kovariatsiya. kovariatsiya koeffiseinti

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1576504252.doc [ ] ) ( ( ) ( ) , cov( j j i i j i ij m m m x x x x x x s - × - = = i x j x i x j x j i j i ij m m m x x x x s - = ) ( = + - - = - - = ) ( ) )( ( j i j i i j j i j j i i ij m m m m m m m m x x x x x x x x x x x x s j i j i j i j i i j j i m m m m m m m m m m x x x x x x x x x x x x - = + - - = ) ( ) ( 1 …

1 x s x x h m - = 2 2 2 2 x s x x h m - = ÷ ÷ ø ö ç ç è æ - - = 2 1 2 1 2 2 1 1 , , cov x x x x s x x s x x r m m 2 1 2 1 ) , cov( 2 1 , x x x x s s x x r = 2 1 2 1 , 2 1 ) , cov( x x x x s s r x x = 1 x 2 x 0 2 1 , = x x r 0 ) , cov( 2 1 = x x ) 1 , 0 ( ~ 1 n x 1 ) ( 2 - = x x f 1 ) ( 2 1 1 2 - = = x x x f 1 …

kovariasiyasi yoki korrelyatsion moment deyiladi. unga va larning aralash markaziy momentga ham deyiladi. 1-teorema: kovariasiyani hisoblash uchun quyidagi formula o`rinli: isboti: 2-teorema: agar va lar bog`lanmagan bo`lsa, . bu teorema isboti 1-teoremadan, va larning bog`lanmaganligidan ni e`tiborga olsak, kelib chiqadi. agar bo`lsa, va tasodifiy miqdorlar bog`liq bo`ladi. kovariatsiya ta`rifi va matematik kutilma xossalaridan quyidagilar kelib chiqadi. (*) bu yerda va lar o`zgarmaslar. tasodifiy vektorning kovariasion matrisasi deb, matrisaga aytiladi, bu yerda kovariasiya xossalaridan kovariasion matrisa simmetrikligi va uning diognal elementlari tasodifiy miqdorlarning dispyersiyalari ekanligi kelib chiqadi. kovariasion matrisa determinantiga umumlashgan dispyersiya deyiladi. umumlashgan dispyersiya n o`lchovli tasodifiy miqdor tarqoqlik darajasini xaraktyerlaydi. kovariasion matrisa va o`rtachalar vektori tasodifiy vektorning asosiy sonli xarakteristikalaridir. quyidagi teoremani isbotsiz keltiramiz. 3-teorema: agar tasodifiy miqdorlar uchun mavjud bo`lsa, u holda har qanday o`zgarmaslar uchun , tasodifiy miqdorlarning kovariasiyalar mavjud va va larning kovariasion matrisalar va lar tenglik bilan bog`langan, bu yerda esa matrisa transponyerlangani. bu teoremadan …

sifat xarakteristikasi sifatida korrelyasiya koeffisienti dan foydalaniladi, u va normallangan tasodifiy miqdor kovariasiyasiga teng. . demak korrelyasiya koeffisienti (4). bu yerdan (5) bog`lanmagan tasodifiy miqdor , lar uchun , chunki . teskari tasdiq o`rinli emas, bog`liq tasodifiy miqdorlarning ham korrelyasiya koeffisienti nolga teng bo`lishi mumkin. masalan; agar , , bo`lsa, va bog`langan, ularning korrelyasiya koeffisienti nolga tengligini ko`rsatamiz. bo`lgani uchun, , bundan demak, . va tasodifiy miqdorlarning korrelyasiya koeffisienti nolga teng bo`lsa, ular korrelyasion bog`lanmagan deyiladi. 4-teorema: korrelyasiya koeffisienti absalyut qiymati bo`yicha birdan katta bo`lolmaydi: (5). isboti: bo`lsin. u holda o`rinli bo`lganligi uchun va hamda va dan bu tengsizlikdan va larga ega bo`lamiz. bulardan esa teoremaning isboti kelib chiqadi. 5-teorema: bo`lishi uchun va ( ) o`zgarmaslar topilib bo`lishi zarur va yetarli. jumladan bo`lsa , bo`lsa, . isboti: bo`lsin, u holda demak , dispyersiya xossasiga asosan (c-o`zgarmas). va demak ya`ni bundan bu yerdan bunda . agar bo`lsa, bo`ladi, bundan kelib chiadi. …

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1576504252.doc [ ] ) ( ( ) ( ) , cov( j j i i j i ij m m m x x x x x x s - × - = = i x j x i x j x j i j i ij m m m x x x x s - = ) ( = + - - = - - = ) ( ) )( ( j i j i i j j i j j i i ij m m m m m m m m x x x x x x x x x x x x s j i j i j i j i i j j i m m m …

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