ࡱ> RTQ_ tbjbj.bbQ%+! 5! A!A!A!U!U!U!8!"TU!"\A("c(c(c(>)>)>)$3RФA!>)>)>)>)>)ФA!A!c(c(u^:^:^:>)L A!c(A!c(^:>)^:^: c(qV5Z0@ 9A!>)>)^:>)>)>)>)>)ФФ9j>)>)>)>)>)>)>)>)>)>)>)>)>)>)>)>)? :Maruza. Tasodifiy miqdorlarning matematik kutilmasi va uning xossalari. Reja. Matematik kutilma tushunchasi. Matematik kutilmaning ehtimollik manosi. Matematik kutilmaning xossalari. Tayanch tushunchalar: Matematik kutilma. Matematik kutilma tushunchasi. Tasodifiy miqdor haqida toliq malumotni uning taqsimot funksiyasi yordamida olish mumkinligi bizga malum. Haqiqatan ham taqsimot funksiya tasodifiy miqdorning qaysi qiymatlarni qanday ehtimolliklar bilan qabul qilishini aniqlashga imkon beradi. Lekin bazi hollarda tasodifiy miqdor haqida kamroq malumotlarni bilish ham yetarli boladi. Ehtimolliklar nazariyasi va uning amaliyotdagi tadbiqlarida tasodifiy miqdorlarning taqsimot funksiyalari orqali malum qoidalar asosida topiladigan bazi ozgarmas sonlar muhim rol oynaydilar. Bunday sonlar orasida tasodifiy miqdorlarning umumiy miqdoriy xarakteristilalarini bilish uchun matematik kutilma, dispersiya va turli tartibdagi momentlar juda muhimdir. "asodifiy miqdorning biz dastlab tanishadigan asosiy sonli Earakteristikasi uning matematik kutilmasidir.  EMBED Equation.DSMT4  diskret tasodifiy miqdor  EMBED Equation.DSMT4  qiymatlarni  EMBED Equation.DSMT4  ehtimolliklar bilan qabul qilsin. Unda, . 1-tarif.  diskret tasodifiy miqdorning matematik kutilmasi deb, ushbu  EMBED Equation.DSMT4  tenglik bilan aniqlanuvchi songa aytiladi. Diskret tasodifiy miqdorlarning mumkin bolgan qiymatlari soni cheksiz bolishi ham mumkin. Bu holda  EMBED Equation.DSMT4  va matematik kutilmani tariflash uchun  EMBED Equation.DSMT4  (1) qatordan foydalaniladi. Bazi misollarni qarab chiqamiz. 1-misol. A hodisaning roy berish ehtimolligi p ga teng bolsa, bitta tajribada A hodisa roy berish sonining matematik kutilmasini toping. Yechish. Bitta tajribada A hodisaning roy berish sonini  deb belgilaylik. U holda  EMBED Equation.DSMT4 , bu erda  EMBED Equation.DSMT4  va 1-tarifga asosan,  EMBED Equation.DSMT4  . 2-misol.  EMBED Equation.DSMT4  parametrli binomial qonun bilan taqsimlangan tasodifiy miqdorning matematik kutilmasini toping. Yechish:  orqali A hodisaning n ta bogliqsiz tajribalarda roy berish sonini belgilasak,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  tenglik orinli ekani bizga malum.Matematik kutilma tarifiga kora  EMBED Equation.DSMT4  3-misol. Puasson qonuni bilan taqsimlangan tasodifiy miqdorning matematik kutilmasini toping. Yechish:  EMBED Equation.3  tenglik orinli ekani bizga malum. Uning taqsimot qonunini ushbu jadval korinishida yozamiz. Ei012& m& pi EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4 &  EMBED Equation.DSMT4 &  Matematik kutilmasi uchun quyidagiga ega bo lamiz:  EMBED Equation.DSMT4  Qavs ichidagi qator  EMBED Equation.DSMT4  funksiyaning Makloren qatoriga yoyilmasidir. Shuning uchun matematik kutilma  EMBED Equation.DSMT4 . Shunday qilib, biz Puasson taqsimot qonuniga kirgan  EMBED Equation.DSMT4  parametrning ehtimolliy manosini topdik:  EMBED Equation.DSMT4  parametr tasodifiy miqdorning matematik kutilmasiga teng.  uzluksiz tasodifiy miqdorning zichlik funksiyasi p(x) bolsin. 2-tarif. Uzluksiz tasodifiy miqdorning matematik kutilmasi deb, ushbu  EMBED Equation.DSMT4  (2) integralga (agar bu integral absolyut yaqinlashuvchi bolsa) aytiladi. 4-misol.  EMBED Equation.DSMT4  oraliqda tekis taqsimlangan  tasodifiy miqdorning matematik kutilmasi quyidagicha topiladi:  EMBED Equation.DSMT4 . 2.Matematik kutilmaning ehtimollik manosi  tasodifiy miqdor ustida n ta bog liqsiz tajriba o tkazilgan bo lsin. "ajriba natijalari ushbu jadvalda keltirilgan:  EMBED Equation.DSMT4  Yuqori satrda  miqdorning kuzatilgan qiymatlari, pastki satrda esa mos qiymatlarning chastotalari ko rsatilgan, ya ni n1 son n1 ta tajribada  miqdor E1 ga teng qiymat qabul qilganligini bildiradi va hakozo.  EMBED Equation.DSMT4  orqali kuzatilgan barcha qiymatlarning o rta arifmetigini belgilaylik, u holda,  EMBED Equation.DSMT4 , yoki  EMBED Equation.DSMT4 . Bu yerda  EMBED Equation.DSMT4   mos ravishda x1, x2, & , xk qiymatlarning nisbiy chastotalari. "ajribalar soni yetarlicha katta bo lganda  EMBED Equation.DSMT4  bo ladi. Shuning uchun  EMBED Equation.DSMT4 , ya ni  tasodifiy miqdorning matematik kutilmasi uning kuzatiladigan qiymatlari o rta arifmetigiga taqriban teng. 3.Matematik kutilmaning xossalari. Matematik kutilma quyidagi Eossalarga ega: 1-Eossa. O zgarmas sonning matematik kutilmasi shu sonning o ziga teng. Isboti. c o zgarmas sonni faqat bitta c qiymatni 1 ehtimollik bilan qabul qiluvchi tasodifiy miqdor deb qarash mumkin. Shuning uchun  EMBED Equation.DSMT4  2-Eossa.  EMBED Equation.DSMT4  tengsizlik o rinli. Bu Eossaning isboti matematik kutilmaning ta rifidan bevosita kelib chiqadi. 3-Eossa.  EMBED Equation.DSMT4  va  EMBED Equation.DSMT4  larning iEtiyoriy ikkitasi mavjud bo lsa, u holda ushbu  EMBED Equation.DSMT4  tenglik o rinli bo ladi. Isboti. Isbotni diskret hol uchun keltiramiz. Faraz qilaylik,  tasodifiy miqdor x1, x2, , xk, qiymatlarni mos ravishda p1, p2, , pk, , ehtimolliklar bilan,  EMBED Equation.DSMT4  tasodifiy miqdor esa y1, y2, yk, ..., qiymatlarni mos ravishda q1, q2, qk, , ehtimolliklar bilan qabul qilsin, u holda  EMBED Equation.DSMT4  yigindining qabul qiladigan qiymatlari  EMBED Equation.DSMT4  korinishdagi sonlardan iborat.  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mHCsHC%jhnh(-CJ UaJ mHCsHC)jبhnh(-CJ EHUaJ mHCsHCEk va  EMBED Equation.DSMT4  ning  EMBED Equation.DSMT4  qiymatlarni qabul qilish ehtimolligini belgilaymiz. U holda to la ehtimollik formulasiga asosan  EMBED Equation.DSMT4  1-natija.  EMBED Equation.DSMT4  tasodifiy miqdorlar yig indisining matematik kutilmasi shu tasodifiy miqdorlar matematik kutilmalarining yig indisiga teng, ya ni  EMBED Equation.DSMT4 . 4-Eossa. O zgarmas sonni matematik kutilma ishorasidan tashqariga chiqarib yozish mumkin:  EMBED Equation.DSMT4 . Isboti. Isbotni diskret va uzluksiz tasodifiy miqdorlar uchun alohida-alohida keltiramiz. 1-ta rifdan va (1) dan foydalanib, diskret tasodifiy miqdor uchun ushbu  EMBED Equation.DSMT4  natijani hosil qilamiz. (2) formulaga asosan uzluksiz tasodifiy miqdorlar uchun ushbu  EMBED Equation.DSMT4  5-Eossa.  EMBED Equation.DSMT4  va  EMBED Equation.DSMT4 tasodifiy miqdorlar o zaro bog liq bo lmasin. Agar  EMBED Equation.DSMT4  va  EMBED Equation.DSMT4  mavjud bo lsa, u holda  EMBED Equation.DSMT4  mavjud bo ladi va  EMBED Equation.DSMT4 . Isboti. Faraz qilaylik,  EMBED Equation.DSMT4  tasodifiy miqdor  EMBED Equation.DSMT4  qiymatlarni mos ravishda  EMBED Equation.DSMT4  ehtimolliklar bilan,  EMBED Equation.DSMT4  tasodifiy miqdor  EMBED Equation.DSMT4  qiymatlarni mos ravishda  EMBED Equation.DSMT4  ehtimolliklar bilan qabul qilsin.  EMBED Equation.DSMT4  va  EMBED Equation.DSMT4  tasodifiy miqdorlar o zaro bog liqsizligidan  EMBED Equation.DSMT4  tasodifiy miqdor  EMBED Equation.DSMT4  ko rinishdagi qiymatlarni  EMBED Equation.DSMT4  ehtimollik bilan qabul qiladi, natijada  EMBED Equation.DSMT4  teoremaning teskarisi doim ham to g ri emas, ya ni  EMBED Equation.DSMT4  ekanligidan  EMBED Equation.DSMT4  va  EMBED Equation.DSMT4  ning o zaro bog liq bo lmasligi kelib chiqmaydi. 6-Eossa. Agar  EMBED Equation.DSMT4  bo lsa,  EMBED Equation.DSMT4 . 7-Eossa. Agar nomanfiy  EMBED Equation.DSMT4  tasodifiy miqdor uchun  EMBED Equation.DSMT4  bo lsa, u holda  EMBED Equation.DSMT4  tenglik 1 ehtimollik bilan bajariladi.  SoongT.T.Fundamental so fprobability and statistics for engineers.p.259-277.     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