
V (X) = 500
W = (X
1
, X
2
, , X
n
) X
1
, X
2
, , X
100
E(X
i
) = E(X) = 1065, V (X
i
) = V (X) = 500
2
, ∀i = 1, 100
X =
1
100
(X
1
+ X
2
+ + X
100
)

= 2500
W = (X
1
, X
2
, , X
n
) X
i
V (X) = V

X
1
+ + X
n
n

=
1
n
2
[V (X
1
) + + V (X
n
)]
=
1
n
2

=
0, 5
10, 9445
= 0, 0456
W = (X
1
, X
2
, X
3
, X
4
, X
5
)
X
x =
x
1
+ + x
5
5
=
10 + 12 + 16 + 18 + 19
5
= 15
f =
3
50
= 0, 06

=
1
2000[f(x
d
)]
2
X
′
V (X
′
) =
V (X)
320
=
σ
2
320
EF =
V (
X
′
)
V (X
d
)
=
σ
2
320
1

=
6, 25
2π
< 1
X ∼ N(µ, σ
2
)
W = (X
1
, X
2
, , X
n
)
X
d
V (X
d
) =
1
4n[f(x
d
)]
2
=
2π
4
V (
X)
X V (X)

E(Y ) = 4.
1
2
+ 6.
1
2
= 5
E(Z) = 32
1
8
+ 40
1
8
+ 44
1
4
+ 48
1
8
+ 60
1
8
+ 66
1
8
= 50
X ∼ A(p)
W
1
= (X

V (f
1
) =
p(1 − p)
n
1
=
p(1 − p)
3
; V (f
2
) =
p(1 − p)
n
2
=
p(1 − p)
4
V (f
2
) < V (f
1
) f
2
f
1
f
2
f
1

V (f
1
) + (1 − α)
2
V (f
2
)
= α
2
p(1 − p)
3
+ (1 − α)
2
p(1 − p)
4
= (7α
2
− 6α + 3)
p(1 − p)
12
g(α) = 7α
2
− 6α + 3 g(α) g(α)
α = 6/14 = 3/7 V (θ)
3
7
f
1
+
4

2
) =
3
4
E(U) +
1
4
E(V ) = X
(iii) W
3
= 1.U + 0.V ⇒ E(W
3
) = E(U) = X
W
1
, W
2
, W
3
W
1
=
1
2
U +
1
2
V
V (W
1

) =
9
16
V (U) +
1
16
V (V ) =
9σ
2
16
+
9σ
2
16
=
18
16
σ
2
W
3
= 1.U + 0.V
V (W
3
) = V (U) = σ
2
V (W
3
) < V (W
2

2
, x
3
, λ) = f(x
1
, λ)f(x
2
, λ)f(x
3
, λ) = e
−3λ
λ
x
1
+x
2
+x
3
x
1
!x
2
!x
3
!
x
1
= 15, x
2
= 8, x

15!8!13!
= 5, 0075.10
−4
L(λ = 15) = e
−45
15
36
15!8!13!
≈ 1, 9043.10
−4
L(λ = 20) = e
−60
20
36
15!8!13!
≈ 1, 8328.10
−6
L(λ = 25) = e
−75
25
36
15!8!13!
≈ 1, 728.10
−9
λ = 12 =
x
1
+x
2
+x

n

i=1
x
i
− ln(x
1
!x
2
! x
n
!)
∂ ln L(x
1
, x
2
, , x
n
, λ)
∂λ
= −n +
1
λ
n

i=1
x
i
∂ ln L(x
1

2
n

i=1
x
i
⇒
∂
2
ln L(x
1
, x
2
, , x
n
, λ)
∂λ
2




λ=
x
= −
n
x
x
2
< 0

2
, x
3
, x
4
, x
5
, λ) =
5

i=1
f(x
i
, λ) = λ
5
e
−λ
5

i=1
x
i
x
1
= 1, 2; x
2
= 7, 5; x
3
= 1, 8; x
4

−15.0,3
= 0, 3
5
e
−4,5
≈ 26, 9947.10
−6
• λ = 0, 4 ⇒ L(λ = 0, 4) = 0, 4
5
e
−15.0,4
= 0, 4
5
.e
−6
≈ 25, 3824.10
−6
• λ = 0, 5 ⇒ L(λ = 0, 5) = 0, 5
5
.e
−15.0,5
= 0, 5
5
e
−7,5
≈ 17, 2839.10
−6
L(λ = 0, 3)
L(x
1

n
, λ)
∂λ
=
n
λ
−
n

i=1
x
i
=
n
λ
− n
x
∂ ln L(x
1
, , x
n
, λ)
∂λ
= 0 ⇔
n
λ
− n
x = 0 ⇔ λ =
1
x

=

1
3

5
e
−15.(1/3)
=

1
3

5
e
−5
≈ 27, 7282.10
−6
X ∼ B(n, p)
W = (X
1
, , X
m
)
X
i
X
i
∼
A(p), ∀i = 1, n

5

i=1
f(x
i
, p) =
5

i=1
p
x
i
(1 − p)
1−x
i
= p

5
i=1
x
i
(1 − p)
5−

5
i=1
x
i
= p
x

−5
L(0, 3) = 0, 3
3
.0, 7
2
= 1323.10
−5
L(0, 4) = 0, 4
3
.0, 6
2
= 2304.10
−5
L(0, 5) = 0, 5
3
.0, 5
2
= 3125.10
−5
L(0, 6) = 0, 6
3
.0, 4
2
= 3456.10
−5
L(0, 7) = 0, 7
3
.0, 3
2
= 3087.10


i=1
p
x
i
(1 − p)
1−x
i
= p

m
i=1
x
i
(1 − p)
m−

m
i=1
x
i
= p
x
(1 − p)
m−x
ln L(x
1
, , x
m
, p) = ln

p(1 − p)
= 0 ⇔ p =
x
m
= f
∂
2
ln L(x
1
, , x
m
, p)
∂p
2
=
2xp − x − mp
2
[p(1 − p)]
2
∂
2
ln L(x
1
, , x
m
, p)
∂p
2



µ
σ
2

X −
σ
√
n
u
α/2
;
X +
σ
√
n
u
α/2

x = 8, 5; σ = 2; n = 600;
α = 0, 05 ⇒ u
α/2
= u
0,025
= 1, 96

8, 5 −
2
√
600
· 1, 96; 8, 5 +

n = 30; α = 0, 05 ⇒ t
(n−1)
α/2
= t
(29)
0,025
= 2, 045
x =
304
30
= 10, 1333
s
2
=
1
29
(3082, 14 − (10, 1333)
2
.30) = 0, 0554 ⇒ s =

0, 0554 = 0, 235

10.1333 −
0, 235
√
30
· 2, 045; 10.1333 +
0, 235
√
30

538
25
= 21, 52
s
2
=
1
24
(11716 −21, 52
2
.25) = 5, 76
s =

5, 76 = 2, 4

21, 52 −
2, 4
√
25
· 2, 064; 21, 52 +
2, 4
√
25
· 2, 064

= (20, 53; 22, 51)
µ σ
2

X −

24
(37706, 25 −38, 5
2
.25) = 27, 0833
s =

27, 0833 = 5, 204

38, 5 −
5, 204
√
25
· 2, 064; 38, 5 +
5, 204
√
25
· 2, 064

= (36, 352; 40, 648)
µ σ
2

X −
S
√
n
· t
(n−1)
α/2
;

√
n
· t
(n−1)
α/2
;
X +
S
√
n
· t
(n−1)
α/2

n = 25; α = 0, 05 ⇒ t
(n−1)
α/2
= t
(4)
0,025
= 2, 776
x =
1
5
5

i=1
x
i
= 2, 018

= (2, 0124; 2, 0236)
µ σ
2

X −
S
√
n
· t
(n−1)
α/2
;
X +
S
√
n
· t
(n−1)
α/2

n = 100; α = 0, 05 ⇒ t
(n−1)
α/2
= t
(99)
0,025
≈ u
0,025
= 1, 96
x =

√
100
· 1, 96

= (11, 727; 11, 783)
µ σ
2

X −
S
√
n
· t
(n−1)
α/2
;
X +
S
√
n
· t
(n−1)
α/2

n = 100; α = 0, 05 ⇒ t
(n−1)
α/2
= t
(99)
0,025


X −
σ
√
n
u
α/2
;
X +
σ
√
n
u
α/2

n = 36, α = 0, 05 ⇒ u
α/2
= u
0,025
= 1, 96

x −
3
6
· 1, 96;
x +
3
6
· 1, 96


2
x
= 1, 2
2
ε =
σ
√
n
u
α/2
ε = 0, 3 1 − α = 0, 95
n 

σ
2
ε
2
u
2
α/2

=

1, 2
2
0, 3
2
· 1, 96
2



= [6, 83]
X ∼ N(µ, σ
2
) µ, σ
2
µ σ
2
n < 30 µ

x −
S
√
n
t
(n−1)
α
, +∞

(1 − α) µ x −
S
√
n
t
(n−1)
α
x =
265
25
= 10, 6

1
24
(8244 − 25.18
2
) = 6 ⇒ s =
√
6 = 2, 45
1 − α = 0, 95 ⇒ t
(n−1)
α
= t
(24)
0,05
= 1, 711
µ (1 − α)

− ∞;
x +
S
√
n
t
(n−1)
α

µ
x +
s
√
n

σ
1
, σ
2

(
x
1
− x
2
) −

S
2
1
n
1
+
S
2
2
n
2
t
(k)
α/2
; (
x
1
− x

s
2
1
=
1
3
(330
2
+ 360
2
+ 400
2
+ 350
2
− 4.360
2
) =
2600
3
⇒ s
1
= 29, 4392
s
2
2
=
1
3
(290
2

=
2600
3.4
2600
3.4
+
3400
3.4
=
2600
6000
=
13
30
k =
(n
1
− 1)(n
2
− 1)
(n
2
− 1)c
2
+ (n
1
− 1)(1 − c)
2
=
3.3

2600
3.4
+
3400
3.4
· 2, 447

= (−24, 72; 84, 72)
X
1
X
2
X
1
∼ N(µ
1
, σ
2
1
); X
2
∼ N(µ
2
, σ
2
2
)
L = 90(µ
1
− µ

µ
1
− µ
2
(1 − α) µ
1
− µ
2

(
X
1
− X
2
) −

S
2
1
n
1
+
S
2
2
n
2
t
(k)
α/2

=
16 + 19 + 12 + 11 + 22
5
= 16
s
2
1
=
(9 − 11)
2
+ (12 − 11)
2
+ (8 − 11)
2
+ (10 − 11)
2
+ (16 − 11)
2
4
= 10
⇒ s
1
=
√
10 = 3, 16
s
2
2
=
(16 − 16)

2
n
2
=
10
5
10
5
+
21,5
5
= 0, 3175
k =
(n
1
− 1)(n
2
− 1)
(n
2
− 1)c
2
+ (n
1
− 1)(1 − c)
2
=
4.4
4.0, 3175
2

2
n
2
t
(k)
α/2
; (
x
1
− x
2
) +

s
2
1
n
1
+
s
2
2
n
2
t
(k)
α/2

=


2
2
)
σ
2
1
, σ
2
2
(1 − α) µ
1
− µ
2

(
X
1
− X
2
) −

S
2
1
n
1
+
S
2
2

s
2
1
=
1
n
1
− 1

(x
1i
−
x
1
)
2
=
1
7
· 1805 = 275, 875
x
2
= 99
s
2
2
=
1
n
2

256
46
≈ 6, 148
1 − α = 0, 95 ⇒ α/2 = 0, 025
c =
s
2
1
n
1
s
2
1
n
1
+
s
2
2
n
1
=
257,875
8
257,875
8
+
256
46
= 0, 8528

− x
2
) −

s
2
1
n
1
+
s
2
2
n
2
t
(k)
α/2
; (
x
1
− x
2
) +

s
2
1
n
1

µ
D

D −
S
D
√
n
t
(n−1)
α/2
;
D +
S
D
√
n
t
(n−1)
α/2

⇒ 1 − α = 0, 95 ⇒ α/2 = 0, 025 ⇒ t
(n−1)
α/2
= t
(5)
0,025
= 2, 571
d
1

5
= 320 ⇒ s
D
=
√
320 = 17, 89
µ
D

d −
s
D
√
n
t
(n−1)
α/2
;
d +
s
D
√
n
t
(n−1)
α/2

=

20 −

;
D +
S
√
n
t
(n−1)
α/2

x
i
y
i
d
i
d
2
i
d =
1
n

d
i
=
9
5
= 1, 8
s
2

1

d −
s
√
n
t
(n−1)
α/2
;
d +
s
√
n
t
(n−1)
α/2

=

1, 8 −
4, 025
√
5
· 2, 776; 1, 8 +
4, 025
√
5
· 2, 776


+
S
2
2
n
2
t
(k)
α/2
; (
X
1
− X
2
) +

S
2
1
n
1
+
S
2
2
n
2
t
(k)
α/2

49
30
+
36
30
= 1, 6833
c =
s
2
1
n
1
s
2
1
n
1
+
s
2
2
n
2
=
49
30
49
30
+
36

0,025
≈ u
0,025
= 1, 96

(
x
1
− x
2
) −

s
2
1
n
1
+
s
2
2
n
2
t
(k)
α/2
; (
x
1
− x

Y ∼ N(µ
2
, σ
2
= 0, 485);
y = 1, 512; n
2
= 302

σ
2
1
n
1
+
σ
2
2
n
2
=

0, 517
2
230
+
0, 485
2
302
= 0, 044057

u
α/2
; (
x − y) +

σ
2
1
n
1
+
σ
2
2
n
2
u
α/2

=

(1, 317 − 1, 512) − 0, 044057.1, 96; (1, 317 − 1, 512) + ·0, 044057.1, 96

= (−0, 28135; −0, 10865)
0, 10865 < µ
2
− µ
1
< 0, 28135
X ∼ N(µ

n
2
=

8, 2
2
300
+
5, 3
2
400
= 0, 543
c =
s
2
1
n
1
s
2
1
n
1
+
s
2
2
n
2
=

299.399
399.0, 76143
2
+ 299.0, 23857
2
= 480, 4
k = 481; 1 − α = 0, 95 ⇒ t
(k)
α/2
= t
(481)
0,025
≈ u
0,025
= 1, 96

(
x − y) −

s
2
1
n
1
+
s
2
2
n
2

< 10, 263
X ∼ N(µ
1
, σ
2
1
); n
1
= 7; x = 17, 5; s
1
= 3, 2
Y ∼ N(µ
2
, σ
2
2
); n
2
= 11;
y = 15, 3; s
2
= 2, 9

s
2
1
n
1
+
s

2
=
3,2
2
7
3,2
2
7
+
2,9
2
11
= 0, 657
k =
(n
1
− 1)(n
2
− 1)
(n
2
− 1)c
2
+ (n
1
− 1)(1 − c)
2
=
6.10
10.0, 657

t
(k)
α/2
; (
x − y) +

s
2
1
n
1
+
s
2
2
n
2
t
(k)
α/2

=

(17, 5 − 15, 3) −1, 493.2, 179; (17, 5 − 15, 3) + 1, 493.2, 179

= (−1, 053; 5, 453)
X ∼ N(µ
1
, σ
2

1
+
s
2
2
n
2
= 0, 3363
c =
s
2
1
n
1
s
2
1
n
1
+
s
2
2
n
2
=
0,4547
6
0,4547
6

0,025
= 2, 228
µ
1
− µ
2

(
x − y) −

s
2
1
n
1
+
s
2
2
n
2
t
(k)
α/2
; (
x − y) +

s
2
1

u
α/2

f =
100−10
100
= 0, 9
1 − α = 0, 95 ⇒ α/2 = 0, 025 ⇒ u
α/2
= u
0,025
= 1, 96
′

0, 9 −
√
0, 9.0, 1
10
· 1, 96; 0, 9 +
√
0, 9.0, 1
10
· 1, 96

= (0, 84; 0, 96)
1 − α

f −

f(1 − f)


f(1 − f)
√
n
u
α/2

=

0, 86 −
√
0, 86.0, 14
√
1000
· 1, 96; 0, 86 +
√
0, 86.0, 14
√
1000
· 1, 96

= (0, 8385; 0, 8815)
1 − α

f −

f(1 − f)
√
n
u


= (0, 01284; 0, 067)
1 − α

f −

f(1 − f)
√
n
u
α/2
; f +

f(1 −f)
√
n
u
α/2

1 − α = 0, 09 ⇒ u
α
= u
0,025
= 1, 96
n = 200; f =
150
200
= 0, 75

0, 75 −


f(1 −f)
√
n
u
α
= 0, 05 +
√
0, 05.0, 95
√
400
· 1, 64 = 0, 06787
1 − α
f −
√
f(1−f)
√
n
u
α
n = 1600; f =
960
1600
= 0, 6
1 − α = 0, 99 ⇒ u
α
= u
0,01
= 2, 33
p > f −


f(1 − f)
√
n
u
α
= 0, 05 +
√
0, 4.0, 6
√
400
· 1, 64 = 0, 44017
I
0
= 0, 02
⇒ f = 0, 9; u
α/2
= u
0,025
= 1, 96
n 

4f(1 − f)
I
2
0
u
2
α/2


) − s
f
u
α/2
; (f
1
− f
2
) + s
f
u
α/2

f
1
= 0, 43; f
2
= 0, 38
s
f
=

0, 43.0, 57
1500
+
0, 38.0, 62
1000
= 0, 02
1 − α = 0, 95 ⇒ u
α/2

; (f
1
− f
2
) + S
f
u
α/2

n
1
= 100; f
1
=
7
100
= 0, 07
n
2
= 150; f
2
=
12
150
= 0, 08
1 − α = 0, 95 ⇒ u
α/2
= u
0,025
= 1, 96

− f
2
) − s
f
u
α/2
; (f
1
− f
2
) + s
f
u
α/2

=

(0, 07 − 0, 08) −0, 03379.1, 96; (0, 07 −0, 08) + 0, 03379.1, 96

= (−0, 07623; 0, 05623)
p
1
, p
2
p
1
− p
2
p
1

2
= 800; f
2
=
15
800
= 0, 01875
1 − α = 0, 95 ⇒ u
α/2
= u
0,025
= 1, 96
s
f
=

0, 03.0, 97
1000
+
0, 01875.0, 98125
800
= 0, 0722

(f
1
− f
2
) − s
f
u

1
− f
2
) − S
f
u
α/2
; (f
1
− f
2
) + S
f
u
α/2

n
1
= 162; f
1
=
61
162
= 0, 3765
n
2
= 189; f
2
=
106

) + s
f
u
α/2

=

(0, 3765 −0, 5608) −0, 05246.1, 96; (0, 3765 − 0, 5608) + 0, 05246.1, 96)
= (−0, 2873; 0, 0815)
p
1
−p
2
p
1
− p
2
1 − α

(f
1
− f
2
) − S
f
u
α/2
; (f
1
− f


0, 2.0, 8
150
+
0, 12143.0, 87857
140
= 0, 043

(f
1
− f
2
) − s
f
u
α/2
; (f
1
− f
2
) + s
f
u
α/2

=

(0, 2 − 0, 12143) − 0, 043.1, 96; (0, 2 − 0, 12143) + 0, 043.1, 96

= (−0, 00524; 0, 1624)