PA · PU . + P( − P( = PA + (1 − PA · PU ) − 1 = PA · (1 − P = P

ËØÓ×Ø×ËÒÐÙÒËÝ×ØÑ ÙÏÖ×ÒÐØ×ØÓÖ ÅÙ×ØÖÐ×ÙÒÖÃÐÙ×ÙÖÚÓÑ
P(ÈØÒØÖÐØ)ÞÒÏÖ×ÒÐØ××ÖÈØ ÈØÒØ×ØÖØ ÒØÖÐØÙÒP(ÈØÒØ×ØÖØ)ÏÖ×ÒÐØ××Ö
P(ÈØÒØÖÐØ) =
P(ÈØÒØ×ØÖØ)
= PA ·PU
= 1−PA ·PU.
P(ÐÐØÙ×ÈØÒØÖÐØ) = P(ÐÐØÙ×∩ÈØÒØÖÐØ)
P(ÈØÒØÖÐØ)
P(ÐÐØÙ×∩ÈØÒØÖÐØ) =
ËÓÑØ×Ø P(ÐÐØÙ×ÈØÒØÖÐØ) = P(ÐÐØÙ×∩ÈØÒØÖÐØ)
P(ÈØÒØÖÐØ)
P(ÐÐØÙ×)
+ P(ÈØÒØÖÐØ)
− P(ÐÐØÙ×∪ÈØÒØÖÐØ)
= PA +(1−PA ·PU)−1
= PA ·(1−PU).
= PA
×ØÖØÛÒÒ×ÖÛÙÒ××Ý×ØÑÒØÚÓÖÒÒ×Ø
MÞÒ×ÖÒ×××ÖÈØÒØ×ØÖØÛÒÒ×ÖÛ ÙÒ××Ý×ØÑÚÓÖÒÒ×ØOÞÒ×ÖÒ×××ÖÈØÒØ
·(1−PU)
.
1−PA ·PU
P(M) = PA ·PU
= 2·P 2 A
= 0.005.
ÚÓÒ
P(O) = PA
= 0.05.
⇒ P(M) < P(O).

ËØÓ×Ø×ËÒÐÙÒËÝ×ØÑ ÂØÞØÛÖÔÖØÛÓÙ×ÐÐÛÖ×ÒÐØÚÓÒÑ ÅÙ×ØÖÐ×ÙÒÖÃÐÙ×ÙÖÚÓÑ
PUÒÏÖ×ÒÐØ×ØÒÒ×ÒØÖÖÐ× ÖÛÙÒ××Ý×ØÑÑØÒÖÏÖ×ÒÐØÚÓÒ1ÛÒÒ× Ù×ÐÐØ ÛÖÒ
×ÖÒ××ÖÛÙÒ××Ý×ØÑÐÐØÙ××ØÞØ×Ù×ÞÛ
Ù××ÚÓÑÖØÞØ×ØÑÑØÒÙÖÛÒÒPU =
ÌÐÖÒ××ÒÞÙ×ÑÑÒU2ÐÐØ×ÖÙ×ÛÒÒÀÙÔØÓÑÔÓ
1ØÛÖ×ÞÙÑËÒÒÖËØÖÛÖ×ÒÐØÖÒ
ÒÒØÚÖ×ØÙÒØÓÒÖØÀÙÔØÓÑÔÓÒÒØÐÐØ×ÓÑÔÐØØ
ËÓÐÒ×ÖÛÙÒ××Ý×ØÑÒÙ×ÐÐÛÖ×ÒÐØPU <
ËÝ×ØÑÒÙÖÒÒÙ×ÛÒÒÆÒÓÑÔÓÒÒØÒÚÖ×ÒÖ Ù×ÐÐÛÖ×ÒÐØPU2ÖØ×ÑØ
ÑØÙ×ÐÐÛÖ×ÒÐØ×ÒÙÒËÝ×ØÑ×ÖÒÖÐ× ×ÐØÒËÝ×ØÑ××ØÑÙ××ÐØÒ
ÖÛÙÒ××Ý×ØÑ×ÒÑÙ××ÑØÙ×××ØÑÑØ
P(M) ≥ P(O)
PA ·PU ≥ PA
PU ≥ 1.
PU2 = PHK +(1−PHK)P 2 NK
= PHK +P 2 NK −PHKP 2 NK
= P 2 NK +(1−P 2 NK)PHK.
PU2 = P2 NK +(1−P 2 NK)PHK < PU
ÚÓÒ
PHK < PU −P 2 NK
1−P 2 NK
PHK < 0.1−0.04
1−0.04
PHK < 1
16

ËØÓ×Ø×ËÒÐÙÒËÝ×ØÑ ÅÙ×ØÖÐ×ÙÒÖÃÐÙ×ÙÖÚÓÑ
PU2
= P2 NK +(1−P 2 NK)PHK
= 0.04+0.96·0.1 = 0.136
P(ÖÐÒÑØU2) = P(ÖÐÒÑØU) =
ÏÖ×ÒÐØÞÙÖÐÒ×ÒØÑØÑÒÙÒÖÛ ÙÒ××Ý×ØÑÙÑØÛ
P(ÖÐÒÑØU2) P(ÖÐÒÑØU)
ÚÓÒ
1−PA ·PU2
= 1−0.05·0.136 = 0.9932.
1−PA ·PU
= 1−0.05·0.1 = 0.995.
0.9932
= = 0.9982.
0.995
ÈÙÒØ

ËØÓ×Ø×ËÒÐÙÒËÝ×ØÑ ÙÙÐÐ×ÚÖÐÒ ÅÙ×ØÖÐ×ÙÒÖÃÐÙ×ÙÖÚÓÑ
ÊÒØÒÖÒÒ×ÙÖÅÖÒÐ×ÖÙÒÞÙ
fX(x) =
=
∞
−∞
∞
−∞
fXY(x,y)dy
e −x
2
2a
e−x2
dy =
2a
a
0
dy = e−x2
2a
e−x2
a =
2 ,
∞
fY(y) = fXY(x,y)dx
−∞
∞
e
=
−∞
−x ∞
2 1
dy = e
2a 2a 0
−x 2dx = 1 −
−2e
2a
x∞
1
2 = 0 a
• X×ØÜÔÓÒÒØÐÐÚÖØÐØÑØÈÖÑØÖλ = 1
2×ËÖÔØËØÓÒ2ÂÒÒØÓÒ
Ùa = 0,b =
• Y×ØÐÚÖØÐØÙÑÁÒØÖÚÐÐ[0,a]
×ÑØ Ù×ÖÍÒÒØÚÓÒXÙÒYÓÐØ××XÙÒYÙÒÓÖÖÐÖØ
E[X]E[Y]ÅØÒÖÒ××ÒÙ×ÖØ XÙÒY×Ò×ØØ×Ø×ÙÒÒfX(x)fY(y) = fXY(x,y) ×ÒÙÒ×ÓÑØE[XY] =
E[XY] = E[X]E[Y] = a ÙÖÍÒÒØÚÓÒX1ÙÒX2ÐØÖÏÖ×ÒÐ
2 = a.
2
X1ÙÒX2ÒØ×ÚÖØÐØ×ÒÐØÙÖÑ
Ø×ÖÒ×××{Z < z}
P({Z < z}) = P({X1 < z})∩P({X2 < z})
= P({X1 < z})·P({X2 < z}).
P({X1 < z})·P({X2 < z}) = [P({X1 < z})] 2 = [FX(z)] 2 ÑØÖØ×FZ(z) = P({Z < z}) = [FX(z)] 2 ÙÒÙÖÐØÒÃØØÒÖÐ×ÐÐ
fZ(z) = d
dz FZ(z) = d
dz [FX(z)] 2 ÚÓÒ
= 2FX(z)fX(z).

ËØÓ×Ø×ËÒÐÙÒËÝ×ØÑ X1ÙÒX2ÐÚÖØÐØ×ÒÙ[0,1]ÐØÖÖÎÖØÐÙÒ× ÅÙ×ØÖÐ×ÙÒÖÃÐÙ×ÙÖÚÓÑ
ÙÒØÙÒØÓÒFX(x) =
ÙÖÒ×ØÞÒÒÐÙÒÙ×ÌÐÙÖÐØÑÒ
ÖÖÛÖØÙÒ×ÛÖØE[Z]ÖÒØ×ÒØ×ÔÖÒÞÙ
ÐÒÖÐ×0.5×ØÐÒÙÐÐÐ×Ó ÏÖ×ÒÐØ××max{X1,X2}ÖÖÐ××ØÒØ×ÔÖØ
X1
ÖÖÏÖ×ÒÐØ××X2ÖÖÐ×0.5×ØÒÙÖ
= 0.5ÒÒØ×Ø×ØÏÖ×ÒÐØ××max{X1,X2}
ÖÖÔÖÎÖØÐÙÒ×ÙÒØÓÒÖØ×Ð×ÓÞÙ
E[Z] =
∞
fX(x) =
fZ(z) =
−∞
x, 0 ≤ x ≤ 1
0, ×ÓÒ×Ø.
zfZ(z)dz =
1, 0 ≤ x ≤ 1
0, ×ÓÒ×Ø.
2z, 0 ≤ z ≤ 1
0, ×ÓÒ×Ø.
1
FZ(z |X1 = 0.5) = 0, Öz
0
2z 2 dz =
<
2
3 z3
1
0.5.
0
= 2
3 .
FZ(z |X1 = 0.5) = FX2 (z) = FX(z) = z, Ö0.5 ≤ z ≤ 1.
FZ(z |X1 = 0.5)
1
0.5
−0.5 0 0.5 1 1.5
ÚÓÒ
ÑØÒÖÍÒ×ØØØz= 0.5
ÈÙÒØ
z

ËØÓ×Ø×ËÒÐÙÒËÝ×ØÑ ÙËØÓ×Ø×ÈÖÓÞ××ÙÒÐÒÖËÝ×ØÑ ÅÙ×ØÖÐ×ÙÒÖÃÐÙ×ÙÖÚÓÑ
ÖÖÛÖØÙÒ×ÛÖØÚÓÒX(n)ÖÒØ×Ð×
ÅÓÑÒØÒÙÒØÓÒ ÇÖÒÙÒ×ØÒÙÖ
ÖÙÐÐ×ÔÖÓÞ××X(n)×ØÒÙÖ
X(n) = A1e j(ω0n+φ1) +A2e j(ω0n+φ2)
2
E[X(n)] = E[Am]e
m=1
jω0n
(E[cos(φm)] +jE[sin(φm)] )
=0 =0
= 0
rXX(n+κ,n) = E[X(n+κ)X(n) ∗ ]
2
= E Ame
m=1
jω0(n+κ)
2
jφm e ·
l=1
2 2
= E[AmAl]e
m=1 l=1
jω0κ
·E e j(φm−φl) 0×ÓÒ×Ø
1ÐÐ×m = l
2
= E A 2
jω0κ
m e
m=1
= (1+σ 2 A1 +σ2 A2 )ejω0κ
Ale −jω0n e −jφl
= rXX(κ). ÀÖÒÙØÞÒÛÖE A 2
m = E[Am] 2 +σ 2 Am . ÑÛØÖÒËÒÒ
rXX(κ)×ØX(n)×ØØÓÒÖ E[X(n)]ÓÒ×ØÙÒrXX(n+κ,n) = Ö×Ä×ØÙÒ×Ø×ÔØÖÙÑSXX(e jω )ÐØ
SXX(e jω ∞
) = rXX(κ)e −jωκ
κ=−∞
∞
= (1+σ
κ=−∞
2 A1 +σ2 A2 )e−j(ω−ω0)κ
= 2π(1+σ 2 A1 +σ2 ÚÓÒ)δ(ω
−ω0), Ö−π A2
≤
ω < π

ËØÓ×Ø×ËÒÐÙÒËÝ×ØÑ ÅÙ×ØÖÐ×ÙÒÖÃÐÙ×ÙÖÚÓÑ
CXX(ejω )
ÈËÖÖÔÐÑÒØ× 2π(1+σ 2 A1 +σ2 A2 )
ω
−π ω0 π
ÅØH(e jω ) = 1+ae −jω +be−j2ωÓÐØ )δ(ω −ω0)
|1+ae −jω0 −j2ω0 2 2 +be | ·2π(1+σ A1 +σ2 A2 )ÐÐ×ω =
CYY(e jω ) = |1+ae −jω +be −j2ω | 2 ·2π(1+σ 2 A1 +σ2 ÙÒ
A2
=
CXY(e jω ) = (1+ae jω +be j2ω )·2π(1+σ 2 A1 +σ2 ×ÐØ
A2
=
CZZ(e jω ) = CYY(e jω )+CVV(e jω ×ÓÑØ×Ø
),
CZZ(e jω ) =
0 ×ÓÒ×Ø.
)δ(ω −ω0)
(1+ae jω0 j2ω0 2 +be )·2π(1+σ A1 +σ2 A2 )ÐÐ×ω =
0 ×ÓÒ×Ø.
ω0
ω0
σ2 V +|1+ae−jω0 −j2ω0 2 2 +be | ·2π(1+σ A1 +σ2 A2 )ÐÐ×ω =
σ2 ×ÓÒ×Ø.
V
ÚÓÒ
ÈÙÒØ
ω0

ËØÓ×Ø×ËÒÐÙÒËÝ×ØÑ ÅÙ×ØÖÐ×ÙÒÖÃÐÙ×ÙÖÚÓÑ
CZZ(e
ÈËÖÖÔÐÑÒØ× 1+6π
jω )
1
ÙÇÔØÑÐÐØÖ −π ω0 π
cXY(κ) = E[X(n+κ)Y(n)]
CYY(e
= E[X(n+κ)(X(n)+W(n)+α·X(n−n0))]
= E[X(n+κ)X(n)]+E[X(n+κ)W(n)]+α·E[X(n+κ)X(n−n0)]
= cXX(κ)+cXW(κ)+α·cXX(κ+n0)
= cXX(κ)+α·cXX(κ+n0).
cYY(κ) = E[Y(n+κ)Y(n)]
= E[(X(n+κ)+W(n+κ)+α·X(n+κ−n0))
(X(n)+W(n)+αX(n−n0))]
= E[X(n+κ)X(n)]+E[X(n+κ)W(n)]+α·E[X(n+κ)X(n−n0)]
+E[W(n+κ)X(n)]+E[W(n+κ)W(n)]
+α·E[W(n+κ)X(n−n0)]+E[X(n+κ−n0)X(n)]
+E[X(n+κ−n0)W(n)]+α·E[X(n+κ−n0)X(n−n0)]
= cXX(κ)+cXW(κ)+α·cXX(κ+n0)
+cWX(κ)+cWW(κ)+α·cWX(κ+n0)
+α·cXX(κ−n0)+α·cXW(κ−n0)+α 2 cXX(κ)
= cXX(κ)(1+α 2 )+α·(cXX(κ−n0)+cXX(κ+n0))+cWW(κ).
ÚÓÒ
jω ) = CXX(e jω )(1+α 2 +α(e −jωn0 +e jωn0 ))+CWW(e jω )
ω

ËØÓ×Ø×ËÒÐÙÒËÝ×ØÑ ÅÙ×ØÖÐ×ÙÒÖÃÐÙ×ÙÖÚÓÑ
CXY(e jω ) = CXX(e jω )(1+αe +jωn0 ×ÏÒÖÐØÖ×ØÓÔØÑÐÒÑËÒÒ×××ÒÑØØÐÖÒÕÙ
ÐÖØ×ÒÑÙ××ÑØÑÓØØÒÒÒ××ÒÐy(n) ÖØ×ÒÐÖÅËÑÒÑÖØ ×ÇÖØÓÓÒÐØØ×ÔÖÒÞÔ×Ø×××ÐÖ×ÒÐε(n)ÙÒÓÖÖ
)
cεY(κ) = 0,∀κ ∈ Z,
ε(n)
n0 = 0ÖCXX(e
= X(n)− ˆ X(n)
= X(n)−
+∞
m=−∞
cεY(κ) = E[ε(n+κ)Y(n))]
= E
= cXY(κ)−
= 0.
(X(n+κ)−
+∞
m=−∞
CXY(e jω ) = H(e jω )CYY(e jω )
H(e jω ) = CXY(e jω )
=
jω )
CWW(ejω )
lim
C XX (e jω )
C WW (e jω ) →∞
CYY(e jω )
hW(m)Y(n−m).
+∞
m=−∞
hW(m)Y(n+κ−m))Y(n)
hW(m)cYY(κ−m)
CXX(e jω )(1+αe −jωn0 )
CXX(e jω )(1+α 2 +α(e −jωn0 +e jωn0))+CWW(e jω )
→ ∞ØÖÐØÖÞÙ
H(e jω ) = lim
C XX (e jω )
C WW (e jω ) →∞
= lim
C XX (e jω )
C WW (e jω ) →∞
=
1
1+α .ÚÓÒ
CXX(e jω )(1+α)
CXX(e jω )(1+2α+α 2 )+CWW(e jω )
1+α
1+2α+α 2 + CWW(e jω )
CXX(e jω )

ËØÓ×Ø×ËÒÐÙÒËÝ×ØÑ ×ÐØÖÛÖØ×ÓÑØÛÒÐÐÔ×××ËÒÐÞÙÊÙ×ÚÖÐØÒ× ÅÙ×ØÖÐ×ÙÒÖÃÐÙ×ÙÖÚÓÑ
ÖÓ×ØÐ××Ø×ÐØÖ×ÒÒ××ÒÐÙÒÚÖÒÖØÙÖ ÖCXX(e jω 0Ð××Ø×ÐØÖÒØ×ÙÖ
→
)
CWW(ejω )
lim
C XX (e jω )
C WW (e jω ) →0
H(e jω ) = lim
C XX (e jω )
C WW (e jω ) →0
CXX(e jω )(1+α)
CXX(e jω )(1+2α+α 2 )+CWW(e jω )
= lim
CXX (ejω )
CWW (ejω ) →0
1
1+2α+α 2 + CWW(ejω )
CXX(ejω )
×ÐØÖÑÙ××Ù×Ð×ÒÙÑÒÖÈÖÜ×Ò×ØÞØÛÖÒÞÙ ×ÐØÖÛÖØ×ÓÑØÛÒÐÐ×ØÓÔÛÐ×ËÒÐÞÙÊÙ×ÚÖÐØÒ× ×ÖÐÒ×ØÙÒ×ÊÙ×ÒÐÓÖØÛÖÒÑÙ××
= 0.
ÒÒÒÙÖÑÑÙ××ÁÑÔÙÐ×ÒØÛÓÖØÒÐ×Ò
Æ
ÚÓÒ
ÈÙÒØ