.?
n:
n0?
p?
Bernoulli?
X=infn1:
nn1,?
14?
X?
EX.?
01?
0?
1/q1(?
n?
),?
1?
1/p,?
1/p+1/q1.6.(Coupon?
)?
N?
n?
N?
n?
n?
Bi?
i?
1.2.1,P(n)=PNi=1Bi!
=NXk=1
(1)k1?
Nk?
P(ki=1Bi)=NXk=1
(1)k1?
Nk?
(Nk)nNn.?
E=Xn0P(n)=NXk=1
(1)k1?
Nk?
Nk.?
1=1,i?
i1?
i?
i?
(N(i1)/N?
=PNi=1i,?
E=PNi=1N/(Ni+1)=PNk=1N/k.?
NXk=1
(1)k1?
Nk?
Nk=NXk=1Nk.7.?
n?
(1)?
(2)?
1/n?
1,n?
n?
(n+1)/2.8.?
X?
Poisson?
E(X(X1)(Xk)=k+1,k=0,1,2.?
X?
9.?
1,n?
n?
1510.8?
7?
15?
(?
M,F?
MMMFFFMFM-MMMFFF?
=5.)?
E.(?
1?
P(1=1).)?
i?
i?
i+1?
114.?
P(i=1)=27813!
/15!
=8/15.?
E=E1+E14=148/15.11.?
?
(1),?
(?
)f,g,?
Ef()g()=Ef()Eg().
(2)?
?
f,g,?
f(),g()?
12.?
?
(?
)?
?
?
A,B?
E=P(AB),E=P(A),E=P(B),?
P(AB)=P(A)P(B),?
A,B?
?
13.?
10%?
i?
i?
EP8i=1i=89/107.?
P(P8i=1i8)0.14.?
E.15.?
1,2,n?
n?
k?
i?
i?
E(1+k)=k(n+1)/2.16.?
A?
n?
B?
n?
k?
A?
12n(1+(12n)k).?
k?
k?
A?
k1=m?
k?
(1m/n)2,(m/n)2,1(1m/n)2(m/n)2?
m+1,m1,m?
E(k|k1=m)=(m+1)(1m/n)2+(m1)(m/n)2+m(1(1m/n)2(m/n)2)=(12/n)m+1.?
Ek=PmE(k|k1=m)P(k1=m)=1+(12/n)Ek1.?
0=n.17.?
p?
Xn?
n?
16?
EXn=Pnk=1pk.?
P(n)=pn.EXn=nXk=1E(Xn|=k)P(=k)+E(Xn|n)P(n)=nXk=1qpk1(k+EXn)+npn=nXk=1pk1+(1pn)EXn.18.?
A,B,C?
A,B,C?
A+B+C1.R?
P(A=1)=|R|2,?
|R|2/3.19.n?
m?
E.E(n|n1)=(n11)n1m+n1(1n1m)=(11m)n1.?
En=(11/m)En1=m(11/m)n.20.?
1,2,n?
1,2,n?
X?
E(X|)=n+12,?
EX=n+12E=?
n+12?
2.21.?
1,k?
Emin(1,k)=Xn1(Xmnpm)k?
pm=P(1=m).22.?
(1)?
N0?
E(|;|N),?
E|n)=0.
(1)E|=E(|;|N)+E(|;|N)N+E(|;|N)n.?
|n|,?
na.s.0,?
limnE|n|=0.?
1723.?
E|?
Pn1P(|n).?
nP(n|n+1)E(|;n|n+1)(n+1)P(n|n+1),?
Xn0nP(n|n+1)E|1+Xn0nP(n|n+1).?
3?
Pn0nP(n|0,?
g(x)0,?
P(x)Eg()g(x).27.?
x0,P(x)11+x2.?
x0,?
P(x)=11+x2.?
a0,?
E(+a)2E(+a)2;x)(a+x)2P(x),?
P(x)(1+a2)/(x+a)2.?
x,?
a=1/x?
1/(1+x2).?
18?
(06?
):
?
p=P(x),?
E2E(2;0)+E(2;x)E(2;0)+x2p?
Cauchy-Schwarz?
E(2;0)E(|;0)2P(0)E(|;0)21p,?
0=EE(;0)+E(;x),?
E(|;0)E(;x)xp,?
1=E
(2)E(2;0)+x2px2p21p+x2p,?
p1/(1+x2).?
(06?
):
?
p=P(x);?
Chebyshev?
Cauchy?
x2p2E(;x)2=E(;x)2E(2;x)P(x)=(1E(2;x)2(1p)(1x2p)(1p),?
p1/(1+x2).28.?
f?
xR:
f(x)6=0?
?
f?
Ef()=Ef().?
ab,?
fn1(a,b),?
P(ab)=E1(a,b)()=E1(a,b)()=P(ay)=0;
(2)E=Z0P(y)dyZ0P(y)dy.?
(1)?
Fubini?
E=EZ0dy=EZ01ydy=Z0P(y)dy.
(2)?
limyyP(y)=0,?
E=E+E,?
y0?
P(+y)=P(y),P(y)=P(x)dx.10.?
F?
F()?
0,1?
x(0,1),?
F1(x):
=supy:
F(y)=x,?
F?
F1(x)R?
F(F1(x)=x.?
P(F()x)=P(F1(x)=F(F1(x)=x,?
11.?
(0,1),z?
-?
infyx:
P(xy)=1=2z2.?
z.?
P(z/20,?
P(xa/2x+a/2)?
x=0?
0yx2z/2,?
P(xy)P(yx2yx2)P(z/2z/2)=1.12.?
(1x1x3)e12x22(1(x)0.?
2213.?
X?
a0.?
limx+P(Xx+ax|Xx)=ea.Lhospital14.?
U?
0,1?
(1)?
X=nU+1?
(2)?
X=logUlogq+1?
0q0?
=?
(?
x?
x?
).20.?
F?
limx+xR+x1ydF(y)=0.xRxy1dF(y)RxdF(y)=1F(x).21.?
F?
a0.?
Z+(F(x+a)F(x)dx=a.leftside=limnZnn(F(x+a)F(x)dx=limn?
Zn+anF(x)dxZn+anF(x)dx?
=a.22.?
Esin.Esin=/(1+2).?
232.2?
1.?
?
P(=)=E()=XxRP(=x)P(=x),?
(y):
=P(=y);?
P(=)=0.?
(1)P(=)=limn+Xk=(F(k+1)/2n)F(k/2n)(G(k+1)/2n)G(k/2n),?
(?
),?
P(=)=0.
(2)?
D=(x,x):
xR,?
Fubini?
P(=)=E1D(,)=ZRE1D(,x)dF(x)=XxP(=x)P(=x).2.
(1)?
1,2,3,4?
1,2?
3,4?
(2)?
X1,X2,Y1,Y2?
X1,X2?
F?
f,Y1,Y2?
G?
g,?
(X1,X2)?
(Y1,Y2)?
13+E?
(F(X1)G(X1)2+(F(Y1)G(Y1)2?
.
(1)?
4!
=24?
8/24.3.?
X,Y?
F,?
a,b0,inf(aX,bY)?
aba+bX?
F?
4.?
(X,Y)?
2?
2?
Q,(X,Y)Q?
(X,Y)?
(?
(X,Y)?
)?
P(X=0,Y=0)=0,?
P(Y=0)=1.?
XY?
X|Y|?
(1)?
A?
0?
x?
Q?
QAQA=,00)+P(XzY;Y0)+P(XzY;Y0).?
(X,Y)?
(X,Y)?
P(XzY;Y0)=P(XzY;Y0)=P(XzY;Y0).(3)?
arctan(Y/X)?
(/2,/2)?
Y/X?
Cauchy?
(?
2.1.7).?
0dxRzx02e(x+y)dy=z1+z.7.?
(X,Y)?
Z=YX?
8.?
(X,Y)?
D?
D?
(0,0),(1,0),(1,1)?
(X,Y).9.?
0,1?
U:
=
(1),V=1U,?
VU?
2/(1+x)2,x1.10.?
AB?
X,Y,Z,?
AX,AY,AZ?
(2)?
(1)(x,y,z):
x,y,z0,1,x+yz,y+zx,z+xy?
1/2.
(2)A?
P(A)=P(A;)+P(A;).?
P(A,)=P(1,+
(1),
(1)+()=P(1/2,1/2,1?
n,N:
=infn:
n+11,?
N?
n,Nn?
1in,i+1n)=P(i+11,1in)=n!
(n+1)!
=1/(n+1).20.(?
)?
P,P?
r,?
21r2.?
0,?
4(1r2)=22cos,?
P(x)=P(coscosx)=P(2r21cosx)=P(r21+cosx2)=112(1+cosx)=12(1cosx).21.?
(X1,Xn)?
(X1,Xn)?
e1=(1,0,0).?
cos=X1|X|.?
cos?
x(1,1),P(|X1|X|x)=ZZ|y1|x|y|,yRn
(2)n/2exp12|y|2dy.?
|y1|py22+y2nx1x2.?
x?
2
(2)n/2(1x2)3/2Ze12(y22+y2n)(y22+y2n)1/2dy2dyn=Cn(1x2)(n3)/2,?
1/Cn=R10(1x2)(n3)/2dx.?
X1|X|?
n=3?
2822.?
(X,Y)?
2?
EX=EY=0,EX2=EY2=1,(X,Y)=.(a)?
X?
YX12?
(b)?
(X,Y)?
p122(12sincos),00,Y0).(d)?
Z=max(X,Y),?
EZ=p
(1)1,EZ2=1.
(1)?
(X,Y1)(?
Y1=YX12)?
DY1=EY21=112EY22EXY+2EX2=1?
cov(X,Y1)=0.
(2)?
tan0=12,P(X0,Y0)=Zx0,y012p12exp?
x2+y22xy2(12)?
dxdy=Zx0,12y+x012exp12(x2+y2)dxdy=Z/20dZ012rer22dr=12(/20)=12(/2+arcsin),?
(x,y)7(x,yx12).(3)?
(2)?
EZ=2Zyxyf(x,y)dxdy=1Z12yx+x(p12y+x)e12(x2+y2)dxdy=1p12I(r11+)+1(I(r1+1)=p
(1)/.?
(?
):
?
0?
I()=Zyxye12(x2+y2)dxdy=r21+2.?
29?
EZ2=1Zy1sqrt12x(p12y+x)2e12(x2+y2)dxdy,?
J():
=12Zyxy2e12(x2+y2)dxdy,0.?
J()=12RR(x)2e12(1+2)x2xdx=0,?
J()?
J()=J(0)=1/2.?
Zyxxye12(x2+y2)dxdy=0,?
EZ2=1(2(12)12+2212)=1.23.
(1)?
O?
A,B,?
OAB?
(2)?
A,B,C,?
ABC?
(1)?
(,)?
SOAB=|sin|,?
211x2,?
2/.
(2)?
A?
(1,0)?
1,2?
y?
AB,AC?
(0,)?
SABC=2|sin1sin2sin(21)|.?
ESABC=3/2.24.?
1,2,m+n(nm)?
S=1+2+n?
T=m+1+m+2+m+n?
(S,T)=1m/n.25.?
E2.?
|?
26.?
1,n?
F?
i?
(i)?
Fi(x)=n!
(i1)!
(ni)!
ZF(x)0ti1(1t)nidt.?
30?
F?
(0,1)?
(i)?
i,ni+1?
x(0,1)?
0in,?
n!
(i1)!
(ni)!
Zx0ti1(1t)nidt=nXk=i?
nk?
xk(1x)nk.?
ni?
Zx0iti1(1t)nidt=?
ni?
xi(1x)ni+?
ni+1?
Zx0(i+1)ti(1t)ni1dt,?
27.?
1,n?
F?
(a)?
(1),(n)?
(b)?
i?
(i)(i1):
1in(?
(0):
=0)?
(c)?
F?
ij,?
(i)/(j)?
(j)?
(1)?
n!
f(x1)f(xn)1x1x2xi,1in)=n!
ZZyiyi1xi,1iney1yndy1dyn=n!
exnZZyiyi1xi,1in1ey1yn22yn1dy1dyn1=exp(nXi=1(ni+1)xn).(3)(k)?
n!
10x1xnc0,?
(4)cC0.?
AB?
AB:
ya=baba(xa),?
c0=a+baba(ca).?
c0=0,?
b=cbcaa,?
b1.?
c0=1c,?
b=cbcaa+(1c)baca,?
32?
b2.?
b,c?
Z1b0Z1c0|cc0|dbdc=Zb10Z1c0(cc0)+Zb2b1Zc00(c0c)+Zb2b1Z1cc0(cc0)+Z1bb2Z1c0(c0c)dbdc=.?
1/72?
Mathematica?
6?
30.?
?
+?
?
f(y):
=E|+y|,?
Fubini?
Ef()=E|+|,?
f()?
y,?
f(y),?
E|E|+y|+|y|0,Y0).?
P(X0,Y0)=1/4,?
P(X+Yz,X0,Y0)=12Z0ey22dyZzy0ex22dx,?
(?
).?
E(X+Y|X0,Y0)=4Zx0Zy0(x+y)12ex2+y22dxdy=4/2.?
335.?
(X,Y)?
D?
D?
(0,0),(1,0),(1,1)?
E(Y|X).6.?
N(a,2),?
E().(?
U,?
()=P(U|).)7.?
(,)?
E(|)=ZRsf|(s|)ds.8.?
(X,Y)?
EX=EY=0,EX2=2,EY2=2,(X,Y)=.?
(a)E(X|Y)=Y;(b)E(X|X+Y)=2+2+2+2(X+Y).
(1)?
X,Y?
fX|Y(x|y)=f(x,y)/f(y)=1p2(12)exp(xy)22(12).?
X|Y=y?
N(y,12)?
E(X|Y)=Y.
(2)?
(1).E(X|X+Y)=(X,X+Y)X+YX+Y.9.?
X?
n,U?
U?
0,1?
X?
0kn,P(X=k)=E(P(X=k|U)=E?
nk?
Uk(1U)nk=Z10?
nk?
xk(1x)nkdx=?
nk?
B(k+1,nk+1)=1n+1.10.?
X?
f,?
E(X|X|).?
x0,E(X|X|=x)=lim0E(X|X|(x,x+)=lim0R|y|(x,x+)yf(y)dyR|y|(x,x+)f(y)dy?
34=xf(x)xf(x)f(x)+f(x),?
E(X|X|)=|X|f(|X|)|X|f(|X|)f(|X|)+f(|X|).?
g?
E(X|X|)=g(|X|),?
g.11.?
Yk:
1kn?
Xi:
=Pnj=1cijYj,?
cij?
E(Xj|Xk)=?
PicjickiPic2ki?
Xk.1
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