Ehtimollar nazariyasi va matematik statistika

Transcript

1 O ZBEKISTON RESPUBLIKASI OLIY VA O RTA MAXSUS TA LIM VAZIRLIGI Toshket Molya Isttut E. Mamurov T. Adrov Ehtmollar azaryas va matematk statstka o quv qo llama Toshket-005

2 E. Mamurov, T. Adrov. Ehtmollar azaryas va matematk statstka. O quv qo llama. Toshket Molya sttut, b. Ushbu o quv qo llama O zbeksto Respublkas Oly va o rta masus ta lm vazrlg tomoda tasdqlaga «Bzes va boshqaruv» ta lm sohasdag barcha bakalavrat yo alshlar uchu ta lm stadartlar talablarga muvofq ehtmollar azaryas va matematk statstka kurs bo ycha yozlga. Uda asosy e tbor talabalarg ushbu fa to lqroq o zlashtrshlar uchu yordam bershga qaratlga. O quv qo llama Toshket Molya sttut qoshdag Oly o quv yurtlararo lmy-usluby Kegash majlsda muhokama qlga va ashrga tavsya etlga Taqrzchlar: TAYI «Oly matematka» kafedrasg mudr, professor M.U.G ofurov Fzka-matematka falar omzod, dotset Hamdamov I. Toshket Molya sttut, 005

3 -.Faga krsh. Dastlabk tushuchalar. Ehtmollk. Ehtmolg turl ta rflar. Ehtmollar azaryas va matematk statstka fag qtsody jarayolar o rgashdag ahamyat. Ehtmollar azaryas fag dastlabk tushuchalar shakllaga davr XVI- XVII asrlar bo lb, Kardao, Gyuyges, Paskal, Ferma va Yakov Berull kab olmlarg omlar bla bog lqdr. Ehtmollar azaryasg paydo bo lshga qmor o ylarg matematk modellar va azaryas yaratsh yo ldag zlashlar turtk bo ld. Ehtmollar azaryasg keyg yutuqlar Muavr, Laplas, Puasso kab olmlarg omlar bla bog lq. Ehtmollar azaryasg yag samaral rvoj Chebshev, Markov, Lyapuov kab rus olmlarg lmy zlashlar bla bog lq bo ld. Fag mustaql fa bo lb uyg ulashshda va keyg rvojda Bershtey, Romaovsky, Kolmogorov, Xch, Gedeko, Smrov va boshqalarg zmatlar katta bo ld. Ehtmollar azaryas va matematk statstka fag rvojda S. X. Srojddov, T. A. Sarmsoqov kab zabardast o zbek olmlarg ham muosb hssalar bor. Hozrg kuda bu kk olmg shogrdlar tomoda O zbekstoda ham ehtmollar azaryas va matematk statstka fa bo ycha ham azary, ham amaly tadqqotlar davom ettrlmoqda. Ehtmollar azaryasg dastlabk tushuchalar tajrba, hodsa, elemetar hodsa, ehtmollk, sby chastota kab tushuchalar bo lb, ular bayo qlshga o tamz. Tajrba hodsa ro yobga keltruvch tay shartlar to plam S g bajarlshda boratdr. Hodsa esa tajrba atjas sfatda qaraymz. Masala, tajrba taga muayya sharotda tashlashda borat bo ls. Taga va u tashlash S shartlar to plam tashkl etsa, tajrba atjalar tagag gerb yok raqam tomolar bla tushsh hodsalardr. Bz kuzatga hodsalar uch turga ajratsh mumk: muqarrar, ro y bermaydga va tasodfy hodsalar. 3

4 Muqarrar hodsa deb, tajrba atjasda albatta ro y beradga hodsaga aytlad va bz buday odsa Ω (omega) harf bla belglaymz. Mumk bo lmaga hodsa deb, tajrba atjasda mutlaqo ro y bermaydga hodsaga aytlad va bu hodsa belgs bla belglaymz. Tasodfy hodsa deb, tajrba atjasda ro y bersh ham, ro y bermaslg ham mumk bo lga hodsaga aytlad. Tasodfy hodsalar A, V, S, katta lot harflar bla belglaymz. Msol: O y kubg br marta tashlaad. Bu holda Ω { tushga ochko 6 da katta emas} muqarrar hodsa; {tushga ochko 0 ga teg} mumk bo lmaga hodsa; A {tushga ochko juft so} tasodfy hodsalardr. Albatta bu tajrbaga mos bo lga boshqa ko plab hodsalar ta rflashmz mumk. Elemetar hodsa deb, tajrbag har qaday atjasga aytlad, hamda ω harf bla belglaad. Tajrba atjasda ro y bersh mumk bo lga barcha elemetar hodsalar to plam elemetar hodsalar fazos deylad. Elemetar hodsalar fazos Ω kab belglaad. Msollar:. Tajrba taga kk marta tashlashda borat bo ls. Buda elemetar hodsalar quydagcha bo lad: ω (gg), ω (gr), ω 3 (rg), ω 4 (rr). Elemetar hodsalar fazos Ω to rt elemetda borat:. Agar taga uch marta tashlasa, u holda ω (ggg), ω (ggr), ω 3 (grr), ω 4 (rrr) ω 5 (rrg), ω 6 (rgg), ω 7 (rgr), ω 8 (grg). 3. Tajrba o y kubg kk marta tashlashda borat bo ls. Bu holda ω j (,j) bo lb, -brch tashlashda tushga ochko bldrad. 4

5 Ω{ω j },,6, j,6 va elemetar hodsalar so 36 ga teg. 4. Tajrba uqta [a;b] kesmaga tashlashda borat bo ls. Buda Ω[a;b] to plamda boratdr. Bz yuqorda hodsalar uch turga bo lga edk. O z avbatda tasodfy hodsalar ham quydag turlarga ajratamz. Brgalkda bo lmaga hodsalar deb, btta tajrbada brg ro y bersh qolgalarg ro y bersh yo qqa chqaradga hodsalarga aytlad. Agar tajrba atjasda br echta hodsalarda bttas va faqat bttasg ro y bersh muqarrar hodsa bo lsa, u holda bu hodsalar yagoa mumk bo lga hodsalar deylad. Agar br echta hodsalarda hech br boshqalarga sbata ro y bersh mumkroq deyshga asos bo lmasa, ular teg mkoyatl hodsalar deylad. Bz qzqtrayotga hodsag ro y bershga olb keladga elemetlar hodsalar bu hodsag ro y bershga qulaylk tug druvch deb ataymz. Ehtmol tushuchas asosy tushuchalarda br bo lb, ug br echta ta rf mavjud. Umumy qlb aytgada, ehtmol - tasodfy hodsag ro y bersh mkoyat mqdory jhatda arakterlovch sodr. Quyda ehtmolg klassk ta rf keltramz. Ta rf. A hodsag ehtmol deb, bu hodsa ro y bershga qulaylk tug druvch elemetar atjalar sog tajrbag yagoa mumk bo lga va teg mkoyatl elemetar atjalar jam soga sbatga aytlad hamda R(A) m formula bla aqlaad. Ehtmolg klassk ta rfda bevosta quydag ossalar kelb chqad. -ossa. Muqarrar hodsag ehtmol ga teg. Haqqata ham, bu holda m va demak. m P(Ω) 5

6 -ossa. Mumk bo lmaga hodsag ehtmol olga teg, bu holda m 0 m0 va P( ) 0 3-ossa. Tasodfy hodsag ehtmol ol va br orasda yotuvch sodr. 0<R(A)< Shuday qlb, stalga hodsag ehtmol quydag muosabat qaotlatrad. 0 R(A) Ehtmolg yuqorda keltrlga klassk ta rf cheklaga bo lb, hamma masalalarga ham qo llalavermayd. Jumlada, elemetar atjalar so cheksz yok elemetar atjalar teg mkoyatl bo lmaga tajrbalarda klassk ta rf qo llab bo lmayd. Shu sababl klassk ta rf bla br qatorda hodsag ehtmol sfatda sby chastota yok uga yaqroq so olb, statstk ta rfda ham foydalalad. Statstk ta rf sby chastotag turg ulk hossasga asoslaad. Bu ossa shuda boratk, ko p sodag tajrbalar seryas uchu A hodsag ta tajrbada v ro y bershlar sby chastotas deb ataluvch W ( A) sbat deyarl o zgarmas mqdor bo lb qolaverad. Bu erda ν - A hodsag ta tajrbada ro y bershlar so. Nsby chastotag turg ulk ossas brch bor demografk harakterdag hodsalarda ochlga. Bzg eramzda 000 yllar buru qadmy Xtoyda o g l bolalar tug lshlar sog jam tug lga bolalar soga sbat deyarl / ga teg ekalg hsoblaga. Bu sog barcha davrlar uchu o zgarmay qolsh statstk ma lumotlar tasdqlayd. Nsby chastotag turg ulk ossasga yaa br msol sfatda taga tashlash tajrbas ko ramz. Taga tashlash tajrbalar ko p marta o tkazlb, ularda «gerb» tomo tushsh so saalga. Br echta tajrbalarg atjalar quydagcha bo lga 6

7 Taga tashlashlar so Gerb tomo tushshlar so Nsby chastota Bu tajrbalarda W(A) sby chastota o zgarmas r0.5 so atrofda tebraayapt, shu 0,5 so taga tashlashda «gerb» tomo tushsh hodsasg ehtmol sfatda olsh tabydr. Umuma, agar tajrbalar so etarlcha ko p bo lb, shu tajrbalarda qaralayotga A hodsag ro y bersh sby chastotas W(A) bror o zgarmas r [0;] so atrofda turg u ravshda tebrasa, shu R so A hodsag ro y bersh ehtmol deb qabul qlamz. Buday usulda aqlaga ehtmol hodsag statstk ehtmol deylad. Ba za geometrk mulohazalarga asoslaga masalalarda ehtmolg geometrk ta rf qo llalad. Ushbu ta rf bayo qlshga o tamz. Bror G soha berlga bo lb, bu soha g soha o z chga ols. G sohaga tavakkalga tashlaga uqtag g sohaga am tushsh ehtmol topsh talab etls. Bu erda Ω elemetar hodsalar fazos G g barcha uqtalarda borat va chekszdr. Shug uchu, bu holda klassk ta rfda foydalaa olmaymz. Tashlaga uqta G ga tushsh ehtmol shu g qsmg o lchovga (uzulgga, yuzga, hajmga) proportsoal bo lb, g g shaklga va g G sohag qaerda joylashgalgga bog lq bo lmas. Bu shartlarda qaralayotga hodsag ehtmol G g ulchov R G g ulchov formula yordamda aqlaad. Bu formula yordamda aqlaga R ehtmollk ehtmolg barcha ossalar qaoatlatrad. Msol. Radus R bo lga dora chga tavakkalga uqta tashlaga. Tashlaga uqta doraga chk chzlga: 7

8 a) kvadrat chga: b) mutazam uchburchak chga tushsh ehtmol topg. Nuqtag yass fguraga tushsh ehtmol bu fgurag yuzga proportsoal bo lb, ug joylashshga esa bog lq emas deb faraz qlad. Echlsh. a) geometrk ehtmollar ta rfga ko ra zlaayotga ehtmollk Kvadratg yuz R P Dorag yuz π R π b) Bu olda, mutazam uchburchak yuz 3 3R 4 ekalg hsobga olsak: Uchburchak yuz 3 3 R 3 3 P dora yuz 4π R 4π Ehtmollar azaryas fa - matematk fa bo lb, ug predmet br l shart sharotlarda ko p marta takrorlauvch tasodfy hodsalarg ehtmoly qouyatlar o rgashda borat. Tasodfy hodsalar bo ysuadga qouyatlar blsh, shu hodsalarg qaday kechsh avvalda ko ra blsh mko berad. Ehtmollar azaryas fag metodlar hozrg davrda amalyotg turl sohalarda, jumlada, qtsodyot sohasda ham keg samaral qo llalmoqda. Tasodfylk bla bog lq bo lga qtsody jarayolar tadqq etshda, bu jarayolarg kechsh bashorat qlshda, hamda ma qul qtsody echmlar qabul qlshda ehtmollar azaryas va matematk statstka fag ahamyat kattadr. Ehtmollar azaryas va matematk statstka fa usullar makro va mkroqtsodyot rejalashtrsh va tashkl etshda, turl teologk jarayolar tahll etshda, mahsulot sfat azorat qlshda, ommavy zmat ko rsatsh azaryasda va boshqa ko plab sohalarda o z tadbqlar topmoqda. 8

9 O z-o z tekshrsh uchu savollar.. Hodsalarg turlar aytg va ularga dor msollar keltrg.. Elemetar atja ta rf berg. 3. Tasodfy hodsalarg turlar aytg. 4. Ehtmollkg klassk va statstk ta rflar keltrg. Ularg farq mada? 5. Nsby chastotag turg ulk ossas mada borat? 6. Geometrk ehtmol ta rf aytg. Tayach boralar Tasodfy hodsa, muqarrar hodsa, mumk bo lmaga hodsa, brgalkda bo lmaga hodsalar, yagoa mumk bo lmaga hodsalar, teg mkoyatl hodsalar, ehtmolg klassk ta rf, sby chastota. Mustaql shlash uchu msollar.. Taga kk marta tashlagada aqall br marta gerbl tomo bla tushsh ehtmol topg.. Ikkta o y soqqas tashlaad. Chqqa ochkolar yg dsg 7 ga teg bo lsh ehtmol topg. 3. Yashkda 5 ta detal bo lb, ularda 0 tas bo yalga. Yashkda tavakkalga 3 ta detal old. Olga detallarg bo yalga bo lsh ehtmol topg. 4. Uch marta taga tashlaga. Ikk marta «gerb» tomo bla tushsh ehtmol topg. Adabyotlar. [] (4-30) [] (-33) [3] (8-5) [4] (5-7) [5] (9-35) [7] (5-8) [] (63-74) 9

10 -. Hodsalar ustda amallar. Shartl ehtmollk. Ehtmollar qo shsh va ko paytrsh teoremalar. Ehtmollar azaryasda hodsalar ustda qo shsh va ko paytrsh amallar bla sh ko rshga to g r kelad, quyda shu amallar ta rflaymz. Ta rf. Ikkta A va V hodsalarg yg ds (brlashmas) deb, A yok V g, yok kkalasg ham ro y bershda borat SA+V hodsaga aytlad. Qsqacha qlb aytgada, A+V yg d A va V hodsalarg kamda bttasg ro y bersh fodalayd. Xudd yuqordag ta rf kab A + A A yg d degada, A, A,...A hodsalarg kamda bttasg ro y bersh tushulad. Masala. A{I mergag shoga tekkzsh}, V{II mergag shoga tekkzsh} bo ls. U holda, A+V hodsa, yok I mergag, yok II mergag, yok kkalasg ham shoga tekkzshda borat hodsa bldrad. Agar A va V hodsalar brgalkda bo lmasa, u holda A+V yg d shu hodsalarda qayss bo lsa ham, brg ro y bershda boratdr. Ta rf. A va V hodsalarg ko paytmas (kesshmas) deb, shu hodsalarg brgalkda ro y bershda borat SA. V hodsaga aytlad. Ushbu ta rf kktada ortq br echta hodsalar ko paytmas uchu ham yuqordagdek umumlashtrlad. Yuqorda keltrlga msolda AV hodsa kkala mergag ham shoga tekkzsh bldrad. Hodsalar ustda bajarladga qo shsh va ko paytrsh amallar quydag shaklda geometrk zohlash mumk. 0

11 А+В,(А В) А В А В (А В) В A hodsaga qarama-qarsh hodsa deb, A hodsag ro y bermaslgda borat hodsaga aytlad va Ā kab belglaad. Qarama-qarsh A va Ā hodsalar uchu A А + A Ω А muosabat o rl ekalg tushush qy emas. Elemetar hodsalar tlda Ā hodsa A ga krmaga barcha elemetar hodsalar to plamda borat bo lad, qarama-qarsh hodsalar geometrk tasvrlash mumk. A A Msol. A hodsa kubk br marta tashlagada «6» ochko tushsh bldrs. U holda Ā hodsa «6» ochko tushmaslg bldrad Ba za A hodsag ehtmol bror V hodsa (R(V)>0 deb faraz qlad) ro y bergada so g hsoblashga to g r kelad. Ta rf. A hodsag V hodsa ro y bergalg shartda hsoblaga ehtmolga shartl ehtmol deylad va R V (A) yok R(A/V) kab belglaad. Xudd shuga o shash R A (V) shartl ehtmol ta rflaad. Msol. Ikkta kubk tashlaayotga bo ls. A{tushga ochkolar yg ds 8 ga teg bo lsh} va V{tushga ochkolar juft so bo lsh} hodsalar uchu

12 R(A)5/36, R(V)8/36 bo lsh ravsha. Ed, masala, R V (A) shartl ehtmol topsak: R V (A)5/8 Shartl ehtmol yordamda hodsalarg bog lqszlg tushuchas krtamz. Ta rf. Ikkta A va V hodsalar uchu R V (A)R(A) va R A (V)R(V) bo lsa, A va V hodsalar bog lqmas (erkl) hodsalar deylad. Aks holda, hodsalar bog lq deylad. Soddaroq qlb aytgada, kkta hodsada tyory brg ro y bersh ehtmol kkchsg ro y bersh yok ro y bermaslgga bog lq bo lmasa, bu hodsalar bog lqmas deylad. Msol. Qutda 6 ta oq va 9 ta qora shar bor. Tavakkalga btta shar olad. Olga sharg oq bo lsh (A hodsa) ehtmol klassk ta rfga ko ra R(A)6/5ga teg. Olga shar qutga solad va sash takrorlaad. Ikkch olshda oq shar chqsh (V hodsa) ehtmol, avvalgdek yaa 6/5ga teg va brch sash atjasga bog lq emas. Shuday qlb, bu holda V hodsa A hodsaga bog lq emas. Agar olga brch shar qutga qaytarb solmasda kkch shar olsa, V hodsa A hodsaga bog lq bo lad, chuk R A (V)5/4 va R A (V)6/4. Ed hodsalar ehtmollar qo shsh va ko paytrsh teoremalar bayo qlshga o tamz. -Teorema. Brgalkda bo lmaga kkta hodsada qayss bo lsa ham brg ro y bersh ehtmol shu hodsalar ehtmollarg yg dsga teg: R(A+V)R(A)+R(V) Isbot. -sashg mumk bo lga elemetar atjalar jam so bo ls; m -A hodsaga qulaylk tug dradga atjalar so; m -V hodsaga qulaylk tug dradga atjalar so.

13 Yo - A hodsa, yok V hodsa ro y bershga qulaylk tug druvch atjalar so m + m ga teg. Buda esa muosabat hosl qlamz. m + m P(A+V) P( A) P( B ) m m + Natja. Xar kktas brgalkda bo lmaga br echta hodsalarda qayss bo lsa am, brg ro y bersh ehtmol shu hodsalar ehtmollarg yg dsga teg: R(A + A A )R (A ) + R (A ) +...+R(A ) Msol. Yashkda 30 ta shar bo lb, ularda 0 tas qzl, 5 tas ko k va 5 tas oq. Tavakkalga olga btta sharg ragl shar bo lsh ehtmol topg. Echsh. Ragl shar chqsh yo qzl, yok ko k shar chqsh bldrad. Qzl shar chqsh (A hodsa) ehtmol 0 P ( A) 30 Ko k shar chqsh (V hodsa) ehtmol 5 P ( A) 30 A va V hodsalar brgalkda emas (br ragl shar chqsh boshqa ragl shar chqsh yo qqa chqarad), shug uchu qo shsh teoremasga ko ra: 3 6 P ( A+ B) P( A) + P( B) A va V hodsalar bog lqmas bo lb, ularda har brg ehtmol ma lum bo lsa, A va V hodsalarg brgalkda ro y bersh ehtmol qaday topsh mumk? Bu savolga quydag ko paytrsh teoremas javob berad. -Teorema. Ikkta bog lqmas hodsag brgalkda ro y bersh ehtmol shu hodsalar ehtmollarg ko paytmasga teg: + R(A. V) R(A). R(V). 3

14 Ko paytrsh teoremas br echta hodsalarga umumlashtrsh uchu brgalkda bog lqmaslk tushuchas krtamz. Br echta hodsalarda har br va qolgalarg stalga kombatsyas bog lqmas bo lsa, u holda bu hodsalar brgalkda bog lq emas deylad. Shu ta kdlash lozmk, br echta hodsalarg juft-juft bog lq emaslgda ularg brgalkda bog lq emaslg kelb chqmayd. Shu ma oda brgalkda bog lq emaslg talab juft-juft bog lqmaslk talabda kuchlroqdr. Ed ko paytrsh teoremasda kelb chqadga atja keltramz. Natja. Brgalkda bog lq bo lmaga br echta hodsalarg brgalkda ro y bersh ehtmol shu hodsalarg ehtmollar ko paytmasga teg: R(A. A,.... A )R (A ). R (A ),.... R(A ) Eslatma. Agar A, A,.... A hodsalar brgalkda bog lqmas bo lsa, u holda ularga qarama-qarsh bo lga A, A,.... A hodsalar ham brgalkda bog lqmas bo lad. Ikkta bog lq A va V hodsalar uchu ko paytrsh teoremas quydagcha bayo qlad. 3-Teorema. Ikkta bog lq hodsag brgalkda ro y bersh ehtmol ularda brg ehtmol kkch hodsag shartl ehtmolga ko paytmasga teg: 4

15 R(A. V)R(A). R A (V) Isbot. Belglashlar krtamz: -sashg A hodsa ro y beradga yok ro y bermaydga elemetar atjalar jam so; - A hodsa ro y bershga qulaylk tug druvch atjalar so ( <). m-sashg A hodsa ro y berd dega farazda V hodsa ro y beradga elemetar atjalar so, ya bu atjalar AV hodsag ro y bershga qulaylk tug drad. A va V hodsalarg brgalkda ro y bersh ehtmol: qlamz: m P( A) va P ( B ) A m m P( A B) 5 ekalg e tborga olb quydag hosl R(A. V)R(A). R A (V) Shu ta kdlab o tamzk, AVVA bo lgalg uchu teorema VA hodsa uchu qo llab quydag teglk hosl qlamz. R(A. V)R(A). R A (V)R(V). R V (A) Natja. Br echta bog lq hodsalarg brgalkda ro y bersh ehtmol ularda brg ehtmol qolgalarg shartl ehtmollarga ko paytmasga teg, buda har br keyg hodsag ehtmol uda oldg hamma hodsalar ro y berd dega farazda hsoblaad. ( A A A 3 A A A ( A ) P A A... A ) P( A) P ( A ) P ( )... A P... Yuqorda brgalkda bo lmaga hodsalar uchu qo shsh teoremas (- teorema) keltrlga ed. Ed brgalkda bo lga hodsalar uchu qo shsh teoremas keltramz. 4-Teorema. Brgalkda bo lga kkta hodsada kamda bttasg ro y bersh ehtmol shu hodsalarg ehtmollar yg dsda ularg brgalkda ro y bersh ehtmol ayrlgaga teg: R(A+V)R(A)+R(V)-R(AV)

16 Isbot. Ta rfga ko ra A+V hodsa yo AV, yo AV yok AV hodsag ro y bershda borat, ya A+VAV+AV+AV Ed AV va AV hodsalar brgalkda emas. Shug uchu, R(A+V)R(AV)+R(AV)+R(AV) (*) AAV+AV, R(A)R(AV)+R(AV ), VAV+AV, R(A)R(AV)+R(AV) muosabatlarda R(AV)R(A)-R(AV) va R(AV)R(V)-R(AV) teglklar hosl qlamz. Bu teglklar (*) fodaga qo yb teglk hosl qlamz. R(A+V)R(A)+R(V)-R(A. V) Msol. I va II to plarda o q otshda shoga tekkzsh ehtmollar mos ravshda r 0,8 va r 0,9. Br yo la otshda to plarda kamda brg shoga tekzshlar ehtmol topg.to plarg tekkzshlar br-brga bog lq emas. Shug uchu A{ I to pg shoga tekkzsh} va V{II to pg shoga tekkzsh} hodsalar erkldr. Buda esa Izlaayotga ehtmol quydagga teg: R(AV)R(A). R(V) R(A+V)R(A)+R(V)-R(A. V) O z-o z tekshrsh uchu savollar.. Hodsalar yg ds va ko paytmas amallar ta rflag.. Qarama-qarsh hodsalar ta rf berg. 3. Bog lqmas hodsalar ta rf berg. 4. Shartl ehtmollk ta rf berg. 5. Ehtmollar qo shsh teoremalar aytg. 6. Ehtmollar ko paytrsh teoremalar keltrg. 6

17 Tayach boralar Qarama-qarsh hodsalar, bog lqmas hodsalar, bog lq hodsalar, shartl ehtmol, brgalkda bo lga hodsalar. Mustaql echsh uchu masalalar.. Guruhda 0 ta talaba bo lb, ularg 7 afar a lochlar. 4 ta talaba dekaatga chaqrtrld. Ularg barchas a loch bo lsh ehtmol topg.. Talaba programmadag 30 ta savolda 5 tas blad. Talabag mtho oluvch taklf etga uchta savol blsh ehtmol topg. 3. Brch yashkda 4 ta oq va 8 ta qora shar bor. Ikkch yashkda 0 ta oq va 6 ta qora shar bor. Har qays yashkda bttada shar olad. Ikkala sharg ham oq chqsh ehtmol topg. 4. Brch yashkda 5 ta oq va 0 ta qzl shar bor. Ikkch yashkda 0 ta oq va 5 ta qzl shar bor. Agar har br yashkda bttada shar olsa, hech bo lmagada btta sharg oq bo lsh ehtmol topg. 5. Mergag uchta o q uzshda kamda btta o q shoga tekkzsh ehtmol 0,875 ga teg. Ug btta o q uzshda shoga tekkzsh ehtmol topg. 6. To rtta o q uzshda kamda btta o q shoga tekkzsh ehtmol 0,3 ga teg. Mergalar avbat bla o q uzadlar, lek har br kktada o q uzad. Brch bo lb o q tekkzga merga mukofot olad. Mergalarg mukofot olshlar ehtmol topg. 7

18 [] (3-47) [] (33-5) [3] (5-5) [4] (7-4) [5] (37-44) [7] (4-6) [8] (70-80) Adabyotlar 8

19 3-.To la ehtmol va Bayes formulalar. To la ehtmol va Bayes formulalar keltrshda avval, bu formulalarda foydalaladga ba z tushuchalar keltramz. Ta rf: Hodsalarg to la guruh deb, sashg yagoa mumk bo lga hodsalar to plamga aytlad. Bu ta rfga boa, agar A, A,.A hodsalar hodsalarg to la guruh tashkl etsa, u holda bu hodsalar uchu A +A +. +A Ω, A A j, ( j) muosabatlar o rl bo lad. Msol. Taga br marta tashlaad. (Taga qrras bla tushmayd deb faraz qlad) bu sovda A{taga «gerb» tomo bla tushad} V{taga «raqam» tomo bla tushad} hodsalar to la guruh tashkl etad. Hodsalarg to la guruh tashkl etuvch A, A,.A hodsalar uchu muhm bo lga quydag teorema keltramz Teorema: To la guruh tashkl etuvch A, A,.A hodsalarg ehtmollar yg ds brga teg, ya R(A )+R(A )+..+R(A ) Isbot. To la guruh tashkl etuvch hodsalarda brg ro y bersh muqarrar. Muqarrar hodsag ehtmol esa brga teg bo lga uchu R(A +A +..+A p ) To la guruhg kkta hodsas brgalkda emaslg sababl, qo shsh teoremas qo llash mumk. Ta rf: Qarama-qarsh hodsalar deb, to la guruh tashkl etuvch kkta hodsaga aytlad. Yuqordag teoremaga asosa qarama-qarsh hodsalar ehtmollarg yg ds brga teg. 9

20 R(A)+R(Ā) Shu alohda eslatb o tamzk, A hodsag ehtmol topshga dor ko pga masalalarda ko pcha qarama-qarsh A hodsasg ehtmol hsoblash acha oso bo lad, key esa zlaayotga ehtmol quydag formula orqal topsh qulay bo lad. R(A)-R(Ā) Msol. Yashkda 0 ta detal bo lb, ularda tas yaroql. Tavakkalga olga 5 ta detal orasda kamda ta yaroql detal bo lsh ehtmol topg. Echsh: A{olga detallar chda kamda btta yaroql} A{olga detallar orasda btta ham yaroql detal yo q} hodsalar qarama-qarsh hodsalardr. Buda R(A) ehtmol topsh osoroq. 5 т С8 R(A) 5 п С Buda esa zlaayotga ehtmol topsak: 0 5 С8 R(A)-R(A)- 5 С Ed «to la ehtmol» formulas keltramz. Faraz qlaylk, A hodsa to la guruh tashkl etuvch hodsalarda bttasg ro y bergalk shart ostda ro y bers. U holda, A hodsag ehtmol quydagcha toplad. R(A)R(V ) R(A/V )+R(V ) R(A/V )+ +R(V p ) R(A/V p ). Bu formula «to la ehtmol» formulas deb atalad. Shu formula keltrb chqaraylk. A hodsas ro y bersh uchu brgalkda bo lmaga. AV, AV,..AV. hodsalarda bror bttas ro y bersh zarur va etarl. Boshqacha aytgada AAV + AV +. +AV. Buda AV (,) hodsalar brgalkda bo lmagalg uchu 0 0

21 R(A)R(AV + AV +. +AV )R(AV )+R(AV )+... +R(AV ) R(V )R(A/V )+R(V )R(A/V )+... +R(V )R(A/V ) Odatda, bu formula shartlarda A hodsag V, V,...V hodsalarg qays br bla ro y bersh oldda oma lum bo lgalg uchu, V, V,...V hodsalar g p o t e z a l a r deb ham atalad. Faraz qlaylk, sash o tkazlga bo lb, ug atjasda A hodsa ro y berga bo ls. Gpotezalarg ehtmollar qaday o zgargalg (A hodsa ro y bergalg sababl) aqlash masalas ko raylk. Boshqacha qlb aytgada, R(V /A), R(V /A),.. R(V /A) shartl ehtmollar zlaymz. Ko rsatlga ehtmollarda, masala, R(V /A) qaraylk. Ko paytrsh teoremasga ko ra R(AV )R(A)R(V /A) R(V ) R(A/V ) Buda esa, R(V /A) Р ( В ) Р ( А / В ) Р ( А) Bu muosabatda marajdag R(A) ehtmol, ug to la ehtmollk formulasdag fodas bla almashtrb, quydag hosl qlamz: R(V /A) Р ( В ) Р ( А / В ) п Р ( В ) P( A/ B Qolga gpotezalarg ham shartl ehtmollar ham udd shuga o shash keltrb chqarlad. Shuday qlb, tyory V k (k,) gpoteza uchu R(V k /A) P( B ) P( A/ B ) k P( B ) P( A/ B ) Bu formulalar Bayes formulalar deb atalad. Bayes formulalar tajrba atjasda A hodsas ro y bergalg ma lum bo lgada so g, V k (k,) gpotezalar ehtmollar qayta baholashga mko berad. To la ehtmol formulas va Bayes formulalarg qo llashga dor quydag msol ko ramz. k )

22 Msol. Brch qutda ta oq, 6 ta qora, kkch qutda esa, 4 ta oq, ta qora shar bor. Brch qutda tavakkalga ta shar olb, kkch qutga solad, shuda key kkch qutda tavakkalga btta shar olad. A) olga sharg oq bo lsh ehtmol topg. V) kkch qutda olga shar oq bo lb chqd; brch qutda olb, kkch qutga solga la sharg oq bo lsh ehtmol maga teg. Echsh: A) Quydag belglashlar krtamz: A {Ikkch qutda olga shar oq}. V {Brch qutda kkch qutga ta oq shar solga}. V {Brch qutda kkch qutga ta oq, ta qora shar solga}. V 3 {brch qutda kkch qutga ta qora shar solga}. V, V, V 3 - hodsalar hodsalarg to la guruh tashkl etad. U holda, to la ehtmol formulasga ko ra, A hodsag ehtmol quydagga teg: R(A)R(V ) R(A/V ) +R(V ) R(A/V )+R(V 3 ) R(A/V 3 ) Buda, masalag shartda R(V ) C C 8 8 R(A/V ) 4 3, CC R(V ) C R(A/V ) 8 5 U holda, C R(V 3 ) C R(V ) R(A/V 3 ) b) R (V /A) ehtmol esa Bayes formulasda foydalab, topamz. 9 6 R (V /A) Р В ) Р ( А / В ) ( Р ( А)

23 O z - o z tekshrsh uchu savollar.. Hodsalar to la guruh ta rf berg.. To la ehtmollk formulasda qaday shartlar talab qlad? 3. Bayes formulas va to la ehtmollk formulalar orasdag umumy, hamda farq qluvch jhatlar aytg. Tayach boralar. Hodsalarg to la gruppas, to la ehtmol formulas, gpotezalar, Bayes formulas. Mustaql echsh uchu masalalar.. Yashkda zavodda tayyorlaga ta detal, -zavodda tayyorlaga 0 ta detal va 3-zavodda tayyorlaga 8 ta detal bor. -zavodda tayyorlaga detalg a lo sfatl bo lsh ehtmol 0,9 ga teg. -zavodda va 3-zavodda mos ravshda 0,6 va 0,9 ga teg. Tavakkalga olga detalg a lo sfatl bo lsh ehtmol topg.. Brch dshda 0 ta shar bo lb, ularg 8 tas oq, kkch dshda 0 ta shar bo lb, ularg 4 tas oq. Har br dshda tavakkalgabttada shar olb, key bu kk sharda yaa btta shar tavakkalga old oq shar olgalk ehtmol topg. 3. Ikkta yashkda radolampalar bor. Brch yashkda ta lampa bo lb, tas yaroqsz, kkch yashkda 0 ta lampa bo lb, ularg bttas yaroqsz. Brch yashkda btta lampa olb, kkch yashkka solad. Ikkch yashkda tavakkalga olga lampag yaroqsz bo lsh ehtmol topg. Adabyotlar. [] (48-55) [] (5-60) [3] (7-30) [4] (-6) [5] (44-50) [7] (0-3) [] (80-83) 3

24 4-.Erkl sovlar ketma-ketlg. Berull formulas. Eg ehtmoll so. Takrorlaadga sovlarda har brg u yok bu atjasg ehtmollg boshqa sovlarda qaday atjalar bo lgalgga bog lq bo lmasa, ular erkl sovlar ketma-ketlg hosl qlad deylad. Har l erkl sashlarda A hodsa yo har l ehtmolga, yok br l ehtmolga ega bo lsh mumk. Bz buda key A hodsa br l ehtmolga ega bo lga erkl sashlar tekshramz. Faraz qlaylk, ta o zaro erkl sash o tkazlayotga bo lb, ularg har brda A hodsa yo ro y bersh, yok ro y bermaslg mumk bo ls. A hodsag ehtmol har br sashda br hl, chuoch r ga teg deb hsoblaymz, ro y bermaslk ehtmol esa q-p ga teg. Sovlarg buday eg sodda ketmaketlgga Berull semas deylad. Masala, o y soqqas tashlashda borat tajrba o tkazlmoqda. Har br tashlashda tay ochko tushsh, boshqalarda qaday ochko chqqalgga bog lqmaslg ravsha, bobar bz bu erda erkl sovlar ketma-ketlgga egamz. ta sashda A hodsag rosa k marta ro y bersh, va demak, -k marta ro y bermaslk ehtmol soblash ko rb chqaylk. ta sashda A hodsag rosa k marta ro y bersh va -k marta ro y bermaslgda borat bo lga btta murakkab hodsag ehtmol erkl hodsalar ehtmol ko paytrsh teoremasga ko ra p k. q -k ga teg. Buday murakkab hodsalar ta elemetda k tada echta gruppalash tuzsh mumk bo lsa, shucha, ya C k ta bo lad. Izlaayotga ehtmollk P (k) bla belglaymz. U holda: Р ( k) C k р Hosl qlga formula Berull formulas deylad. Msol. Har br detalg stadart bo lsh ehtmol r0,8 bo lsa, tavakkalga olga 5 ta detalda rosa tasg stadart bo lsh ehtmol topg. к q k 4

25 Echsh. Izlaayotga ehtmol 5, m, p0,8, va q0, da Berull formulasda topamz Р () 0,8 0, 5! 0,005 3!! 3 5 C 5 0,05 Berull formulasg tatbqga dor yaa btta msol keltramz. Taga 0 marta tashlaad. Gerb tomog aq 3 marta tushsh ehtmol qachaga teg? Echsh. Bu hodsag har br tajrbadag ehtmol ga teg. Buda, Р 3 3 0! ( 3) C0 0 3!7! 0 A hodsag o tkazlayotga ta erkl takrory sov davomda kamda k marta ro y bersh ehtmol Р ( к) + Р ( k + ) +... P ( ) + ko p bla k marta ro y bersh ehtmol esa P formulalar bla hsoblaad. ( 0 ) + P ( ) +... P ( k ) + Agar ta erkl sovda hodsag k 0 marta ro y bersh ehtmol sovg boshqa mumk bo lga atjalar ehtmollarda kchk bo lmasa, u holda k 0 so eg ehtmoll so deb atalad va quydag qo sh tegszlk bla aqlaad: p q к p + 0 Eg ehtmoll so aqlash uchu hamma ehtmollar hsoblab chqmasda sovlar so, har br sovda A hodsag ro y bersh ehtmol blsh kfoya eka. Haqqata ham, eg ehtmoll soga mos keluvch ehtmol P (k o ) bla belglasak, yuqordag formulada P k0 ko k 0 ( k ) C p q Eg ehtmoll so ta rfda P o o 7 P k!( k )! ( k ) P ( k ) o o o k p o q k o 5 8 P ( k ) P ( k + ) o o 5

26 Bu tegszlklarga mos ravshda P (k o ), P (k o -), P (k o +) larg qymatlar qo yb quydaglarga ega bo lamz.! k!( k o o P )! k o q k o ( k o ko k! P q )!( k 0 o +, +)!! k!( k o o P )! k o q k o ( k o k o! P q )! ( k k 0 o + p + )! Bu tegszlklar k 0 ga sbata echamz va quydaglarga ega bo lamz: к p + p ; к p q o o Ohrg kk tegszlk brlashtrb, eg ehtmoll so aqlovch qo sh tegszlkka ega bo lamz: p q кo < p + Bu tegszlk aqlovch tervalg uzulg p + p ( p q) p + q ekalg va hodsa ta sov atjasda butu so marta ro y bersh hsobga olsak, eg ehtmoll so k 0 quydag shartlar qaotlatrad: a) agar p-q so kasr bo lsa, u holda btta eg ehtmoll k 0 so mavjud bo lad. b) agar p-q butu so bo lsa, u holda kkta k 0 va k 0 + eg ehtmoll solar mavjud bo lad; v) agar p butu so bo lsa, u holda eg ehtmoll so k 0 p bo lad. Msol. Taga 6 marta tashlaad. Gerbl tomo tushshlarg eg ehtmoll so topg. Echsh. Berlga masalag shartlarga asosa, 6, pq/. U holda gerbl tomo tushshlarg eg ehtmoll so k 0 quydag qo sh tegszlkda foydalab topamz: 6 k o 6 +,5 ko 3,5 p k o + q k o 6

27 Demak, eg ehtmoll so 3 eka. K 0 p3 ekalgda foydalasak ham bo lad. Shuday qlb, eg ehtmoll so aqlash jarayoda bz p sog Berull semasda masus ahamyatga ega ekalgga shoch hosl qlsh mkoga ega bo ldk. Bu shuda borat bo ldk, p soga eg yaq bo lga kkta butu solarda br (ba za esa kkalas ham) eg ehtmoll so bo lad. p so yuqordagda boshqa uga sbata muhmroq bo lga talqga am ega eka. Chuoch, p ma lum ma oda ta tajrbalardag muvaffaqyatlarg o rtacha so deb qarash mumk. Qsqalk uchu tajrbag marta takrorlash serya deb ataymz. Faraz qlaylk, bz bror soga teg, aytaylk, N ta serya o tkazga bo laylk. Brch seryada k muvaffaqyat, kkchsda k ta va.k. N-seryada esa k N ta muvaffaqyat olga bo ls. Bu solarg o rta arfmetg tuzamz: k + k k W ortsh bla ko rsatlga o rta arfmetk bror o zgarmas qymatga yaqlashar eka. Buga shoch hosl qlsh maqsadda org muosabat k + k k N ko rshda yozb olamz; so gra quydag hol e tborga olamz. N ta serya o tkazsh bla bz qaralayotga tajrba N marta amalga oshramz. Yuqorda yozlga N marajl kasr aa shu N ta tajrbalardag muvaffaqyatlar umumy sog barcha tajrbalar soga sbatda boshqa arsa emas. N g o ssh (demak, N ham o ssh) bla bu kasr muvaffaqyatg ehtmol bo lga R soga yaqlashad. Demak, N k + k +... N N + k foda p soga yaqlashad. Aa shu hosl qlsh talab qlga ed. 7

28 Msol. Ma lum koroag sharotda yaroqszlkka yo l qo ysh ehtmol 0,05 ga teg. 00 ta mahsulot orasdag yaroqsz mahsulotg o rtacha so maga teg? Echsh. Izlaayotga so p ga teg bo lad. Polomal sema Bu sema bomal semag (Berull semasg) umumlashmasdr. Agar Berull semasda har br tajrbada faqat ta hodsa: Ā va A qaralga bo lsa, polomal semada har br sovda k ta hodsa qaralad. Tajrba shuda borat bo ladk, ta bog lq bo lmaga sov o tkazlad va ularg har brda to la guruh hosl qladga k ta A, A,...A k hodsag faqat bttas ro y bersh mumk, buda bu hodsalarg ehtmollklar ma lum: R R(A ), R P(A ),...R k P(A k ) A hodsa rosa m marta A hodsa rosa m marta,... A k hodsa rosa m k marta ro y bersh ehtmol! m m P ( m, m,... m k ) P P... P m! m!... m! ususy holda, k bo lgada Berull formulas kelb chqad. k k m k 8

29 O z- o z tekshrsh uchu savollar.. Erkl sovlar ketma-ketlg ta rflag.. Berull formulas ma uchu zmat qlad? 3. Berull formulas keltrb chqarg. 4. Eg ehtmoll so ta rf berg va hsoblash formulas keltrg. 5. Erkl sovlar ketma-ketlgg polomal semas ma? Tayach boralar. Berull formulas, eg ehtmoll so, erkl sovlar ketma-ketlg, Berull semas va polomal sema. Mustaql shlash uchu masalalar.. Taga 0 marta tashlaad. Taga marta gerb tomo bla tushsh ehtmol topg.. Mergag shoga ursh ehtmol 0,6 ga teg. Mergag 6 ta o qda 4 tas shoga ursh ehtmol topg. 3. Taga 5 marta tashlaad. Gerb tomo bla tushshlar sog eg ehtmoll so topg. 4. Nshoga tushsh ehtmol r0,35. Nshoga qarata 0 marta o q uzlad. Nshoga tushshlar eg ehtmoll so va bu sog ehtmol topg. 5. Taga 7 marta tashlaad. Tagag marta raqam tomo bla tushsh ehtmolg topg. 6. Nshoga tegsh ehtmol r0,8. Nshoga otlga 5 ta o qda tasg shoga tegsh ehtmol topg. 9

30 [] (55-63) [] (67-70) [3] (30-35) [4] (6-36) [5] (47-50) [7] (4-6) [] (83-87) Adabyotlar 30

31 5-.Laplasg lokal va tegral lmt teoremalar. Puasso formulas. Lmt teoremalarg amaly ahamyat. Ehtmollar azaryasg tatbqlarda va k larg achaga katta qymatlarda R (k) ehtmollar hsoblash zarurat tez-tez uchrab turad. Masala, quydag masala echsh talab qls. Bror koroada mahsulotg yaroqszlkka yo l qo ysh ehtmol 0,05 ga teg. Tayyor mahsulotda 500 ta buyum tekshrld. Bular orasda rosa 5 tas yaroqsz buyum bo lsh ehtmol topg. Har br alohda buyumg tekshrlsh tajrba sfatda qarab, har brda A hodsag (buyum, yaroqsz deb toplad) yuz bersh ehtmol 0,05 ga teg bo lga 500 ta erkl tajrba o tkazlyapt deb, ayta olamz. Berull formulasga asosa Р hosl qlamz ( 5 ) C 500 (0,05 ) (0,95 ) R 500 (5) g fodas acha murakkab bo lgalg sababl bu foda bevosta hsoblash katta qychlklarga olb kelad: C Shu sababl, va k g katta qymatlar uchu R (k) ehtmollar taqrby formulalar yordamda hsoblash zaruryat tug lad. Bu formulalar Laplasg lokal lmt teoremas va tegral lmt teoremas deb ataluvch kkta teoremada keltrlad. Laplasg lokal teoremas. Agar har br tajrbada A hodsag ro y bersh ehtmol r o zgarmas bo lb, ol va brda farql bo lsa, u holda ta tajrbada A hodsag rosa k marta ro y bersh ehtmol R (k) taqrba ( qacha katta bo lsa, shucha aq). у ϕ ( ) pq pq e π 3

32 fuktsyag k p dag qymatga teg. pq ϕ ( ) e fuktsya argumetg musbat qymatlarga mos π qymatlarda tuzlga jadvallar ehtmollar azaryasga od ko plab adabyotlarda keltrlga. Shugdek, ϕ( ) fuktsya juft, ya ϕ(- ) ϕ( ) bo lgalg uchu bu jadvallarda argumetg qymatlar mafy bo lgada ham foydalalad. Shuday qlb, ta erkl sashda A hodsag rosa k marta ro y bersh ehtmol taqrba quydagga teg. P ( k) ϕ( ) pq Msol. Agar har br sashda A hodsag ro y bersh ehtmol 0, ga teg bo lsa, 400 ta sashda bu hodsag rosa 80 marta ro y bersh ehtmol topg. Echsh. 400, k80, p0,, q-0,8. P400 (80 ) ϕ ( ) ϕ ( ) 400 0, 0,8 8 k p pq , 0 jadvalda ϕ(0)0,3989 ekalg aqlaymz. U holda, zlaayotga ehtmollk 0,3989 P 400 (80 ) 0, Boshqa msol. Mergag o q uzshda shoga tekkzsh ehtmol r0.75. Merga 0 ta o q uzgada 8 ta o q shoga tekkzsh ehtmol topg. Echsh. 0, k8, p0.75, q0.5. Laplasg asmptotk formulasda foydalaamz. P (8 ) ϕ( х) 0,730 ϕ( х) 0 0 0,75 0,5 х g masala ma lumotlar bo ycha aqlaadga qymat hsoblaymz: 3

33 к p х pq 80 0,75 0 0,75 0,5 0,36 jadvalda ϕ(0,36)0,3789 Izlaayotga ehtmol: R 0 (8)0,730. 0,3739 0,73 Berull formulas boshqa atjaga, chuoch R 0 (8)0,8 atjaga olb kelad. Javoblarg buchalk katta farq qlsh bu msolda kchk qymatga egalg bla tushutrlad. Laplasg tegral teoremas. Teorema. Agar har br sashda A hodsag ro y bersh ehtmol r o zgarmas bo lb, ol va brda farql bo lsa, u holda ta sashda A hodsag k da k martagacha ro y bersh ehtmol R (k,k ) taqrba quydag aq tegralga teg: bu erda P k, k,, ) e dy φ ( ) (,,, y, π, k p,, k ва pq p pq φ ( ), Ф 0 ( х ) е dy π у Masus jadvallarda yuqordag tegralg 5 gacha bo lga qymatlar berlga, chuk >5 lar uchu F()0,5 deb olsh mumk. F() fuktsya ko pcha Laplas fuktsyas deb atalad. Laplas fuktsyas jadvalda foydalash uchu u quydagcha o zgartramz. 33

34 P k, k ) df df df df ( Ф(,, 0 e π,, ) Ф( ) f,,,,, + e e e π х 0 f π х f х f π 0 0 Bu jadvallarda argumetg mafy qymatlar uchu ham F() fuktsyag toqlg hsobga olb, (ya F(-) F()) foydalaamz. Shuday qlb, ta erkl sashda A hodsag k da k martagacha ro y bersh ehtmol P,, ( k, k) Ф( ) Ф(, ), k pq p ва,, k pq p Msol. Detal tekavy azorat bo lm tekshrmaga bo lsh ehtmol r0,. Tasodfy olga 400 ta detalda 70 tada 00 tagachas azorat bo lm tekshrmaga bo lsh ehtmol topg. Echsh. r0.. q0,8. 400, k 70, k 00.,,, , 400 0, 0, , 400 0, 0,8,5,75 Shuday qlb, R 400 (70,00) φ(,5)- φ(-,5) φ(,5)+ φ(,5) jadvalda φ(,5) 0,4938; φ(,5) 0,3944 Izlaayotga ehtmol R 400 (70,00)0,4938+0,39440,888 Puassog lmt teoremas. R (k) ehtmolg 34

35 k k k P ( k ) C p ( p) ; ( p q) fodas formal ravshda uchta, p va q o zgaruvchlarg fuktsyas foda qlad. Aytaylk, k taylaga, va p esa o zgarad deb faraz qlamz. Aqrog va p lar mos holda chekszlkka va olga shuday tladk, λ p mqdor chegaralaga bo lb qolaverad: λ p, λ Cost Buday holda quydag teorema o rl bo lad. Teorema. Yuqorda ko rsatlga shartlar bajarlgada ushbu k λ λ P ( k ) e muosabat o rl bo lad. k! Msol. Qo shma koroa ste molchga 5000 ta sfatl mahsulot jo atad. Mahsulotg yo lda shkastlash ehtmol 0,00 ga teg bo lsa, kkta yok uda ortq mahsulotg shkastlash ehtmol topg. Echsh. shkastlaga mahsulotlar so m desak, zlaayotga ehtmol R 5000 (m ) bo lb, u quydagga teg bo lad: R 5000 (m ) R 5000 ()+ R 5000 (3)+...+R 5000 (5000)-( R 5000 (0)+ R 5000 ()) bzg olda sashlar so katta va hodsa ro y bersh ehtmol 0 ga yaq bo lgalg uchu Puasso teoremasda foydalaamz. λ p ,00 5 ekalg e tborga olsak: P 5000 (0 ) e e 0! 5 ; P 5 e e! () 5 U holda, R 5000 (m ) -e -5-5e -5 0,9596 Erkl sashlarda sby chastotag o zgarmas ehtmolda chetlash ehtmol hsoblaymz. Faraz qlaylk, A hodsag ro y bersh ehtmol o zgarmas r ga (0<p<) m teg bo lga ta erkl sash o tkazlayotga bo ls. sby chastotag 5 o zgarmas r ehtmolda chetlash absolyut qymat bo cha avvalda berlga ε>0 35

36 soda katta bo lmaslk ehtmol topsh o z oldmzga maqsad qlb qo yaylk, ya m p ε tegszlkg ro y bersh ehtmol topamz. Bu ehtmol buday belglaymz: P m p ε Yuqordag tegszlk uga teg kuchl bo lga ε m p ε tegszlk bla almashtramz. U musbat pq ko paytuvchga ko paytrsak ε pq m p pq ε pq Laplasg tegral teoremasda foydalab,,,, ε ва ε pq pq deb olb, quydag hosl qlamz: P ε pq m p ε pq pq π ε ε pq pq e pq Z Z dz π ε 0 e dz φ ε pq Nhoyat, qavs chdag tegszlklar ularga teg kuchl bo lga dastlabk tegszlk bla almashtrb, quydag hosl qlamz: 36

37 P m p ε φ ε pq Xulosa qlb aytgada. m p ε tegszlkg ro y bersh ehtmol taqrba Laplas fuktsyasg ε dag pq kklaga qymatga teg eka. O z-o z tekshrsh uchu savollar.. Laplasg lokal teoremas ta rflag.. Laplasg tegral teoremas aytg. 3. Puasso teoremas qaday ollarda qo llalad? 4. Lokal va tegral teoremalarg amaly ahamyat mada borat? Tayach boralar. Laplasg lokal teoremas, Laplasg tegral teoremas, Puasso teoremas. Mustaql echsh uchu masalalar.. Btta o q uzlgada shoga tegsh ehtmol 0,8 ga teg. 00 marta o q uzlgada shoga rosa 75 marta tegsh ehtmol topg.. O y soqqas 0 marta tashlagada uchga karral ochkolar kamda marta, ko p bla besh marta tushsh ehtmol topg. 3. O y soqqas 800 marta tashlagada uchga karral ochko 67 marta tushsh ehtmol topg. 37

38 4. O y soqqas 90 marta tashlashda 3 ga karral sog kamda 00, ko p bla 70 marta chqsh ehtmol topg. 5. Detalg yaroql bo lsh ehtmol 0,97 ga teg. Olga 00 ta detal orasda rosa 00 tasg yaroql bo lsh ehtmol topg. 6. Teologk jarayoga ko ra kalava pg soat davomda uzlsh ehtmol 0, ga teg. Ygruvch ayol 00 ta kalavaga zmat qlad. Ug br soat davomda ko p bla 30 ta p ulash ehtmol topg. Adabyotlar [] (57-63) [] (70-8) [3] (30-35) [4] (43-58) [5] (47-50) [7] (30-33) [](87-30) 38

39 6-.Tasodfy mqdorlar va ularg turlar. Dskret tasodfy mqdor ehtmollarg taqsmot qou. Amalda ko p uchraydga dskret taqsmot qoular. Tasodfy mqdor tushuchas ehtmollar azaryas fag asosy tushuchalarda br soblaad. Ta rf: Tasodfy mqdor deb, tasodfy sabablarg ta sr atjasda mumk bo lga qymatlarda faqat bttas tay ehtmol bla qabul qluvch mqdorga aytlad. Bz tasodfy mqdorlar lot alfavtg bosh harflar X, Y, Z, bla, ularg mumk bo lga qymatlar esa tegshl kchk harflar, u, z, bla belglaymz. Odatda tasodfy mqdorlar kk l bo lad: dskret tasodfy mqdorlar va uzluksz tasodfy mqdorlar. Dskret tasodfy mqdorlar deb, mumk bo lga qymatlar ayrm ajralga solarda (bu mumk bo lga qymatlar chekl yok cheksz bo lsh mumk) borat mqdorga aytlad. Msol. X-tasodfy mqdor 00 ta buyumda borat guruhdag yaroqsz buyumlar so. Bu mqdorg mumk bo lga qymatlar quydagcha bo lad: 0,, 3., 0 00 Shuday qlb, dskret tasodfy mqdor tasvrlash uchu eg avvalo ug barcha mumk bo lga qymatlar ko rsatsh lozm. Ammo, X tasodfy mqdor uchu ug faqat mumk bo lga qymatlar, ga emas, balk { }, { }, hodsalarg ehtmollar ham, ya P P(X ), P P(X ), ham ko rsatsh lozm. Ta rf. Tasodfy mqdorg qymatlar bla ularg ehtmollar orasdag bog lash tasodfy mqdorg taqsmot qou deb atalad. Dskret tasodfy mqdor taqsmot qou fodalash usullar va shakllar turlcha bo lsh mumk. 39

40 X dskrekt tasodfy mqdor taqsmot qou berlshg eg sodda shakl jadval bo lb, buda tasodfy mqdorg barcha mumk bo lga qymatlar va ularga mos ehtmollklar ko rsatlga bo lad: X:... p: p p... p... qymatlar odatda ortb borsh tartbda yozlad. Buda tashqar, {X } hodsalarg har kktas brgalkda emaslg sababl r +r + +r p teglk har dom o rl bo lad. Ba za dskret tasodfy mqdorg taqsmot qou grafk usulda taqsmot ko pburchag yordamda ham berlad. Taqsmot ko pburchag hosl qlsh uchu, abstsssalar o qda tasodfy mqdorg mumk bo lga qymatlar, ordatalar o qda esa ularga mos ehtmollar qo ylad, key esa ( ;r ), ( ;r ) uqtalar kesmalar bla tutashtrlad. Taqsmot qou formula (aaltk) usulda ham berlad. Msol. Taga 5 marta tashlaad. Gerb tomog tushsh so X tasodfy mqdor. Bu X tasodfy mqdorg mumk bo lga qymatlar 0,,, 3, 4, 5, solarda borat bo lad. Bu qymatlarg ehtmollar Berull formulas yordamda hsoblaad. Masala, Р ( Х 3) С

41 U holda X: P: ko rshdag jadval hosl qlamz. Amalda ko p uchraydga dskret taqsmot qoular Bomal taqsmot va Puasso taqsmot hsoblaad. Bomal taqsmot. marta erkl tajrba o tkazlad. Ularda har brda bror A hodsa br l R ehtmol bla yuz bersh mumk. ta tajrbada A hodsag yuz bersh soda borat X tasodfy mqdor qaralad. Bu tasodfy mqdorga mos jadval X: 0 - ko rshda bo lb, buda P: P (0) P () p () P (-) P () P (k)c k p k q -k, (k0,,, ) Bu bevosta Berull formulasda kelb chqad. Bu jadval bla harakterlaadga taqsmot qou bomal taqsmot qou deb atalad. Agar X tasodfy mqdorga mos jadval X: 0 k R: r 0 r r r k ko rshda bo lsa, u holda X tasodfy mqdor Puasso qou bo ycha taqsmlaga tasodfy mqdor deylad. Jadvalda k λ λ Pk e, (k 0,,,, ) k! Budag λ taylaga musbat so ( λ g har l qymatlarga turlcha Puasso taqsmot mos kelad). Ehtmollar azaryasg tatbqlarda Puasso taqsmot boshqa ko plab dskret taqsmotlarga sbata ko proq uchragalg sababl u muhm aamyat kasb etad. Masala, bomal ehtmollarg 4

42 P (k)c k p k q -k fodasdag r taylab qo yb, tajrbalar so chekszlkka, R ehtmol esa va r larg ko paytmas uchu prcost shart bajarladga qlb olga tltrsak, u holda lm p k e ( k) λ k! muosabatga ega bo lamz. Org muosabatda ko rb turbdk, yuqordag lmtga o tsh atjasda bomal taqsmotg jadval Puasso taqsmotg jadvalga o tad. Shuday qlb, Puasso taqsmot bomal taqsmot uchu yuqordag shartlar bajarlgada lmt taqsmot bo lar eka. Puasso taqsmotg bu hossas tajrbalar so katta bo lb, ehtmol esa kchk bo lgada bomal taqsmot fodalash bla u tez-tez shlatladga syrak voqealar om bog lq ekalg ta kdlab o tamz. Geometrk taqsmot qou deb ataluvch qou R(Xk)q k- p, (p+q, k,, ) formula shaklda berlsh yok X: 3 k P: p qp q p q k- p jadval ko rshda berlsh mumk. Msol. X btta kubk tashlashda brch marta «6» ochko tushgucha o tkazladga tajrbalar so bo ls. Ravshak, bu holda X dskret tasodfy mqdor bo lb, r/6 parametrl geometrk taqsmot qouga bo ysuad. Ya X: 3 k k r: Msol. Talabag mtho bletdag savollarg har brga to g r javob bersh ehtmol 0,7 ga teg. Ug mtho bletdag 4 ta savolga berga to g r javoblar sog taqsmot qou tuzg. k 4

43 Echsh: X tasodfy mqdor orqal talabag to g r javoblar so belglasak, ug qabul qladga qymatlar 0; ; 3 ; 4 3; 5 4; Ko rb turbdk, 4; p0.7; q0.3 va X tasodfy mqdorg yuqordag qymatlar qabul qlsh ehtmollar Berull formulas orqal toplad: R R 4 (0) S 0 4 (0,7) 0 (0,3) 4 0,008 R R 4 () S 4 (0,7) (0,3) 3 0,0756 R 3 R 4 () S 4 (0,7) (0,3) 0,646 R 4 R 4 (3) S 3 4 (0,7) 3 (0,3) 0,46 R 5 R 4 (4) S 4 4 (0,7) 4 (0,3) 0 0,40 U holda X tasodfy mqdorg taqsmot qou quydagcha bo lad: X P 0,008 0,0756 0,646 0,46 0,40 Tekshrsh: 0,008+0,0756+0,646+0,46+0,40 O z -o z tekshrsh uchu savollar.. Tasodfy mqdor ta rf berg.. Tasodfy mqdorg qaday turlar bor? Ularga msollar keltrg. 3. Dskret tasodfy mqdor taqsmot qou deb maga aytlad? 4. Taqsmot qou qaday shakllarda berlsh mumk? 5. Amalda ko p uchraydga dskret taqsmot qoularga msollar keltrg. Tayach boralar. Tasodfy mqdor, dskret tasodfy mqdor, dskret tasodfy mqdorg taqsmot qou, bomal taqsmot qou, Puasso taqsmot qou, geometrk taqsmot qou. Mustaql echsh uchu msollar.. Nshoga qarata 4 ta o q uzlad, buda har qays o q uzshda shoga tegsh ehtmol R0,8 ga teg. Quydaglar topg: 43

44 a) shoga tegshlar soga teg bo lga X dskret tasodfy mqdorg taqsmot qou; b) Х 3 ва X >3 hodsalarg ehtmol; v) Taqsmot ko pburchag chzg.. Yashkda 5 ta oq va 5 ta qora shar bor. Yashkda ta shar old. X- tasodfy mqdor olga oq sharlar so bo lsa, ug taqsmot qou tuzg. 3. Uchta merga shoga qarata o q uzshd. Nshoga tekkzsh ehtmol brch merga uchu 0,8 ga, kkch merga uchu 0,6 ga, uchchs uchu 0,5 ga teg. Nshoga tekka o qlar soda borat bo lga X tasodfy mqdorg taqsmot qou topg. 4. Ichda 5 ta oq va 7 ta qora shar solga dshda 4 ta shar olad. Olga oq sharlar soda borat bo lga X tasodfy mqdor taqsmot qou tuzg. Adabyotlar. [] (64-74) [] (86-94), (40-49) [3] (37-4) [4] (36-58) [5] (5-69) [7] (39-44) [] (30-30) 44

45 7-. Dskret tasodfy mqdorg sol arakterstkalar va ularg ossalar. Shu ta kdlash jozk, X tasodfy mqdorg taqsmot qou blsh ehtmollk uqta-azarda X mqdor haqda to lq ma lumot berad. Amalyotda esa ko pcha buda acha kam arsa blsh kfoya qlad, chuoch taqsmot arakterlaydga ba z solarga blsh kfoyadr, bular tasodfy mqdorg sol arakterstkalar deb atalad va ularg vazfas tasodfy mqdorg eg muhm ususyatlar qsqa shaklda fodalashdr. Eg muhm sol arakterstkalar qatorga matematk kutlsh va dspersya krad. Ushbu dskret tasodfy mqdor berlga bo ls. X: х... R: r r... r Ta rf. X dskret tasodfy mqdorg matematk kutlsh M(X) deb, X mqdorg mumk bo lga qymatlar mos ehtmollarga ko paytmalar yg dsga teg soga aytlad, ya M( X) p + p p X tasodfy mqdorg mumk bo lga qymatlar so cheksz, ya X tasodfy mqdor X: х х... х.... R: r r... r.... taqsmotga ega bo lga holda ug matematk kutlsh M ( X) p + p p +... formula bla aqlaad, buda org qator absolyut yaqlashad deb faraz qlad. Aks holda, bu tasodfy mqdor matematk kutlshga ega bo lmayd. Msol. Ushbu tasodfy mqdorg matematk kutlsh topg. p p 45

46 X: R: Echsh. M(), , ,5 6 Msol. Puasso qou bo ycha taqsmlaga X dskret tasodfy mqdorg matematk kutlsh topg. Echsh. Ma lumk, Puasso qou quydag jadval bla arakterlaad. X: k p: е λ U holda λе λ М( Х) λе! λ к λ к е к! 3 λ λ е 3! λ λе... λ к 0 к к λ е к! λ к λ е ( к)! λ λе λ е λ λ Shuday qlb, Puasso taqsmot arakterlovch parametr X tasodfy mqdorg matematk kutlshda boshqa arsa emas eka. X tasodfy mqdor ustda ta sov o tkazlga bo ls. Sov atjalar quydagcha bo ls. X: х х... х k p:... k Yuqor satrda X mqdorg kuzatlga qymatlar, pastk satrda esa mos qymatlarg chastotalar ko rsatlga. X orqal kuzatlga barcha qymatlarg o rta arfmetg belglaylk, u holda k k Х yok k X k ν + ν kν k Bu erda ν, ν,....,ν K - mos ravshda х, х,... х qymatlarg sby chastotalar.

47 Demak, Х M(X) ya X tasodfy mqdorg matematk kutlsh ug kuzatladga qymatlar o rta arfmetgga taqrba teg. Matematk kutlshg ossalar. -ossa. O zgarmas mqdorg matematk kutlsh shu o zgarmasg o zga teg, ya M(S)S. Isbot. S o zgarmas mqdor yagoa S qymat ga teg ehtmol bla qabul qladga tasodfy mqdor deb qarash mumk. Shug uchu, M(S)S. S -ossa. Chekl sodag tasodfy mqdorlar yg dsg matematk kutlsh ular matematk kutlshlarg yg dsga teg, ya M(X +X X )M(X ) + M(X )+...+M(X ) 3-ossa. Chekl sodag bog lqmas tasodfy mqdorlar ko paytmasg matematk kutlsh ular matematk kutlshlarg ko paytmasga teg, ya M(X. X..... X )M(X ). M(X )..... M(X ) 4- ossa. M(aX+b) am(x)+b, (a, b cost) Isbot. M(aX+b)M(aX)+M(b)aM(X)+ b 5-ossa. M(X-M(X))0 X-M(X) tasodfy mqdor X tasodfy mqdor o zg matematk kutlshda chetlash (og sh) deb atalad. Shuday qlb, tasodfy mqdor chetlashg matematk kutlsh olga teg. Tasodfy mqdor dspersyas. Ko pchlk holatlarda, tasodfy mqdorg matematk kutlsh blsh u etarl darajada arakterlash uchu kfoya qlmayd. Masala. X: -0,7 0,0 0 0,

48 r: 0, 0, 0,4 0, 0, Y: p: 0,3 0, 0, 0, 0,3 M(X)0 va M(Y)0 ekalg ko rb turbd. Ammo bu tasodfy mqdorlar taqsmotlarg mohyat turlcha: X mqdorg mumk bo lga qymatlar ug matematk kutlshda kam farq qlad, shu bla br vaqtda Y mqdorg qymatlar ug matematk kutlshda katta farq qlad. Boshqacha aytgada, matematk kutlsh blsh uda qaday chetlashlar bo lsh mumklg haqda ukm yurtshga mko bermayd. Ta rf. X tasodfy mqdorg dspersyas D(X) deb, ug chetlash kvadratg matematk kutlshga aytlad, ya D(X)M(X-M(X)) Dskret tasodfy mqdor uchu bu formula ushbu ko rsh olad: D( X ) ( M ( X )) p Ta rf. X tasodfy mqdorg o rtacha kvadratk chetlash σ(x) deb, dspersyada olga kvadrat ldzg qymatga aytlad, ya σ(x) D (X ) Msol. Agar A hodsag ro y bersh ehtmol r ga teg bo lsa, u holda A hodsag btta sovda ro y bersh sog matematk kutlsh, dspersyas va o rtacha kvadratk chetlash topg. Echsh. Taqsmot qou quydagcha bo lad: X: 0 r: q p U holda, M(X)0. q+pp D(X)(0-p). q+(-p). pqp +pq (p+q)qp 48

49 σ(x) pq ma qul: Dspersya hsoblash uchu ko pcha quydag formulada foydalaga D(X)M(X )-(M(X)) Dspersyag ossalar. -ossa. O zgarmas mqdorg dspersyas olga teg, ya D(S)0 Isbot. S o zgarmas mqdor S qymat ehtmol bla qabul qlad deb qarash mumk. U holda M(S)S va D(S)(S-S). 0 -ossa. O zgarmas ko paytuvch kvadratga ko tarb dspersya belgsda tashqarga chqsh mumk. D(S. X)S D(X) 3-ossa. Chekl sodag bog lqmas tasodfy mqdorlar yg dsg dspersyas ular dspersyalarg yg dsga teg: D(X +X X )D(X )+D(X )+... +D(X ) Msol. Quydag taqsmot qou bla berlga X dskeret tasodfy mqdorg matematk kutlsh, dspersyas va o rtacha kvadratk chetlash hsoblag. X: r: 0.4 0, 0, 0,3 Echsh. M(X)-. 0,4+. 0,+30,+6. 0,3,5 D(X)M(X-M(X)) (--,5). 0,4+(-,5). 0.+(3-,5). 0,+(6-,5). 0,3,5 σ ( X ) D( X ),5 3,36 Bz yuqorda dspersya ta rf bo ycha hsobladk. Ed D(X)M(X )-M (X) formula bo ycha hsoblaylk. Bug uchu dastlabk X tasodfy mqdorg taqsmot qou tuzb olamz. X :

50 r: 0,4 0, 0, 0.3 D(X)M(X )-M (X)3,5-,5,5 Ta rf. X va Y tasodfy mqdorlarg korrelyatsya momet (yok kovaratsyas) deb, quydag soga aytlad. K ku M[(X-M(X))(Y-M(Y))] Dskret X va Y tasodfy mqdorlar uchu bu formula ushbu ko rsh olad: К ху, j ( M( X ))( y j M( Y)) P buda R j P(X ;Yy j ) Korrelyatsya momet fodas matematk kutlsh ossalar asosda buday almashtrlsh mumk; M(X-M (Y-M))M[XY-XM(Y)-YM(X)+M(X)M(Y)] M(XY)-M(X)M(Y)-M(Y)M(X)+M(X)M(Y)M(XY)-M(X). M(Y) j Teorema. Bog lqmas tasodfy mqdorlar korrelyatsya momet olga teg. Ta rf: r y σ K y σ y sbat X va Y tasodfy mqdorg korrelyatsya koefftset deb atalad. Agar X va Y tasodfy mqdorlar bog lqmas bo lsa, u holda ularg korrelyatsya koefftset olga teglg tushush qy emas. Quydag teorema tasodfy mqdorlar orasdag bog lash tavsflashda korrelyatsya koefftsetg ahamyat yaa ham batafsl oydlashtrb berad. Teorema. Agar Y tasodfy mqdor X tasodfy mqdorg chzql fuktsyas, ya YaX+b bo lsa, u holda agar a>0 bo lsa, r y agar a<0 bo lsa, u holda r y - bo lad. Isbot. 50

51 K u M[(X-M(X))(Y-M(Y))]M[(X-M(X)(aX+b- M(Y))]M[(X- M(X))(aX+b-aM(X)-b)]aM[X-M(X)] ad(x)aσ aσ ud(y)a D(X)a σ σ y [a] σ r y σ K y σ y aσ a σ,, a > 0 a < 0 O z-o z tekshrsh uchu savollar.. Tasodfy mqdor matematk kutlsh va dspersyas ta rflar aytg.. Matematk kutlsh va dspersya tasodfy mqdorg qays ossalar fodalayd? 3. Matematk kutlsh va dspersyag ossalar keltrg. 4. Kovaratsya ma? Tayach boralar Matematk kutlsh, chetlash, o rtacha kvadratk chetlash, tasodfy mqdor dspersyas, korrelyatsya koefftset. Mustaql echsh uchu masalalar.. 0 ta detalda borat partyada 3 ta yaroqsz detal bor. Tavakkalga ta detal olga. X dskret tasodfy mqdor olga ta detal orasdag yaroqsz detallar so bo lsa, ug matematk kutlsh topg.. Taga 5 marta tashlaad. «Raqam» tomo bla tushshlar sog taqsmot qou tuzg va dspersyas hsoblag. 3. Merga o q shoga tekkucha otad. O qg shoga tegsh ehtmol R ga teg, otlga o qlar sog matematk kutlsh va dspersyas topg. 4. Ichda 4 ta oq va 6 ta qora shar bo lga dshda 5 ta shar olad. X tasodfy mqdor chqqa oq sharlar so. M(X), D(X) va σ(x) lar topg. 5