1nÙ ‘ÅCþ êiA
3.1 êÆÏ"9© ê
3.1.1 êÆÏ"
êÆÏ"´‘ÅCþ ˜‡•Ä êiA "·‚kwXe ˜‡~f
~3.1.1. ˜‡œhúi3˜‡}&‘8¥¤¦^ }Þ±YóŠ2§3§4 ž ŒU5•0.1, 0.7Ú0.2" yTúik10‡da. }Þ§Kd}&‘8Œ±±Yõ•žmº
): <‚˜„¦^\ ²þ
2 · 0.1 + 3 · 0.7 + 4 · 0.2 = 3.1 , Œ±±Y10 · 3.1 = 31 ž"
ùp \ ²þ·‚¡•Ï"Š½öXe‘ÅCþ êÆÏ"µ
X 2 3 4
P 0.1 0.7 0.2
阄 lÑ.©Ù§·‚k
½
½
½Â 3.1.1. X•˜lÑ.‘ÅCþ§Ù©ÙÆ•
P (X = xi) = pi, i = 1, 2, · · · XJP∞
i=1
|xi|pi < +∞§K¡
∞
X
i=1
xipi
•‘ÅCþX êÆÏ"({¡•Ï"½öþŠ), ^ÎÒEXL«X êÆÏ"" eP∞
i=1
|xi|pi = +∞§K¡X êÆÏ"Ø•3"
~3.1.2. r.v. X ©ÙÆ•
P
X = (−1)k2k k
= 1
2k, k = 1, 2, · · · KX êÆÏ"Ø•3"
): du
∞
X
k=1
|(−1)k2k k|1
2k =
∞
X
k=1
1
k = +∞
ÏdX êÆÏ"Ø•3" ¦+
∞
X
k=1
(−1)k2k k
1 2k =
∞
X
k=1
(−1)k1
k = −ln2.
éëY.‘ÅCþ§ÙêÆÏ" ½ÂXe
½
½
½Â 3.1.2. XJëY.‘ÅCþξäk—ݼêp(x), K Z ∞
−∞
|x|p(x)dx < ∞ ž, ·‚òÈ©
Z ∞
−∞
xp(x)dx Š¡•ξ êÆÏ", PŠEξ. XJ
Z ∞
−∞
|x|p(x)dx = ∞, K¡ξ êÆÏ"Ø•3.
~3.1.3. (Cauchy©Ù)
p(x) = 1
π(1 + x2), x ∈ R, K: T©Ù Ï"Ø•3.
):N´wÑ,p(x)šK, ¿…
Z ∞
−∞
p(x)dx = 1 π
Z ∞
−∞
1
1 + x2dx = 1
π arctan x
∞−∞ = 1,
¤±p(x)´˜‡—ݼê(¡•Cauchy©Ù), ´ Z ∞
−∞
|x|p(x)dx = 2 π
Z ∞ 0
x
1 + x2dx = ∞,
¤±Cauchy©Ù Ï"Ø•3. #
˜„§·‚rþãü‡½Â 3˜åµ
½
½
½Â 3.1.3. ξ•˜‘ÅCþ§Ù©Ù¼ê•F (x)§XJR+∞
−∞ |x|dF (x) < ∞§K¡
Z +∞
−∞
xdF (x)
•‘ÅCþξ êÆÏ"§P•Eξ"
êÆÏ" ˜„5ŸXe
b c•~ê, ¿…e¡ 9 Ï"Ñ´•3 "·‚k 1. Ec = c
2. Ecξ = cEξ
3. E(ξ + η) = Eξ + Eη 4. eξ ≥ 0§KEξ ≥ 0 5. eξ ≥ η§KEξ ≥ Eη.
6. g(x)•˜BorelŒÿ¼ê(•) ëY¼ê! F¼ê )§K Eg(ξ) =
Z ∞
−∞
g(x)dFξ(x)
5Ÿ6`² OŽ‘ÅCþ¼ê Ï"† l 5 ‘ÅCþ©ÙÑu.
~3.1.4. r.v. X ∼ N (0, 1)§¦Y = X2+ 1 êÆÏ""
~3.1.5. œÅ|1•ð•þk20 ¦•§lmÅ| k10‡•ÕŒ±e•§e,‡•Õvk<
e•KT•ÕØÊ•" ¦•3z‡•Õe• ŒU5ƒ §±XL«Ê• gꧦEX"
):
Yi=
( 1, 1 i ‡•Õk<e•
0, 1 i ‡•ÕÃ<e• i = 1, · · · , 20.
Kw,X = P20
i=1
Yi§¤±
EX =
20
X
i=1
EYi=
20
X
i=1
P (1 i ‡•Õk<e•)
=
20
X
i=1
[1 − 0.920] = 8.784.
½
½
½nnn 3.1.1. XJξÚη´½Â3Ó˜‡Vǘmþ ƒpÕá ‘ÅCþ, §‚ êÆÏ"Ñ•
3, K§‚ ¦Èξη êÆÏ"••3, ¿…k
Eξη = EξEη.
3.1.2 ^‡Ï"
·‚• ^‡©Ù•´˜‡VÇ©Ù§ÏdaqêÆÏ" ½Â§·‚Œ±½Â^‡Ï""
½
½
½Â 3.1.4. r.v. X3r.v. Y = y ^‡e ^‡©Ù•F (x|Y = y)"K R−∞∞ |x|dF (x|Y = y) <
∞ž§·‚¡
E(X|Y = y) = Z ∞
−∞
xdF (x|Y = y)
•‘ÅCþX3‰½^‡Y = ye ^‡Ï""
Ï"¤äk 5Ÿ^‡Ï"Ó ÷v"
~3.1.6. (X, Y ) ∼ M (N, p1, p2)§ÁOŽE(X|Y = k)"
)µduX|Y = k ∼ B(N − k,1−pp1
2), ¤±d ‘©Ù 5Ÿ•E(X|Y = k) = (N − k)1−pp1
2.
½
½
½nnn 3.1.2. X, Y •ü‡‘ÅCþ§g(X)•ŒÈ ‘ÅCþ"Kk
Eg(X) = E{E[g(X)|Y ]} [ Ï"úª]
Proof. ·‚=3ëY.‘ÅCþ œ/ey²d½n" Y p.d.f •p(y)§X|Y = y p.d.f•q(x|y)"
K
Eg(X) =
Z Z ∞
−∞
g(x)q(x|y)p(y)dxdy
=
Z Z ∞
−∞
g(x)q(x|y)dxp(y)dy = Z ∞
−∞
E[g(X)|Y = y]p(y)dy
= E{E[g(X)|Y ]}
~3.1.7. ˜‡M '3k3‡€ /Op§Ù¥1˜‡€Ï•gd"Ñù€r3‡ žBŒ±£
/¡¶12‡€Ï•,˜‡/ §r5‡ žòˆ£ /O¶13‡€Ï••• / §r7‡ ž
•£ /O"e‡MzgÀJ3‡€ ŒU5oƒÓ§¦¦•¼ gd r ²þžm"
)µ ù‡‡MI‡rX žâU ˆ/¡§¿ Y “L¦zgé3‡€ ÀJœ¹§Y ˆ
±1/3 VÇ Š1§2§3"K
EX = E[E(X|Y )] =
3
X
i=1
E(X|Y = i)P (Y = i)
5¿ E(X|Y = 1) = 3, E(X|Y = 2) = 5 + EX, E(X|Y = 3) = 7 + EX§¤±
EX = 1
3[3 + 5 + EX + 7 + EX]
= EX = 15"
3.1.3 © êÚp© ê
·‚®²• §‘ÅCþξ êÆÏ"Ò´§ ²þŠ§Ïdl˜½¿Âþ§êÆÏ"•x
‘ÅCþ¤ ƒŠ ”¥% ˜”" ´§·‚•Œ±^O êiA 5•x‘ÅCþ /¥%
˜0" ¥ êÒ´ù ˜«êiA " 3Ø•3êÆÏ" ‘ÅCþ§ù«•xóäw c•
-‡§=¦éu•3êÆÏ" ‘ÅCþ§¥ ê•´˜«ƒ k^ êiA "
½
½
½Â 3.1.5. ¡µ•‘ÅCþξ ¥ ê§XJ P (ξ ≤ µ) ≥ 1
2, P (ξ ≥ µ) = 1 2.
~3.1.8. r.v. ξ ∼ B(1,12)§¦ξ ¥ ê"
)µduξ ©Ù¼ê•
F (x) =
0, x ≤ 0
1
2, 0 < x < 1 1, x ≥ 1 d¥ ê ½Â•«m(0,1)S z˜‡êÑ´ξ ¥ ê"
½
½
½Â 3.1.6. 0 < p < 1§¡µp´‘ÅCþξ p© ê§XJ P (ξ ≤ µp) ≥ p, P (ξ ≥ µp) ≥ 1 − p.
3.2 • ! • ÚÝ
3.2.1 ‘ÅCþ ݆•
Ø Ï" §XJ‘ÅCþξ•rgŒÈž§·‚„Œ±•ÄEξr9E|ξ − Eξ|r"©O¡•‘Å Cþξ r :ÝÚ¥%Ý"½ÂXe
½
½
½Â 3.2.1. ‘ÅCþξ rgŒÈ§=
E|ξ|r= Z ∞
−∞
|x|rdFξ(x) < ∞ K·‚¡
Eξr= Z ∞
−∞
xrdFξ(x) E|ξ − Eξ|r=
Z ∞
−∞
|x − Eξ|rdFξ(x)
© O • ‘ Å C þξ r : Ý Ú ¥ % Ý" r = 2ž§¡E(ξ − Eξ)2• ‘ Å C þξ • , P
•D(ξ)½öV ar(ξ)"w,k
D(ξ) = Eξ2− (Eξ)2
é‘ÅCþ • §·‚Œ±
½
½
½nnn 3.2.3. ±e‘ÅCþ 2 Ý•3k•§c•~ê. Kk
1. 0 ≤ D(ξ) = Eξ2− (Eξ)2, ÏdD(ξ) ≤ Eξ2. 2. D(cξ) = c2D(ξ)
3. D(ξ) = 0 …= P (ξ = c) = 1§Ù¥c = Eξ"
4. é?Û~êck§D(ξ) ≤ E(ξ − c)2§Ù¥ Ò¤á …= c = Eξ"
5. XJ‘ÅCþξÚηƒpÕá§a, b•~ê" KD(aξ + bη) = a2D(ξ) + b2D(η)"
•y²þã½n§·‚0 ˜‡Ún"
Ú Ú
Únnn 3.2.1. XJξ•òzu0 ‘ÅCþ§KkEξ2 = 0¶‡ƒ§XJ‘ÅCþξ 2 Ý•3
…Eξ2 = 0§Kξ7•òzu0 ‘ÅCþ.
Proof. XJξ•òzu0 ‘ÅCþ§KkP (ξ = 0) = 1, kEξ2 = 0" ‡ƒ§XJ‘ÅCþξ²
•ŒÈ§¿…Eξ2 = 0§ ´ξØòzu0§KkP (ξ = 0) < 1"@oÒ•3δ > 0 Ú0 < < 1§¦
P (|ξ| > δ) > §u´Eξ2 > δ2" —gñ§¤±ξ7òz 0.
~3.2.1. P ossion©ÙP (λ) • ´λ, ‘©ÙB(n, p) • ´np(1 − p)§ ©ÙN (a, σ2)
• •σ2"
½
½
½Â 3.2.2. šòz ‘ÅCþŒÈ§·‚¡
ξ∗ = ξ − Eξ pD(ξ)
•ξ IOz‘ÅCþ§Ù¥pD(ξ)¤•ξ IO "´„Eξ∗ = 0, D(ξ∗) = 1.
·‚Ú\IOz‘ÅCþ´• žØduOþü ØÓ ‰‘ÅCþ‘5 K•. ~X, ·
‚• < p, @o ,Œ±±’•ü , ξ1, •Œ±±f’•ü , ξ2. u´Òk ξ2 = 100ξ1. @où ˜5, ξ2†ξ1 ©ÙÒk¤ØÓ. ù ,´˜‡ØÜn y–. ´ÏLI Oz, ÒŒ±žØüöƒm O, Ï•·‚kξ∗2 = ξ1∗.
3.2.2 • Ú •
XJξÚη´½Â3Ó˜‡Vǘmþ ü‡‘ÅCþ, …ξ, 粕ŒÈ, @o·‚ÒŒ±
¦ξ + η • :
D(ξ + η) = E ((ξ + η) − E(ξ + η))2= E ((ξ − Eξ) + (η − Eη))2
= E(ξ − Eξ)2+ E(ξ − Eξ)(η − Eη) + E(η − Eη)2
= Dξ + E(ξ − Eξ)(η − Eη) + Dη.
3þª¥Ñy \‘E(ξ − Eξ)(η − Eη), dué?Û¢êx, y, Ñk|xy| ≤ 12(x2+ y2), ¤±·‚k
|E(ξ − Eξ)(η − Eη)| ≤ E|ξ − Eξ||η − Eη| ≤ 1
2(Dξ + Dη).
¤±•‡ξ, 粕ŒÈ, @oÒkE(ξ − Eξ)(η − Eη)•3.
½
½
½Â 3.2.3. ‘ÅCþξ, 粕ŒÈ, ·‚¡
cov(ξ, η) = E(ξ − Eξ)(η − Eη)
•ξ†η • , Ù¥cov´=©ücCovariance .
d • ½Â·‚Œ±
• äkXe5Ÿµ 1. cov(ξ, η) = cov(η, ξ)
2. D(ξ + η) = D(ξ) + cov(ξ, η) + D(η) 3. cov(ξ, η) = E(ξη) − (Eξ)(Eη) 4. é?Û¢êa1, a2, b1, b2§k
cov(a1ξ1+ a2ξ2, b1η1+ b2η2) =
2
X
i=1 2
X
j=1
aibjcov(ξi, ηj)
XJξ1, · · · , ξn´½Â3Ó˜Vǘme ‘ÅCþ§¿…Ù¥z‡‘ÅCþÑ´²•ŒÈ
"¡Ý
Σ = (bij) = (cov(ξi, ξj))
=
D(ξ1) cov(ξ1, ξ2) · · · cov(ξ1, ξn) cov(ξ2, ξ1) D(ξ2) · · · cov(ξ2, ξn)
... ... . .. ... cov(ξn, ξ1) cov(ξn, ξ2) · · · D(ξn)
•ξ1, · · · , ξn • Ý "w,Σ ≥ 0"
~3.2.2. (X, Y ) ∼ N (a, b, σ21, σ22, ρ)§K(X, Y ) • Ý •
Σ = σ21 ρσ1σ2 ρσ1σ2 σ22
!
3.2.3 ƒ'Xê
30 ƒ'Xê ½Âƒc§·‚wXeÚn"
Ú Ú
Únnn 3.2.2. [Cauchy − Schwarz Inequality] ξ, ηþ²•ŒÈ§Kk [Eξη]2≤ Eξ2Eη2
Ò¤á …= P (ξ = t0η) = 1§Ù¥t0•˜~ê"
Proof. ´•, é?Ût ∈ R, Ñk
g(t) := Eη2· t2− 2Eξη · t + Eξ2 = E(ξ − tη)2≥ 0 ,
¤± g¼êg(t) Oª
∆ = 4(Eξη)2− 4Eξ2· Eη2 ≤ 0, Ø ª.
XJ•3t0∈ R, ¦ P (ξ = t0η) = 1, w,Òk (Eξη)2 = Eξ2Eη2.
‡ƒ, XJØ ª Ò¤á, @o•§g(t) = 0k•˜ ¢Št0, =k E(ξ − t0η)2= g(t0) = 0,
u´dÚn3.2.1•ξ − t0η´òzu0 ‘ÅCþ, =kP (ξ = t0η) = 1.
í í
íØØØ 3.2.1. ‘ÅCþξ, 粕ŒÈ, Kk cov(ξ, η) ≤p
Dξ ·p Dη,
¿… Ò¤á, …= •3t0∈ R, ¦ P (ξ = t0η) = 1.
y3·‚5‰Ñƒ'Xê ½Â:
½
½
½Â 3.2.4. ‘ÅCþξ, 粕ŒÈ, ·‚¡
ρξ,η= cov(ξ, η)
√Dξ ·√ Dη,
•ξ†η ƒ'Xê. XJrξ,η = 0, K¡ξ†η؃'.
dd½Â§·‚Œ±á=
ƒ'Xê 5Ÿ
1. eξÚηƒpÕá§Kρξ,η= 0
2. |ρξ,η| ≤ 1, Ò¤á …= ξ, ηƒm•3î‚ ‚5'X§=
ρξ,η= 1, K•3 a > 0, b ∈ R ¦ ξ = aη + b ( ƒ') ρξ,η = −1, K•3 a < 0, b ∈ R ¦ ξ = aη + b (Kƒ')
~3.2.3. X ∼ U (−12,12)§ Y = cosX§K cov(X, Y ) = EXY =
Z 1/2
−1/2
xcosxdx = 0
¤±X, Y ؃'" ´X, Y ƒm•3Xš‚5 ¼ê'X"
d~`²ƒ'Xê´ïþ‘ÅCþƒm ‚5'X .
½
½
½nnn 3.2.4. é?Ûšòz ‘ÅCþξ, 粕ŒÈ, Xeo‡·Kƒp d:
(1) ξ†η؃'; (2) cov(ξ, η) = 0;
(3) Eξη = EξEη ; (4) D(ξ + η) = Dξ + Dη.
·‚5?Ø؃'†Õá5ƒm 'X.
½
½
½nnn 3.2.5. é?Ûšòz ²•ŒÈ ‘ÅCþξ, η ó, XJξ†ηÕá, @o§‚؃'; ´ XJ§‚؃'%™7ƒpÕá.
~3.2.4. Áy²e(X, Y )Ñlü S þ!©Ù§KX, Y ؃' ØÕá"
~3.2.5. ‘ÅCþξÚη ©ÙÆ©O•
ξ ∼ −1 0 1
1 4
1 2
1 4
!
, η ∼ 0 1
1 2
1 2
!
¿…P (ξ · η = 0) = 1. Kξ†ηØÕá, •Øƒ'.
): dξÚη ©ÙÆ, •Eξ = 0, Eη = 1/2. qÏ•P (ξ · η = 0) = 1, ¤±E(ξ · η) = 0. Ï dcov(ξ, η) = E(ξ · η) − Eξ · Eη = 0, •ξ†η؃'. ´
P (ξ = 1) = 1
4, P (η = 1) = 1
2, P (ξ = 1, η = 1) ≤ P (ξ · η = 1) ≤ P (ξ · η 6= 0) = 0,
¤±
P (ξ = 1)P (η = 1) = 1
8 6= 0 = P (ξ = 1, η = 1), Œ„ξ†ηØÕá.
¯¢þ, |^ξ†η > ©ÙÚ^‡P (ξ · η = 0) = 1, ØJ¦Ñ(ξ, η) éÜ©ÙÆ•:
pij -1 0 1 pi·
0 1/4 0 1/4 1/2 1 0 1/2 0 1/2 p·j 1/4 1/2 1/4 1
ù´˜‡|^^‡P (ξ · η = 0) = 1, d> ©ÙƇíéÜ©ÙÆ ~f. Ù‰{´: k3L¥
W\pi·Úp·j, 23ü‡‘ÅCþ ¦È u0 ˜þWþVÇ0, , |^1i1VÇƒÚ upi·, 1j VÇƒÚ up·j, íäÑÙ{ˆ? VÇŠ.
~3.2.6. ‘ÅCþξÚη •Bernoulli‘ÅCþ, k
ξ ∼ 0 1
1 2
1 2
!
, η ∼ 0 1
2 3
1 3
! .
y•Cov(ξ, η) = −16. Á¦(X, Y ) éÜ©ÙÆ.
3.3 Ù¦˜ êiA
• ²þýé E|X − EX|
• Ý1¼êEetX§Ù¥t ∈ R"
• A ¼êEeitX =R eitxdF (x)§Ù¥t ∈ R, i •Jê.
½
½
½Â 3.3.1. F (x)•˜‘©Ù¼ê, ·‚ò f (t) =
Z ∞
−∞
eitxdF (x), t ∈ R
¡•F (x) A ¼ê. XJF (x)´‘ÅCþξ ©Ù¼ê, KTf (t)•¡•ξ A ¼ê, džk f (t) = Eeitξ.
dA ¼ê ½ÂN´ : XJlÑ.‘ÅCþξ ©ÙÆ•P (ξ = an) = pn, n ∈ N, @o
f (t) = Eeitξ =
∞
X
n=1
eitanpn. XJëY.‘ÅCþξ —Ý¼ê•p(x), @o
f (t) = Eeitξ = Z ∞
−∞
eitxp(x)dx.
'u‘ÅCþ‚5¼ê A ¼ê, ·‚kXe(Ø:
1◦. |f (t)| ≤ f (0) = 1, ∀ t ∈ R;
2◦. fa+bξ(t) = eitafξ(bt).
½
½
½nnn 3.3.6. (•˜5½n)©Ù¼êdA ¼ê•˜(½.
(܇üúª(d?Ñ)•§©Ù¼ê†A ¼êƒp•˜(½.
|^A ¼ê§·‚Œ±é•B ),˜©Ù´Ääk2)5"wXe ~fµ
~3.3.1. X ∼ B(n, p), Y ∼ B(m, p)…X, Y ƒpÕá§KX + Y ∼ B(n + m, p)"
Proof. duX ∼ B(n, p)§ X A ¼ê•(q + peit)n"2dX†Y ƒpÕ᧤±X + Y A
¼ê•
Eeit(X+Y ) = EeitXEeitY = (q + peit)n+m ÏdX + Y ∼ B(n + m, p)"
, §|^A ¼ê·‚„Œ±é•B • ù ~„©Ùƒm 'X"'X
• n‡ÕáÓ©ÙB(1, p) 0-1©Ù‘ÅCþƒÚ• ‘©ÙB(n, p);
• k•‡Õá ‘‘ÅCþ(¤õ VǃÓ)ƒÚE• ‘©Ù;
• k•‡Õá P oisson©Ù‘ÅCþƒÚÑlP oisson©Ù§ëêƒ\¶
• r‡ÕáÓ©ÙAÛ©ÙG(p) ‘ÅCþƒÚÑlëê•rÚp P ascal©Ù¶
• ?¿k•‡Õá ©Ù‘ÅCþ ‚5|ÜE,Ñl ©Ù;
~„©Ù A ¼ê
·‚deã ~„©ÙL‰Ñ
L3.1:~„©ÙL ©Ù¶¡ëêVÇ—ÝÏ"•A¼ê5 òz©Ùcc 1 c0eict 'uëêcäk2)5 :©Ùp (0<p<1)
01 qp
! ppqq+peit •n=1‘©Ù ‘©Ù B(n,p)
n≥1 0<p<1
n k
pk qn−k k=0,···,nnpnpq(q+peit )n 'unäk2)5 AÛ©Ùp (0<p<1)qk−1 p,k=1,2,···1 pq p2peit 1−qeitÃPÁ ndk©Ù
r,p r∈N 0<p<1
k−1 r−1
pr qk−r , k=r,r+1,···
r prq p2(peit 1−qeit)r 'uräk2)5 Åt©ÙP(λ)λ(λ>0)
λk k!e−λ , k=0,1,···λλeλ(eit−1) 'uλäk2)5 ‡AÛ©ÙM,N,n∈N(M k)(N−M n−k) (N n)nM NnM N(N−M) NN−n N−1 þ!©Ù U(a,b)a,b(a<b)1 b−aIa<x<ba+b 2(b−a)2 12eitb−eita it(b−a) ©Ù N(a,σ2 )a,σ21 σ√ 2πe−(x−a)2 2σ2aσ2 eiat−1 2σ2t2 'ua,σ2 äk2)5 •ê©Ùλ(λ>0)λe−λx Ix>01 λ1 λ2(1−it λ)−1 χ2 ©Ùn(n≥1)1 2n/2Γ(n/2)xn/2−1 e−x/2 x>0 n2n(1−2it)−n/2 'unäk2)5
ë•©z
[1] €W., VÇØ, ®: ‰Æч , 2004.