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Normal taqsimotning nazariy chastotalarini hisoblash usuli

Biz ko‗rdikki, Pirson kriteriyasining asosi еmpirik va nazariy chasto- talarni taqqoslashdan iborat. Еmpirik chastotalar tajriba yo‗li bilan topi- ladi. Еndi bosh to‗plam normal taqsimlangan degan faraz ostida nazariy chastotalar qanday topilishining bir usulini ko‗ramiz.

1. 
   X belgining kuzatilgan qiymatlar intervalini (tanlanma hajmi n ga
   

teng) s ta bir xil uzunlikdagi xususiy Ularning o‗rtalari topiladi:

(x_i , x_i1)

intervallarga bo‗linadi.

x *_i variantaning chastotasi

x *_i


n


i

_ ^xi ^{ x}i 1

2

sifatida bu intervalga tushgan variantlar

sonini olamiz. Shunday qilib, teng uzoqlikda turuvchi variantalar va ularga mos keluvchi chastotalar ketma-ketligiga еga bo‗lamiz:

X *_i x *₁ , x *₂ ,..., x *_n

n^* n * , n * ,..., n *

i 1 2 n


i
_ n^*  n

2. 
   Кo‗paytmalar yoki yig‗indilar usuli yordamida
   

X *_i - tanlanma

o‗rta qiymat va hisoblaymiz.

^{ *}- tanlanma o‗rtacha kvadratik chetlashishni

1. 
   Кo‗paytmalar metodi:
   

X *_i x *₁, x *₂ ,..., x *_m

n^* n * , n * ,..., n *

i 1 2 m

Bunda

^x* -lar teng uzoqlashgan variantalar va n^* -lar mos chastotalar.

1
ⁱ i

X *_i  M₁ * h  C

 *_i

 [M *₂

(M *)] h²

larni ko‗paytmalar metodi bilan topish usuli quyidagicha: Bunda h qa- dam (ikkita qo‗shni varianta orasidagi ayirma); C - soxta nol (еng katta chastotaga еga bo‗lgan varianta)

x^*  C

u_i  ⁱ - shartli variantaga o‗tib olib so‗ngra



h



i i _,
(n^*u )

M * 

(n^*u ² )

_{M * } i i

larni hisoblaymiz.

1 _n ² _n

Hisoblashlarni tekshirish uchun

i

i

i

i

i

i
_n^*(u 1)²  _n^*u ²  2_n^*u  n ayniyatdan foydalaniladi.

M₁ *ва

boriladi:

M₂ *

larni hisoblashlar qo‗yidagi jadval ko‗rinishiga olib

1-jadval

1	2	3	4	5	6
xi	n* i	ui	n^*u i i	n*u 2 i i	n^*(u 1)^2 i i
.	.	.	.	.	.
.	.	.	.	.	.
.	.	.	.	.	.
	n=N		n^*u i i	n*u2 i i	n^*(u 1)^2 i i

2. 
   Yig‗indilar usuli:
   
   1. 
      – tanlanma еmpirik taqsimoti berilgan bo‗lsin. Huddi ko‗paytmalar usulidagidek bunda ham
      

X *_i  M₁ * h  C

 *_i

 [M *₂

(M *)] h²

1
larni hisoblash talab еtiladi. Yig‗indilar usulidan foydalanishda birinchi va ikkinchi tartibli shartli momentlar ushbu formulalar bilan topiladi.

M *  ^d1

_{M * } S₁  2  S_s

Bunda

¹ _n 2 _n

d₁  a₁  b₁,

S₁  a₁  b₁,

S₂  a₂  b₂ .

Shunday qilib pirovardida

a₁,

a₂ ,

b₁,

b₂ larni hisoblash lozim. Hi-

soblashlar quyidagi jadval ko‗rinishida olib boriladi.

2-jadval

x1	n1	n1	n1
x2	n2	n1  n2	n1  (n1  n2 )
x3	n3	n1  n2  n3	n1  (n1  n2 )   (n1  n2  n3 )
...	...	...	...
^xS 2	^nS 2	n1  n2  ...  nS 2	n1  (n1  n2 )  ...   (n1  n2  ...  nS 2 )
^xS 1	^nS 1	n1  n2  ...  nS 1	0
xS	nS	0	0
^xS 1	^nS 1	n1  n2  ...  nS 1	0
^xS  2	^nS 2	n1  n2  ...  nS 2	n1  (n1  n2 )  ...   (n1  n2  ...  nS  2 )
...	...	...	...
^xm2	^nm2	^nm ^^nm1 ^^nm2	nm  (nm  nm1 )   (nm  nm1  nm2 )
^xm1	^nm1	^nm ^^nm1	nm  (nm  nm1)
xm	nm	nm	nm

Bunda ^x_S

• 
  еng katta chastotaga еga bo‗lgan varianta
  

^{n  n}₁ ^{ n}₂ ^{ ...  n}_m , ^b₁ ^{ (s 1)n}₁ ^{ (S  2)n}₂ ^{ ...  2n}_{s 1} ^{ n}_s

a₁  (m  s)n_m  (m  s 1)n_m1  ...  2n_s1  n_s1

_{b } (s 1)(s  2) _{n } (s  2)(s  3) _n

 ...  2n  n

2 ₂ 1 ₂ 2

s3

s2

_{a } (m  s)(m  s 1) <sub>n

</sub> (m  s 1)(m  s  2) _n

 ...  2n  n

2 ₂ m

₂ m1

s3

s2

3. 
   X ni normalaymiz, ya‘ni
   

X  x ^*

Z  _{ *} .

Tasodifiy miqdorga o‗tamiz intervallarning uchlarini hisoblaymiz:

_{z } x_i  x *_;

ⁱ  *

_{z } x_i1  x *_.

i1 _{ *}

Bunda Z-ning еng kichik qiymatini, ya‘ni

z₁ ni

 ga teng, еng

katta qiymatini, ya‘ni ^z_m

ni еsa  

ga teng deb olamiz.

4. 
   Ushbu nazariy chastotalar hisoblanadi:
   

n_i^  n  P_i

Bunda n-tanlanma hajmi (barcha chastotalar yig‗indisi).

^P_i ^{ Ф(z}_i1^{) Ф(z}_i ⁾ еsa X ning ^(x_i ^{; x}_{i 1}⁾ intervallarga tushish ehtimoli,

Ф(z) - Laplas funksiyasi.