matematik kutilma

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1629206026.doc ( ) n n 2 2 1 1 p x ... p x p x x m × + + × + × = ( ) c c m = ( ) ( ) x cm cx m = ( ) ( ) ( ) y m x m xy m = ( ) ( ) ( ) y m x m y x m + = + m ( ) ) a ( p p , p n m = × = m { } n ,..., 2 , 1 x ( ) ; 2 n 1 m + = x ( ) ( ) [ ] 2 x m x m x d - = х 1 x 2 x 3 x n x 1 p 2 p 3 p n p ( ) [ ] 2 x m x - 2 )] x ( m x …

+ - x ¥ - - x ¥ + ¥ - - x ( ) ( ) ( ) ( ) + + = - + + = ò ò ¥ x ¥ x x 0 0 x 2 dx x m xp dx x m p x m p x 2 1 x ( ) ( ) ( ) ( ) ( ) ) 3 ( m 2 m 2 ) 2 ( 0 dz z p m z dy y p m y x 2 dx x m xp = - - - = - ò ò ò -¥ x ¥ x ¥ x ( ) ( ) ( ) ( ) ( ) ( ) ò ò ò ¥ ¥ - x ¥ - x ¥ x - = - + - = dx x p m x dx x p m x dx x p m …

’ladi va a<0 bо’lgani uchun (integrallashning yangi yuqori chegarasi) , (integrallashning yangi quyi chegarasi) (3) integrallash chegaralarining о’rnini almashtiramiz. bu teoremadan, integralning chiziqliligidan va о’tgan temadagi teorema 2 dan (3) formula tо’la isbot bо’ladi. ya’ni: еndi tasodifiy miqdorning dispersiyasini hisoblash formulasini chiqaramiz. teorema 3. agar - zichlik funksiya bо’lib, bо’lsa, u holda: (11) isboti: avvalo - tasodifiy miqdorning taqsimot funksiyasi - ni topamiz: bо’lgani uchun, larda va shuning uchun: da deb - tasodifiy miqdorning zichlik funksiyasini belgilasak, quyidagi hosil bо’ladi: shunday qilib: еndi - tasodifiy miqdorning dispersiyasini ta’rifga asosan hisoblash mumkin (1): lar uchun bо’lganidan (2): birinchi integralda almashtirish bajarsak va bо’ladi (integrallashning yangi yuqori chegarasi), ikkinchi integralda almashtirish bajarsak va bо’ladi (integrallashning yangi quyi chegarasi). (3): integrallash о’zgaruvchisini yana x deb belgilaymiz (integrallashning о’zgaruvchiga nisbatan invariantligi uchun integral qiymati о’zgarmaydi), ikkinchi integralda еsa integrallash tartibini о’zgartiramiz. shunday qilib (11) formula isbot bо’ldi. misol: tasodifiy miqdorning zichlik funksiyasi berilgan: va …

y miqdorlarning quyidagi ikki sonli xarakteristikalarini ko’rish mumkin. mediana me deb tasodifiy miqdorning shunday qiymatiga aytamizki, bunda bo’ladi.demak tasodifiy tajribada x-tasodifiy miqdor bir xil extimol bilan yoki me dan katta bo’ladi, yoki me dan kichik bo’ladi. bu xarakteristika tasodifiy miqdorning qiymatlari sonlar o’qida qanday joylashganligini xarakterlaydi. normal taqsimlangan tasodifiy miqdor uchun . mediana noparametrik statistikada muhim rol o’ynaydi. moda deb diskret taqsimlangan tasodifiy miqdorlar uchun tasodifiy miqdorning eng katta extimolga ega bo’lgan qiymatiga aytiladi. uzluksiz tasodifiy miqdorlar uchun moda tasodifiy miqdorning zichlik funksiya maksimum qiymatga erishadigan qiymatiga teng. zichlik funksiyasi bitta maksimumga ega bo’lgan taqsimotlar unimodal taqsimotlar, bir necha maksimumga ega bo’lgan taqsimotlar polimodal taqsimotlar deyiladi. moda ham matematik kutilma va mediana kabi tasodifiy miqdorning sonlar o’qidagi joylanishini xarakterlovchi sonli xarakteristikadir. simmetrik, unimodal taqsimotlar uchun bu uchala xarakteristika bir xildir. normal taqsimot uchun moda bilan matematik kutilma teng bo’ladi. uzluksiz tasodifiy miqdorlarning taqsimot qonunlariga misollar. 1. agar - tasodifiy miqdor …

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1629206026.doc ( ) n n 2 2 1 1 p x ... p x p x x m × + + × + × = ( ) c c m = ( ) ( ) x cm cx m = ( ) ( ) ( ) y m x m xy m = ( ) ( ) ( ) y m x m y x m + = + m ( ) ) a ( p p , p n m = × = m { } n ,..., 2 , 1 x ( ) ; 2 n 1 m + = x ( ) ( ) [ ] 2 x m x m x d - = х 1 …

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