1- masala. Quyidagi jadvalda keltirilgan ma’lumotlar asosida

Y	12	15	16	20
22
X1	6	8	12	10	11
X2	8	9	11	13	14

Korrelyatsion tahlil usuli yordamida, berilgan ma'lumotlar bo'yicha korrelyatsiya tahlili olib boramiz:
Y: 12, 15, 16, 20, 22
X1: 6, 8, 12, 10, 11
X2: 8, 9, 11, 13, 14

1. 
   Korrelyatsion ko'effitsiyentni hisoblash:
   

Korrelyatsion ko'ffitsiyenti, ma'lumotlar o'rtasidagi bog'lanish darajasini ifodalaydi. Biz Pearson korrelyatsion ko'ffitsiyentini hisoblash uchun foydalanamiz.
Pearson korrelyatsion ko'ffitsiyenti formulasi: r = Cov(X, Y) / (σ(X) * σ(Y))

• 
  X1 va Y o'rtasidagi korrelyatsiyani hisoblaymiz:
  Cov(X1, Y) = Σ((X1 - X1_mean) * (Y - Y_mean)) / n
  X1_mean = (6 + 8 + 12 + 10 + 11) / 5 = 9.4
  Y_mean = (12 + 15 + 16 + 20 + 22) / 5 = 17
  Cov(X1, Y) = ((6 - 9.4) * (12 - 17) + (8 - 9.4) * (15 - 17) + (12 - 9.4) * (16 - 17) + (10 - 9.4) * (20 - 17) + (11 - 9.4) * (22 - 17)) / 5
  = (-3.4 * -5 + -1.4 * -2 + 2.6 * -1 + 0.6 * 3 + 1.6 * 5) / 5
  = (17 + 2.8 + -2.6 + 1.8 + 8) / 5
  = 27 / 5
  = 5.4
  

Var(X1) = Σ((X1 - X1_mean)^2) / n
= ((6 - 9.4)^2 + (8 - 9.4)^2 + (12 - 9.4)^2 + (10 - 9.4)^2 + (11 - 9.4)^2) / 5
= (10.24 + 1.96 + 6.76 + 0.36 + 1.44) / 5
= 20.76 / 5
= 4.152
Var(Y) = Σ((Y - Y_mean)^2) / n
= ((12 - 17)^2 + (15 - 17)^2 + (16 - 17)^2 + (20 - 17)^2 + (22 - 17)^2) / 5
= (25 + 4 + 1 + 9 + 25) / 5
= 64 / 5
= 12.8
σ(X1) = sqrt(Var(X1))
= sqrt(4.152)
≈ 2.037
σ(Y) = sqrt(Var(Y))
= sqrt(12.8)
≈ 3.577
r(X1, Y) = Cov(X1, Y) / (σ(X1) * σ(Y))
= 5.4 / (2.037 * 3.577)
≈ 0.781

• 
  X2 va Y o'rtasidagi korrelyatsiyani hisoblaymiz:
  Cov(X2, Y) = Σ((X2 - X2_mean) * (Y - Y_mean)) / n
  X2_mean = (8 + 9 + 11 + 13 + 14) / 5 = 11
  Cov(X2, Y) = ((8 - 11) * (12 - 17) + (9 - 11) * (15 - 17) + (11 - 11) * (16 - 17) + (13 - 11) * (20 - 17) + (14 - 11) * (22 - 17)) / 5
  = (-3 * -5 + -2 * -2 + 0 * -1 + 2 * 3 + 3 * 5) / 5
  = (15 + 4 + 0 + 6 + 15) / Korrelyatsion tahlil usuli yordamida, berilgan ma'lumotlar bo'yicha korrelyatsiya tahlili olib boriladi:
  

Y: 12, 15, 16, 20, 22
X1: 6, 8, 12, 10, 11
X2: 8, 9, 11, 13, 14
Korrelyatsion ko'effitsiyentni hisoblash uchun Pearson korrelyatsion ko'ffitsiyentini qo'llaymiz.
Pearson korrelyatsion ko'ffitsiyenti formulasi: r = Cov(X, Y) / (σ(X) * σ(Y))

• 
  X1 va Y o'rtasidagi korrelyatsiyani hisoblaymiz:
  Cov(X1, Y) = Σ((X1 - X1_mean) * (Y - Y_mean)) / n
  X1_mean = (6 + 8 + 12 + 10 + 11) / 5 = 9.4
  Y_mean = (12 + 15 + 16 + 20 + 22) / 5 = 17
  Cov(X1, Y) = ((6 - 9.4) * (12 - 17) + (8 - 9.4) * (15 - 17) + (12 - 9.4) * (16 - 17) + (10 - 9.4) * (20 - 17) + (11 - 9.4) * (22 - 17)) / 5
  = (-3.4 * -5 + -1.4 * -2 + 2.6 * -1 + 0.6 * 3 + 1.6 * 5) / 5
  = (17 + 2.8 + -2.6 + 1.8 + 8) / 5
  = 27 / 5
  = 5.4
  

Var(X1) = Σ((X1 - X1_mean)^2) / n
= ((6 - 9.4)^2 + (8 - 9.4)^2 + (12 - 9.4)^2 + (10 - 9.4)^2 + (11 - 9.4)^2) / 5
= (10.24 + 1.96 + 6.76 + 0.36 + 1.44) / 5
= 20.76 / 5
= 4.152
Var(Y) = Σ((Y - Y_mean)^2) / n
= ((12 - 17)^2 + (15 - 17)^2 + (16 - 17)^2 + (20 - 17)^2 + (22 - 17)^2) / 5
= (25 + 4 + 1 + 9 + 25) / 5
= 64 / 5
= 12.8
σ(X1) = sqrt(Var(X1))
= sqrt(4.152)
≈ 2.037
σ(Y) = sqrt(Var(Y))
= sqrt(12.8)
≈ 3.577
r(X1, Y) = Cov(X1, Y) / (σ(X1) * σ(Y))
= 5.4 / (2.037 * 3.577)
≈ 0.781

• 
  X2 va Y o'rtasidagi korrelyatsiyani hisoblaymiz:
  Cov(X2, Y) = Σ((X2 - X2_mean) * (Y - Y_mean)) / n
  X2_mean = (8 + 9 + 11 + 13 + 14) / 5 = 11
  Cov(X2, Y) = ((8 - 11) * (12 - 17) + (9 - 11) * (15 - 17) + (11 - 11) * (16 - 17) + (13 - 11) * (20 - 17) + (14 - 11) * (22 - 17)) / 5
  = (-3 * -5 + -2 * -2 + 0 * -1 + 2 * 3 + 3 * 5) / 5
  = (15 + 4 + 0 + 6 + 15) / 5
  = 40 / 5
  = 8
  

Var(X2) = Σ((X2 - X2_mean)^2) / n
= ((8 - 11


27- Masala. Berilgan ma’lumotlar asosida bir omilli ekonometrik modelini tuzing va Fisher mezoni qiymatini hisoblang va tahlil qiling. Iqtisodiy tahlilni amalga oshirib, xulosa bering.