❳⑩❈ ❙❯❻❚ ❚❍➮◆● ❑➊
❚!♥ ❚❤➜% ❚&
✣➔ ◆➤♥❣✱ ✷✵✶✾
❚!♥ ❚❤➜% ❚& ✶✴✹✵
❈❤"#♥❣ ✺✳ ()❝ ❧",♥❣ -❤❛♠ 01
✶✳ ,-❝ ❧01♥❣ ✤✐➸♠
✶✳✶ ✣6♥❤ ♥❣❤➽❛
✲ ❈❤♦ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥
{X1, X2, ..., Xn}
+, +-♥❣ +❤➸ ❝0 ♣❤➙♥ ♣❤3✐ ♣❤4 +❤✉5❝ ✈➔♦
+❤❛♠ 93
θ
❝❤:❛ ❜✐➳+✳ ❑❤✐ ✤0✱ ♠5+ ❤➔♠ ✭+❤3♥❣ ❦➯✮
ˆ
θ=ˆ
θ(X1, ..., Xn)
✤:D❝ ❣E✐ ❧➔ ♠5+
!❝ ❧ $♥❣
❝G❛ +❤❛♠ 93
θ
✳
✲ ❱I✐ ♠5+ ♠➝✉ ❣✐→ +KL ❝4 +❤➸
{x1, x2, ..., xn}
+❛ +❤✉ ✤:D❝ ♠5+ ❣✐→ +KL ❝4 +❤➸ ❝G❛
ˆ
θ
✳
❑❤✐ ✤0✱ ❣✐→ +KL
ˆ
θ(x1, ..., xn)
✤:D❝ ❣E✐ ❧➔
!❝ ❧ $♥❣ ✤✐➸♠
❝G❛ +❤❛♠ 93
θ
❞N❛ +K➯♥ ♠➝✉ ❣✐→
+KL
{x1, x2, ..., xn}
✳
❚!♥ ❚❤➜% ❚& ✷✴✹✵
✶✳✷ :❤➙♥ ❧♦↕✐ 0-❝ ❧01♥❣
✲ OI❝ ❧:D♥❣
ˆ
θ=ˆ
θ(X1, ..., Xn)
✤:D❝ ❣E✐ ❧➔
!❝ ❧ $♥❣ ❦❤-♥❣ ❝❤➺❝❤
❝G❛ +❤❛♠ 93
θ
♥➳✉
E(ˆ
θ) = θ
✳
❚K♦♥❣ +K:Q♥❣ ❤D♣ ♥❣:D❝ ❧↕✐ +❤➻ +❛ ❣E✐
ˆ
θ
✤:D❝ ❣E✐ ❧➔
!❝ ❧ $♥❣ ❝❤➺❝❤
✈➔ ❣✐→ +KL
b(θ) =
E(ˆ
θ)−θ
✤:D❝ ❣E✐ ❧➔
✤/ ❝❤➺❝❤ ❝0❛ !❝ ❧ $♥❣
✳
✲ OI❝ ❧:D♥❣
ˆ
θ=ˆ
θ(X1, ..., Xn)
✤:D❝ ❣E✐ ❧➔
!❝ ❧ $♥❣ ❦❤-♥❣ ❝❤➺❝❤ 2✐➺♠ ❝➟♥
❝G❛ +❤❛♠
93
θ
♥➳✉
lim
n→+∞E(ˆ
θ) = θ.
✲ ❈❤♦
ˆ
θ1
✈➔
ˆ
θ2
❧➔ ❤❛✐ :I❝ ❧:D♥❣ ❦❤T♥❣ ❝❤➺❝❤ ❝G❛ +❤❛♠ 93
θ
✳ ❚❛ ♥0✐ :I❝ ❧:D♥❣
ˆ
θ1
❤✐➺✉
5✉↔ ❤7♥
:I❝ ❧:D♥❣
ˆ
θ2
♥➳✉
D(ˆ
θ1)< D(ˆ
θ2)
✳
✲ OI❝ ❧:D♥❣
ˆ
θ
❝G❛
θ
❧➔ :I❝ ❧:D♥❣ ❦❤T♥❣ ❝❤➺❝❤ ✈➔ ❝0 ♣❤:V♥❣ 9❛✐
D(ˆ
θ)
❜➨ ♥❤➜+ ✤:D❝
❣E✐ ❧➔
!❝ ❧ $♥❣ 282 ♥❤➜2
✳
❚!♥ ❚❤➜% ❚& ✸✴✹✵
!❝ ❧$%♥❣ ❦❤*♥❣ ❝❤➺❝❤ ❝,❛ ./✉♥❣ ❜➻♥❤ ✈➔ ♣❤$6♥❣ 7❛✐
✣:♥❤ ❧;✿
❈❤♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
X
❝,❛ ♠/0 01♥❣ 0❤➸ ❝3
E(X) = µ, D(X) = σ2
✳ ●✐↔ 78
{X1, X2, ..., Xn}
❧➔ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥ ❝,❛
X
✳ ❑❤✐ ✤3✿
✐✮
X=1
n
n
P
i=1
Xi
❧➔ ?@❝ ❧?A♥❣ ❦❤C♥❣ ❝❤➺❝❤ ❝,❛
µ
✳
✐✐✮
S2=1
n−1
n
P
i=1
(Xi−X)2
❧➔ ?@❝ ❧?A♥❣ ❦❤C♥❣ ❝❤➺❝❤ ❝,❛
σ2
✳
◆❤➟♥ ①➨.✿
❈3 0❤➸ ❝❤E♥❣ ♠✐♥❤ ✤?A❝ F➡♥❣✿
E(S2
∗) = n−1
nσ2
✈@✐
S2
∗=1
n
n
X
i=1
(Xi−¯
X)2
✣✐➲✉ ♥➔② ❝3 ♥❣❤➽❛
S2
∗
❧➔ ?@❝ ❧?A♥❣ ❝❤➺❝❤ ✭❦❤C♥❣ ❝❤➺❝❤ 0✐➺♠ ❝➟♥✮ ❝,❛
σ2
✱ ❞♦ ✤3 ♥3
✓❦❤C♥❣ ✤?A❝ ❝❤R♥✔ ❧➔♠ ♣❤?U♥❣ 7❛✐ ♠➝✉✳
❚!♥ ❚❤➜% ❚& ✹✴✹✵
!❝ ❧$%♥❣ ❦❤*♥❣ ❝❤➺❝❤ ❝,❛ .➾ ❧➺
✣:♥❤ ❧;✿
❈❤♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
X
❝3 ♣❤➙♥ ♣❤W✐ ❇❡F♥♦✉❧❧✐ ✈@✐ 0❤❛♠ 7W
p
✳ ●R✐
(X1, X2, ..., Xn)
❧➔ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥ ❝,❛
X
✳ ❑❤✐ ✤3✿
ˆ
P=X1+X2+... +Xn
n
❧➔ ♠/0 ?@❝ ❧?A♥❣ ❦❤C♥❣ ❝❤➺❝❤ ❝,❛ 0❤❛♠ 7W
p
✳
B ♥❣❤➽❛✿
◆➳✉ ❞➜✉ ❤✐➺✉
A
❝3 0➾ ❧➺
p
❝❤?❛ ❜✐➳0✱ ✈➔ 0F♦♥❣ ♠➝✉ ✤✐➲✉ 0F❛ ❦➼❝❤ 0❤?@❝
n
❝3
k
❝→ 0❤➸ ♠❛♥❣ ❞➜✉ ❤✐➺✉
A
✱ 0❤➻ 0➛♥ 7✉➜0
fn=k/n
❧➔ ♠/0 ?@❝ ❧?A♥❣ ❦❤C♥❣ ❝❤➺❝❤ ❝,❛
p
✳
❚!♥ ❚❤➜% ❚& ✺✴✹✵
❱➼ ❞G ✶
✲ ❈❤♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
X
❝- ❦/ ✈1♥❣
µ
✈➔ ♣❤45♥❣ 6❛✐
σ2
❝❤4❛ ❜✐➳8✳ ✣➸ 4<❝ ❧4>♥❣
❝❤♦
µ
✈➔
σ2
♥❣4?✐ 8❛ 8✐➳♥ ❤➔♥❤ ✤✐➲✉ 8B❛ ♠➝✉ ❦➼❝❤ 8❤4<❝ ✷✵✵ ✈➔ 8➼♥❤ ✤4>❝ 8B✉♥❣ ❜➻♥❤ ✈➔
♣❤45♥❣ 6❛✐ ❝❤✉➞♥ ♠➝✉✿
¯x= 125,8; s2= 2,76
❑❤✐ ✤-✱ 8❛ ❝- 8❤➸ ①❡♠
µ≈125,8
✈➔
σ2≈2,76
✳
✲ ✣➸ 4<❝ ❧4>♥❣ 8➾ ❧➺
p
❝P 8B✐ Q♥❣ ❤R ❝❤♦ S♥❣ ❝P ✈✐➯♥ ❆✱ ♥❣4?✐ 8❛ ❦❤↔♦ 6→8 ♥❣➝✉ ♥❤✐➯♥
4000
♥❣4?✐ 8❤➻ ❝- ✷✻✹✵ ♥❣4?✐ Q♥❣ ❤R S♥❣ ❝P ✈✐➯♥ ♥➔②✳ ◆❤4 ✈➟②✱ 8❛ ❝- 8❤➸ ①❡♠ 8➾ ❧➺ Q♥❣
❤R S♥❣ ❝P ✈✐➯♥ ❆ ①➜♣ ①➾✿
p≈2640/4000 = 0,66
❚!♥ ❚❤➜% ❚& ✻✴✹✵
X θ
{X1, X2, ..., Xn}L=L(X1, ..., Xn)U=U(X1, ..., Xn)
α∈(0,1)
P(L < θ < U) = 1 −α.
(L, U)θ1−α
α U −L
(L, U)θ
1−α
1−α
α1, α2∈(0,1) α1+α2=α
Gn=Gn(X1, ..., Xn, θ)
GnGn
n→ ∞ θ Gn
θ
a b P (Gn≤a) = α1P(Gn< b) = 1 −α2
a < Gn(X1, ..., Xn, θ)< b θ
L, U
P(L < θ < U)≥1−α.
α1= 0 α2= 0
✷✳✷ ❑❤♦↔♥❣ (✐♥ ❝➟② ❝❤♦ ❦➻ ✈0♥❣ ❦❤✐ ✤➣ ❜✐➳( ♣❤67♥❣ 8❛✐
❇➔✐ (♦→♥
✳ ❈❤♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
X
❝-❛ ♠01 12♥❣ 1❤➸ ❝4 ♣❤➙♥ ♣❤7✐ ❝❤✉➞♥
N(µ;σ2)
✈:✐
µ
❝❤;❛ ❜✐➳1 ✈➔
σ2
✤➣ ❜✐➳1✳ ❚➻♠ ❦❤♦↔♥❣ 1✐♥ ❝➟② ❝❤♦
µ
✈:✐ ♠E❝ F ♥❣❤➽❛
α
✳
●>✐ ?
✳ ◆➳✉
{X1, X2, ..., Xn}
❧➔ ♠➝✉ ♥❣➝✉ ♥❤✐➯♥ ❝-❛
X
1❤➻
Z=X−µ
σ/√n∼N(0; 1).
✲ ❱:✐
α∈(0; 1)
✱ ❝❤M♥
α1=α2=α/2
✈➔
zα/2= Φ−1(1 −α
2)
❧➔
❣✐→ (@A (B✐ ❤↕♥
♠E❝
α/2
❝-❛ ♣❤➙♥ ♣❤7✐ ❝❤✉➞♥ 1➢❝✱ ❣✐↔✐ ❜➜1 ♣❤;P♥❣ 1Q➻♥❤✿
−zα/2< Z < zα/2⇔X−zα/2
σ
√n< µ < X+zα/2
σ
√n
1❛ ✤;S❝ ❦❤♦↔♥❣ 1✐♥ ❝➟② ✤7✐ ①E♥❣✳
✲ ◆➳✉ ❝❤M♥
α1= 0
❤♦➦❝
α2= 0
✳ ▲W❝ ♥➔② 1❛ 1❤❛②
zα/2
❜X✐
zα
✈➔ 1❤✉ ✤;S❝ ❦❤♦↔♥❣ 1✐♥
❝➟② ♠01 ♣❤➼❛✳
❚!♥ ❚❤➜% ❚& ✶✵✴✹✵
❚!♥ ❚❤➜% ❚& ✶✶✴✹✵
❛✳ ❑➳( D✉↔
❱:✐ ✤0 1✐♥ ❝➟②
1−α
✿
✲ ❑❤♦↔♥❣ 1✐♥ ❝➟② ✤7✐ ①E♥❣ ❝-❛
µ
✿
x−zα/2
σ
√n< µ < x+zα/2
σ
√n,
✲ ❑❤♦↔♥❣ 1✐♥ ❝➟② 17✐ ✤❛ ❝-❛
µ
✿
µ < x+zα
σ
√n.
✲ ❑❤♦↔♥❣ 1✐♥ ❝➟② 17✐ 1❤✐➸✉ ❝-❛
µ
✿
µ > x−zα
σ
√n.
❚!♥ ❚❤➜% ❚& ✶✷✴✹✵
❱➼ ❞# ✷
❑❤"✐ ❧%&♥❣ ✭❦❣✮ ❝-❛ ♠01 1❤✐➳1 ❜4 ❝5 ♣❤➙♥ ♣❤"✐ ❝❤✉➞♥
N(µ;σ2)
✈;✐
σ= 0.2
✭❦❣✮✳ ❈❤>♥
♥❣➝✉ ♥❤✐➯♥ ✷✺ 1❤✐➳1 ❜4 ♥❣%C✐ 1❛ 1➼♥❤ ✤%&❝ 1F✉♥❣ ❜➻♥❤ ♠➝✉
x= 65,1
✭❦❣✮✳ ❱;✐ ✤0 1✐♥ ❝➟②
✾✺✪ ❤➣② 1➻♠ ❦❤♦↔♥❣ 1✐♥ ❝➟② ✭✤"✐ ①Q♥❣✮ ❝❤♦ ❦❤"✐ ❧%&♥❣ 1F✉♥❣ ❜➻♥❤ ❝-❛ 1❤✐➳1 ❜4 ♥➔②✳ ❈❤♦
❜✐➳1
z0,025 = 1,96
✳
●✐↔✐✳
✲ ❈→❝ $% ✤➦❝ ()*♥❣ ♠➝✉✿
n= 25; ¯x= 65,1
✳
✲ ✣3 (✐♥ ❝➟②✿
1−α= 0,95 ⇒α= 0,05; zα/2=z0,025 = 1,96
✳
✲ ❙❛✐ $% *9❝ ❧*;♥❣✿
ε=zα/2
σ
√n= 1,96 ∗0,2
√25 = 0,0784
✲ ❑❤♦↔♥❣ (✐♥ ❝➟② ❝❤♦ ❦❤%✐ ❧*;♥❣ ()✉♥❣ ❜➻♥❤
µ
❝C❛ (❤✐➳( ❜E ♥➔②✿
¯x−ε < µ < ¯x+ε⇔65,02 < µ < 65,18
❚!♥ ❚❤➜% ❚& ✶✸✴✹✵
❜✳ ❱➜♥ ✤➲ ❝/ ♠➝✉
❚H ❝I♥❣ (❤J❝ ❦❤♦↔♥❣ (✐♥ ❝➟② ❝❤♦
µ
(❛ (❤➜② )➡♥❣ $❛✐ $% ❝C❛ *9❝ ❧*;♥❣
|x−µ|
❜➨ ❤N♥
❤♦➦❝ ❜➡♥❣
zα/2
σ
√n
✳
✲ ❙❛✐ $% *9❝ ❧*;♥❣✿
ε=zα/2
σ
√n
✲ ✣✐➲✉ ❦✐➺♥ ❤↕♥ ❝❤➳✿
ε < ∆
✈9✐
∆>0
❝❤♦ ()*9❝✳
ε=zα/2
σ
√n<∆
✲ ●✐↔✐ ❜➜( ♣❤*N♥❣ ()➻♥❤ (❤❡♦
n
✿
n > zα/2σ
∆2
.
❚!♥ ❚❤➜% ❚& ✶✹✴✹✵
❱➼ ❞# ✸
❚FT ❧↕✐ ✈;✐ ❱➼ ❞W ✷ ♥➳✉ ②➯✉ ❝➛✉ Y❛✐ Y" %;❝ ❧%&♥❣ ❦❤Z♥❣ ✈%&1 [✉→ ✵✱✵✺ 1❤➻ ✈;✐ ✤0 1✐♥ ❝➟②
✾✽✪ 1❛ ❝➛♥ ❝❤>♥ 1"✐ 1❤✐➸✉ ❜❛♦ ♥❤✐➯✉ 1❤✐➳1 ❜4 ✤➸ ❦❤↔♦ Y→1❄ ❈❤♦ ❜✐➳1
z0,01 = 2,326
✳
●✐↔✐✳
✲ ❙❛✐ $% *9❝ ❧*;♥❣✿
ε=zα/2
σ
√n
✲ ✣✐➲✉ ❦✐➺♥ ❤↕♥ ❝❤➳✿
ε < ∆ = 0,05
✳
✲ ❙✉② )❛✿
n > zα/2σ
∆2
.
✲ ✣3 (✐♥ ❝➟②✿
1−α= 0,98 ⇒α= 0,2; zα/2=z0,01 = 2,326
✳
❉♦ ✤W✿
n > 2,326 ∗0,2
0,05 2
= 86,56
❱➟②✱ ❦➼❝❤ (❤*9❝ ♠➝✉ (%✐ (❤✐➸✉ ❝➛♥ ❦❤↔♦ $→(✿
n= 87
❚!♥ ❚❤➜% ❚& ✶✺✴✹✵