1- masala. Quyidagi jadvalda keltirilgan ma’lumotlar asosida

Yil	2012	2013	2014	2015	2016	2017	2018	2019	2020
2021
Y	550,1	891,5	1387,2	2104,8	1161,4	1165,4	1167,6	550,1	891,5	1387,2
X	66,4	81,3	102,8	134,0	176,2	178,6	181,8	66,4	81,3	102,8

Topshiriq

Y ning X ga ga bog‘liqligini tavsiflash uchun quyidagi funktsiyalar parametrlarini hisoblang:

1. 
   Chiziqli;
   
2. 
   Darajali;
   
3. 
   Har bir aniqlangan modelni R², o‘rtacha aproksimatsiya hatoligi – Ā bilan baholang.
   

Berilgan ma'lumotlar asosida funktsiyalar parametrlarini hisoblaylik:

1. 
   Chiziqli funksiya:
   y = a + bx
   

Ma'lumotlar:
x = [66.4, 81.3, 102.8, 134.0, 176.2, 178.6, 181.8, 66.4, 81.3, 102.8]
y = [550.1, 891.5, 1387.2, 2104.8, 1161.4, 1165.4, 1167.6, 550.1, 891.5, 1387.2]
n = 10
Hisoblanishi kerak bo'lgan qiymatlar:
Σx = 1072
Σy = 10746.6
Σ(x - Ẋ)² = 46560.32
Σ((x - Ẋ)(y - Ȳ)) = 3009655.58
Bu qiymatlarga asosan, parametrlarni hisoblaymiz:
Ȳ = (10746.6) / 10 = 1074.66
Ẋ = (1072) / 10 = 107.2
b = 3009655.58 / 46560.32 = 64.62
a = 1074.66 - (64.62 * 107.2) = 340.83
Shundaylikcha, chiziqli funksiyadagi a va b parametrlari:
a ≈ 340.83
b ≈ 64.62

2. 
   Darajali funksiya:
   y = a * x^b
   

Ma'lumotlar:
ln(x) = [4.194, 4.398, 4.632, 4.897, 5.170, 5.185, 5.201, 4.194, 4.398, 4.632]
ln(y) = [6.309, 6.794, 7.063, 7.652, 7.057, 7.060, 7.062, 6.309, 6.794, 7.063]
n = 10
Hisoblanishi kerak bo'lgan qiymatlar:
Σln(x) = 45.760
Σln(y) = 66.862
Σ(ln(x))^2 = 192.816
Σ(ln(x) * ln(y)) = 278.037
Bu qiymatlarga asosan, parametrlarni hisoblaymiz:
b = (10 * 278.037 - 45.760 * 66.862) / (10 * 192.816 - (45.760)^2) = 1.267
a = exp((66.862 - 1.267 * 45.760) / 10) = 165.148
Shundaylikcha, darajali funksiyadagi a va b parametrlari:
a ≈ 165.148
b ≈ 1.267

3. 
   R2 koeffitsienti va o'rtacha aproksimatsiya hatoligi Ā ni hisoblaymiz:
   

R2 (Chiziqli):
Ŷ1 = 340.83 + 64.62 * 66.4 ≈ 4596.79
Ŷ2 = 340.83 + 64.62 * 81.3 ≈ 5731.08
Ŷ3 = 340.83 + 64.62 * 102.8 ≈ 7233.62
Ŷ4 = 340.83 + 64.62 * 134.0 ≈ 9416.08
Ȳ = (10746.6) / 10 = 1074.66
R2 (Chiziqli):
R2 = 1 - ( (550.1 - 4596.79)^2 + (891.5 - 5731.08)^2 + (1387.2 - 7233.62)^2 + (2104.8 - 9416.08)^2 ) / ( (550.1 - 1074.66)^2 + (891.5 - 1074.66)^2 + (1387.2 - 1074.66)^2 + (2104.8 - 1074.66)^2 )
≈ 1 - ( 11276448.05 + 13316460.13 + 13934497.40 + 17792635.40 ) / ( 2132355.45+ 113827.20 + 135796.76 + 173296.08 )
≈ 1 - 56219940.98 / 760410.49
≈ 0.926
Ā (Chiziqli):
Ā = sqrt( ( (550.1 - 4596.79)^2 + (891.5 - 5731.08)^2 + (1387.2 - 7233.62)^2 + (2104.8 - 9416.08)^2 ) / 10 )
≈ sqrt( ( 11276448.05 + 13316460.13 + 13934497.40 + 17792635.40 ) / 10 )
≈ sqrt( 56219940.98 / 10 )
≈ sqrt( 5621994.10 )
≈ 2372.61
R2 (Darajali):
Ŷ1 = 165.148 * (66.4^1.267) ≈ 718.26
Ŷ2 = 165.148 * (81.3^1.267) ≈ 887.14
Ŷ3 = 165.148 * (102.8^1.267) ≈ 1122.38
Ŷ4 = 165.148 * (134.0^1.267) ≈ 1457.47
Ȳ = (10746.6) / 10 = 1074.66
R2 (Darajali):
R2 = 1 - ( (550.1 - 718.26)^2 + (891.5 - 887.14)^2 + (1387.2 - 1122.38)^2 + (2104.8 - 1457.47)^2 ) / ( (550.1 - 1074.66)^2 + (891.5 - 1074.66)^2 + (1387.2 - 1074.66)^2 + (2104.8 - 1074.66)^2 )
≈ 1 - ( 1014821.06 + 0.02 + 102640.87 + 303444.58 ) / ( 2132355.45 + 113827.20 + 135796.76 + 173296.08 )
≈ 1 - 1434906.53 / 2408275.49
≈ 0.404
Ā (Darajali):
Ā = sqrt( ( (550.1 - 718.26)^2 + (891.5 - 887.14)^2 + (1387.2 - 1122.38)^2 + (2104.8 - 1457.47)^2 ) / 10 )
≈ sqrt( ( 1014821.06 + 0.02 + 102640.87 + 303444.58 ) / 10 )
≈ sqrt( 1434906.53 / 10 )
≈ sqrt( 143490.65 )
≈ 378.93
Jami hisoblangan qiymatlar:

1. 
   Chiziqli funksiyada:
   a ≈ 340.83
   b ≈ 64.62
   R2 ≈ 0.926
   Ā ≈ 2372.61
   
2. 
   Darajali funksiyada:
   a ≈ 165.148
   b ≈ 1.267
   R2 ≈ 0.404
   Ā ≈ 378.93
   

Ushbu qiymatlarga asosan, ma'lumotlarning chiziqli va darajali modellarga bog'liqligi tavsiflanadi. Chiziqli model R2 koeffitsienti 0.926 ga yaqinligi bilan ma'lumotlarni yaxshi ifodalamoqda, darajali model esa R2 koeffitsienti 0.404 bilan ma'lumotlarni to'g'ri ifodalamaydi.


17- Masala. O‘zbekiston Rspublikasi aholisining daromadi – Y ming so‘mni tashkil etib undan X ming so‘mini jamg‘arma sifatida bankka omonat sifatida qo‘yishadi. Ko‘rsatkich qiymatlarining yillar davomida o‘zgarishi quyidagi jadvalda berilgan