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(1)

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"

(2)

~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

(3)

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.

(4)

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"

(5)

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

(6)

½

½

½Â 3.2.1. ‘ÅCþξ rgŒÈ§=

E|ξ|r= Z

−∞

|x|rdFξ(x) < ∞ K·‚¡

r= 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"

(7)

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•·‚kξ2 = 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±

(8)

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≤ Eξ22

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 ,

(9)

¤± g¼êg(t)

∆ = 4(Eξη)2− 4Eξ2· Eη2 ≤ 0, Ø ª.

XJ•3t0∈ R, ¦ P (ξ = t0η) = 1, w,Òk (Eξη)2 = Eξ22.

‡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 · ,

•ξ†η ƒ'Xê. XJrξ,η = 0, K¡ξ†η؃'.

dd½Â§·‚Œ±á=

ƒ'Xê

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"

(10)

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 p

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\pÚp·j, 23ü‡‘ÅCþ ¦È u0 ˜þWþVÇ0, , |^1i1VÇƒÚ up, 1j VÇƒÚ up·j, íäÑÙ{ˆ? VÇŠ.

(11)

~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

(12)

~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

(13)

L3.1:~©ÙL ©Ù¡ëêVÇÝÏ"A¼ê5 òz©Ùcc 1 c0eict 'uëêcäk2)5 :©Ùp (0<p<1)

01 qp

! ppqq+peit n=1©Ù ©Ù B(n,p)

n1 0<p<1

n k

 pk qnk k=0,···,nnpnpq(q+peit )n 'unäk2)5 AÛ©Ùp (0<p<1)qk1 p,k=1,2,···1 pq p2peit 1qeitÃPÁ ndk©Ù

r,p rN 0<p<1

k1 r1

 pr qkr , k=r,r+1,···

r prq p2(peit 1qeit)r 'uräk2)5 Åt©ÙP)λ>0)

λk k!eλ , k=0,1,···λλeλ(eit1) 'uλäk2)5 AÛ©ÙM,N,nN(M k)(NM nk) (N n)nM NnM N(NM) NNn N1 þ!©Ù U(a,b)a,b(a<b)1 baIa<x<ba+b 2(ba)2 12eitbeita it(ba) ©Ù N(a,σ2 )a,σ21 σ 2πe(xa)2 2σ2aσ2 eiat1 2σ2t2 'ua,σ2 äk2)5 ê©Ùλ>0)λeλx Ix>01 λ1 λ2(1it λ)1 χ2 ©Ùn(n1)1 2n/2Γ(n/2)xn/21 ex/2 x>0 n2n(12it)n/2 'unäk2)5

(14)

ë•©z

[1] €W., VÇØ, ®: ‰Æч , 2004.

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