Ehtimollar nazariyasi va matematik
Download 0.54 Mb. Pdf ko'rish
|
ehtimollar nazariyasi va matematik statistika
- Bu sahifa navigatsiya:
- 12-§. Tnlanma korrelyasiya ko’rsatkichi. Chiziqli regressiya tenglamasi. Eng kichik kvadratlar usuli.
|
1. X belgili bosh to‘plamdan olingan tanlanmaning statistik taqsimot berilgan:
∆ [4,1;4,2) [4,2; 4,3) [4,3; 4,4) [4,4;4,5) [4,5;4,6) [4,6;4,7) [4,7;4,8) [4,8;4,9) [4,9;5,0) n i 1 2 3 4 5 8 8 9 10
X belgining taqsimot funksiyasi normal taqsimotga muvofiq yoki muvofiq emasligini 0,05 aniqlik daraja bilan Pirsonning muvofiqlik kriteriysi yordamida aniqlang.
J: Normal taqsimotga mos keladi.
2. X belgili bosh to‘plamdan olingan tanlanmaning statistic taqsimoti berilgan:
∆
[0;10) [10; 20) [20; 30) [30;40)
[40;50) [50;60)
n i 11
14 15
10 14
16
X belgining taqsimot funksiyasi tekis taqsimotga muvofiq yoki muvofiq emasligini 0,05 aniqlilik darajasi bilan Pirsonning muvofiqlik kriteriyasi yordamida aniqlang. J: Tekis taqsimot bilan muvofiqlashadi. 73
Eng kichik kvadratlar usuli. X va Y belgili ikki o‘lchovli bosh to‘plamdan olingan n hajmli tanlanma berilgan bo‘lsin. ) ,
k i y x kuzatilgan qiymatlarini mos chastotalari bilan ushbu korrelyatsion jadvalga joylashtiramiz:
X m y y y ...
2 1
∑ =
j j i n 1
l x x x ...
2 1
m t t t m m n n n n n n n n n ...
... ...
.... ...
... ...
2 1 2 22 21 1 12 11
t x x x n n n ...
2 1
∑ =
j j i n 1
m y y y n n n ...
2 1
n = ∑ = l j 1 ∑ = m j j i n 1
Quyidagi belgilashlarni kiritamiz: y x x x y m i j x l i i j i l i j i x m i i x l i i Y X Y X r Y Y X X n y n Y n x n X n y x n Y X n y n Y n x n X j i i i σ σ σ σ ⋅ ⋅ − = − = − = = = = = = ∑ ∑ ∑ ∑ ∑ = = = = = , , 1 1 , 1 1 , 1 2 2 2 2 2 2 1 2 2 1 2 2 1 1 1
) ( x x r Y Y x y − = − σ σ eng kichik kvadratlar usuli bilan topilgan Y ning X ga to‘g‘ri chiziqli regressiya tenglamasidir. Ko‘pincha bu tenglamani topishni soddalashtirish uchun 2 2
1 ,
C x u h C x u i i i i − = − =
almashtirishlar kiritiladi.
1 va C 2 mos ravishda i x x ≤ ≤ ... 1 va m y y ≤ ≤ ... 1 variatsion qatorlarning o‘rtalarida joylashgan variantalar, h 1 va h 2 lar esa variatsion qatorlar qo‘shni variantalarining ayirmasi.
Yuqoridagi almashtirishlardan foydalanib, chiziqli regressiya tenglamasini topishda quyidagi formulalar ishlatiladi: 74 ν ν ν σ σ ν ν ν σ σ σ σ ν ν σ σ ν ν ν ν u m j j i j i l i y u x u m j y j l i x i m j y j l i x i n u n n u r C h Y C h u X h h u u n n n u n u n n n u n u j i j i ∑ ∑ ∑ ∑ ∑ ∑ = = = = = = ⋅ − = + ⋅ = + ⋅ = = ⋅ = − = − = = = = = 1 1 2 2 1 1 2 1 2 2 2 2 2 2 1 2 2 1 2 2 1 1 , , , , , 1 , 1 1 , 1 Misol. Berilgan korrelyatsion jadval ma’lumotlariga ko‘ra Y ning X ga chiziqli regressiya tenglamasini toping. 1-jadval . X Y 2 7 12 17
22 27
n y 100
1 5 - - - - 6 110
- 5 3 - - - 8 120
- - 3 40 12
- 55
130 - - 2 10
5 - 17 140 - - - 3 4 7 14
n x 1 10 8 53
21 7 100 Hisoblashlarni osonlashtirish maqsadida 120 17
1 = = C va C deb olamiz. U holda 5
= h bo‘lgani uchun: 2 5
27 1 5 17 22 0 1 5 17 12 2 5 17 7 3 5 17 2 6 5 4 3 2 1 = − = = − = = − = − = − = − = − = − = U , U , U U , U , U
bo‘ladi va 10 2 = h bo‘lgani uchun 2 10
140 1 10 120 130
0 1 10 120 110
2 10 120 100 5 4 3 2 1 = − = = − = = − = − = − = − = V V V , V , V
bo‘ladi. Bu topilgan qiymatlarni 1-jadvaldagi j i Y va X lar o‘rniga qo‘yamiz:
75 2-jadval U V -3
-2 -1
0 1 2 n v -2
1 5 - - - - 6 -1
- 5 3 - - - 8 0 - - 3 40 12 - 55 1 - - 2 10
5 - 17 2 - - - 3 4 7 14
n n 1 10 8 53
21 7 100
2-jadvaldan foydalanib hisoblash ishlarini bajaramiz: ( ) ( )
( ) ( ) ( ) ( ) 06 1 28 21 8 40 9 100 1 7 4 21 1 8 1 10 4 1 9 100 1 1 04 0 14 21 8 20 3 100
1 7 2 21 1 8 1 10 2 1 3 100 1 1 2 2 , n u n U , n u n U u u = + + + + = = ⋅ + ⋅ + ⋅ + ⋅ + ⋅ = = = + + − − − = = ⋅ + ⋅ + ⋅ − + ⋅ − + ⋅ − = = ∑ ∑
Xuddi shunga o‘xshash y va , , v , v y v υ υ 2 larni topamiz. ( ) (
( ) ( ) ( ) 05 1 56 17 8 24 100 1 14 4 17 1 8 1 6 4 100
1 1 25 0 28 17 8 12 100 1 14 2 17 1 8 1 6 2 100 1 1 2 2
n v n v , n v n v v v = + + + = = ⋅ + ⋅ + ⋅ + ⋅ = = = + + − − = = ⋅ + ⋅ + ⋅ − + ⋅ − = = ∑ ∑ 99 0 9875 0 0625 0 05 1 25 0 05 1 25 0 05 1 2 2 2 2 2 , , , , , , , , v v v v ≈ = − = − = − = − = σ σ
5 122 120
10 25 0 9 9 99 0 10 2 2 2
, C h v Y , , h v y = + ⋅ = + ⋅ = = ⋅ = ⋅ = σ σ Korrelyatsiya koeffitsiyenti r ni hisoblash uchun avval uv ni topamiz: ( ) ( ) ( ) ( ) ( ) ( )
( ( ) ( )
( ) ) ( ) 78 0 28 8 5 2 3 10 20 6 100 1 7 2 2 4 1 2 5 1 1 2 1 1 3 1 1 5 2 1 5 2 2 1 3 2 100
1 1
n v u n uv j i j i = + + + − + + + = ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ + ⋅ − ⋅ + ⋅ − ⋅ − + + ⋅ − ⋅ − + ⋅ − ⋅ − + ⋅ − + − = = ∑∑
u v u v u r σ σ ⋅ − = ga ko‘ra 75 0 0197 1 77 0 99 0 03 1 25 0 04 0 78 0
, , , , , , , r ≈ = ⋅ ⋅ − =
Korrelyatsiya koeffitsiyenti o‘zgaruvchilarni chiziqli almashtirishga nisbatan invariant bo‘lganligi uchun 75 0, r r r v u y x = = = bo‘ladi. 76
Topilgan ko‘rsatkichlarni chiziqli regressiya tenglamasiga qo‘yib quyidagini topamiz: ( ) (
06 33 92 1 5 122 2 17 15 5 9 9 5 122
, x , , y , x , , , y x x r y y x y − = − − = − − ⋅ = − σ σ
yoki 44 89 92 1 , x , y + = Bu esa Y o‘zgaruvchini X o‘zgaruvchiga chiziqli regressiya tenglamasi bo‘ladi. Javob: 44
92 1
x , y + = 2 17 17 5 04 0 15 5 03 1 5 03 1 0584 1 0016
0 06 1 04 0 06 1 1 1 1 2 2 2 2
, C h u X , , h , , , , , , u u n x n u = + ⋅ = + ⋅ = = ⋅ = ⋅ = ≈ = − = − = − = σ σ σ σ
12-mavzu bo‘yicha topshiriqlar.
Korrelyatsion jadval ma’lumotlari bo‘yicha Y ning X ga chiziqli regressiya tenglamasini toping.
Korrelyatsion jadval ma’lumotlari bo‘yicha Y ning X ga to‘g‘ri chiziqli regressiya tenglamasini eng kichik kvadratlar usuli bilan toping.
1) Y X 5 10 15 20
25 30
n y 45
2 4 - - - - 6 55
- 3 5 - - - 8 65
- - 5 35 5 - 45 75
- - 2 8 17
- 27
85 - - - 4 7 3 14
n x
2 7 12
47 29
3 n = 100
2) Y X 10
15 20
25 30
35 n y 40
2 4 - - - - 6 50
- 3 7 - - - 10 60
- - 5 30 10
- 45
70 - - 7 10
8 - 25 80 - - - 5 6 3 14
n x
2 7 19
45 24
3 n = 100 77
3) Y X 15
20 25
30 35
40 n y 15
4 1 - - - - 5 25
- 6 4 - - - 10 35
- - 2 50 2 - 54 45
- - 1 9 7 - 17 55
- - - 4 3 7 14 n x
4 7 7 63 12 7
4)
Y X 2 7 12 27
22 27
n y 100
1 5 - - - - 6 110
- 5 3 - - - 8 120
- - 3 40 12
- 55
130 - - 2 10
5 - 17 140 - - - 3 4 7 14
n x
1 10 8 53 21 7
5) Y X 5 10 15 20
25 30
n y 10
3 5 - - - - 8 20
- 4 4 - - - 8 30
- - 7 35 8 - 50 40
- - 2 10 8 - 20 50
- - - 5 6 3 14 n x
3 9 13
50 22
3 n = 100
6) Y X 12
14 22
27 32
37 n y 25
2 4 - - - - 6 35
- 6 3 - - - 9 45
- - 6 35 4 - 45 55
- - 2 8 6 - 16 65
- - - 14 7 3 24 n x
2 10 11
57 17
3 n = 100
78 7) Y X 15
20 25
30 35
40 n y 25
3 4 - - - - 7 35
- 6 3 - - - 9 45
- - 6 35 2 - 43 55
- - 12 8 6 - 26 65
- - - 4 7 4 15 n x
3 10 21
47 15
4 n = 100
8) Y X 4 9 14 19
24 29
n y 30
3 3 - - - - 6 40
- 5 4 - - - 9 50
- - 40 2 8 - 50 60
- - 5 10 6 - 21 70
- - - 4 7 3 14 n x
3 8 49
16 21
3 n = 100
9) Y X 5 10 15 20
25 30
n y 30
2 6 - - - - 8 40
- 5 3 - - - 8 50
- - 7 40 2 - 49 60
- - 4 9 6 - 19 70
- - - 4 7 5 16 n x
2 11 14
53 15
5 n = 100
10) Y X 10
15 20
25 30
35 n y 20
5 1 - - - - 6 30
- 6 2 - - - 8 40
- - 40 5 5 - 50 50
- - 2 8 7 - 17 60
- - - 4 7 8 19 n x
5 7 9 52 19 8
11)
Y X 5 10 15 20
25 30
35 40
n y 100
2 1 - - - - - - 3 120 3 4 3 - - - - - 10 140
- - 5 10 8 - - - 23 160 - - - 1 - 6 1 1 9 180
- - - - - - 4 1 5 n x
5 5 8 11 8 6 5 2 n = 50
79 12) Y X 18
23 28
33 38
43 48
n y 125
- 1 - - - - - 1 150 1 2 5 - - - - 8 175 - 3 2 12 - - - 17
200 - - 1 8 7 - - 16 225 - - - - 3 3 - 6 250 - - - - 1 1 - 2 n x
1 6 8 20 10 4 1 n = 50
13) Y X 5 10 15 20
25 30
35 n y 100
- - - - - 6 1 7 120 - - - - - 4 2 6 140 - - 8 10 5 - - 23
160 3 4 3 - - - - 10 180 2 1 - 1 - - - 4 n x
5 5 11
11 5 10 3 n = 50
14) Y X 13
18 23
28 33
n y 25
3 2 - - - 5 35 - 6 4 - - 10 45
- 1 9 5 - 15 55 - 1 2 4 8 15 65
- - 1 - 4 5 n x
3 10 16
9 12
n = 50
15) Y X 30
35 40
45 50
n y 46
2 6 - - - 8 56 2 8 10 - - 20 66
- - 32 3 9 44 76 - - 4 11
6 21
86 - - - 2 5 7 n x
4 14 46
16 20
n = 100
16) Y X 33
38 43
48 53
58 n y 65
4 8 1 - - - 13 75
- 4 4 2 - - 10 85
- 1 6 6 1 - 14 95
- - - 1 5 - 6 105
- - - 1 4 1 6 115
- - - - 2 4 6 n x
4 13 11
10 12
5 n = 55 80
17) Y X 3 7 11 15
19 23
n y 6 5 3 - 2 - - 10 16 7 10 1 2 - - 20
26 2 18 15 20
- - 55 36 - - 30 26
- - 56 46 - - - 19
12 - 31 56 - - - - 21 7 28
n x
14 31 46
69 33
7 n = 200
18) Y X 45
50 55
60 65
70 75
n y 30
- - - - 8 2 1 11
35 - 1 6 22
33 10
3 75
40 1 2 10 48
37 8 1 107 45
- 1 12 11 2 - - 26
50 - - 1 - - - - 1 55 - - 1 - - - - 1 n x
1 6 30
82 80
20 5
19)
Y X 0 1 2 3 4 n y 0 18 1 1 - - 20
3 1 20 - - - 21 6 3 5 10
2 - 20 9 - - 7 12
- 19
12 - - - - 20 20 n x
22 26 18
14 20
n = 100
20) Y X 0 4 8 12
16 n y 7 19 1 1 - - 21
13 2 14 - - - 16 19
- 3 22 2 - 27 25 - - - 15
- 15
31 - - - - 21 21 n x
21 18 23
17 21
n = 100
21) Y X 0 1 2 3 4 n y 10
20 5 - - - 25 20 7 15 3 1 - 26 30
- 3 17 4 - 24 40 - - 8 13
7 28
50 - - - 5 42 47 n x
27 23 28
23 49
n = 150 81
22) Y X 150
165 175
185 195
n y 50
2 2 - - - 4 70 - 2 - - - 2 90
- - 9 2 1 12 110 - - 2 7 9 18 130
- - - 3 11
14 n x
2 4 11
12 21
n = 50
23) Y X 20
25 30
35 40
n y 10
3 7 - - - 10 16 - 12 5 1 - 18 20
- - 6 1 1 8 24 - - - 3 1 4 28
- - - - 1 1 n x
3 19 11
5 3
24)
Y X 25
35 45
55 65
n y 2 5 10 - - - 15
4 - 13 10 10
- 33
6 - - 18 16
- 34
8 - - - 2 2 4 10
- - - - 1 1 n x
5 23 28
28 3
Y X 10
20 30
40 50
n y 10
7 17
10 - - 34 20
- 23
12 5 - 40 30
- 10
5 3 2 20 40
- - 2 2 1 5 50 - - - - 1 1 n x
7 50 29
10 4
26)
Y X 5 15 25 35
45 n y 2 3 14 - - - 17
12 - 16 18 - - 34 22
- - 20 10 11
41 32
- - - 6 2 8 n x
3 30 38
16 13
n = 100
82 27) Y X 1 6 11 16
21 n y 5 3 10 - - - 13
10 4 11 10 - - 25 15
- 5 15 10 - 30 20 - - 11 10
4 25
25 - - - 4 3 7 n x
7 26 36
24 7
28)
Y X 4 6 8 10
12 n y 3 7 21 10
- - 38 8 - 5 15 10
- 30
13 - - 11 10
4 25
18 - - - 4 3 7 n x
7 26 36
24 7
29)
Y X 3 7 11 15
19 n y 2 2 4 - - - 6 6 - 3 5 - - 8 8 - - 5 35
5 45
10 - - 2 8 17 27 12
- - - 4 10
14 n x
2 7 12
47 32
n = 100
30) Y X 2 5 8 11
14 17
n y 1 2 4 - - - - 6 6 - 6 3 - - - 9 11 - - 6 35 4 - 45 16
- - 2 8 6 - 16 21
- - - 14 7 3 24 n x
2 10 11
57 17
3 n = 100
Download 0.54 Mb. Do'stlaringiz bilan baham: 
|