OLIY MATEMATIKA. Ehtimollar nazariyasi va matematik statistika bo yicha mustaqil ishlarni bajarish uchun qo llanma

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1 O ZBEКISTON RESPUBLIКASI OLIY VA O RTA MAXSUS TA LIM VAZIRLIGI Abu Rayho Beruy omdag TOSHКENT DAVLAT TEXNIКA UNIVERSITETI OLIY MATEMATIKA Ehtmollar azaryas va matematk statstka bo ycha mustaql shlar bajarsh uchu qo llama Oly teka o quv yurtlarg bakalavrat ta lm yo alsh talabalar uchu Toshket 0

2 OLIY MATEMATIKA. Ehtmollar azaryas va matematk statstka bo ycha mustaql shlar bajarsh uchu qo llama. R.R. Abzalmov, G.R. Abdurahmoov, A.S. Holmuhamedov. Toshket davlat teka uverstet. 0. O quv- usluby qo llama ehtmollar azaryas va matematk statstka bo ycha mustaql sh masalalarda borat. Nazary ma lumotlar va amua uchu masalalar yechm ko rsatlga. O quv- usluby qo llama oly teka o quv yurtlarg bakalavrat yo alsh talabalar uchu mo ljallaga. Shugdek, bu o quvusluby qo llamada oly teka o quv yurtlarg professoro qtuvchlar ham foydalashlar mumk. Abu Rayho Beruy omdag Toshket davlat teka uverstet lmy-usluby kegashg qarorga muvofq chop etld. Taqrzchlar: A. Djamrzayev-O z.mu, f.-m.f.. dotset; E. Esoov- Tosh. DTU, f.-m.f.. dotset. Toshket davlat teka uverstet 0.

3 So z bosh. Zamoavy kadrlar yetshtrsh borasda respublkamz oly ta lm tzmda tub o zgarshlar amalga oshrlmoqda. Buga sabab, «Ta lm to g rsda»g qou va «Кadrlar tayyorlash mlly dastur»g qabul qlsh va ularda lmy-teka taraqqyot yutuqlar alq o jalgga tatbq qlsh, jtmoy-qtsody rvojlash bla uzvy bog lq еkalgg aq ko rsatlshdr. Buda shuday ulosa chqarsh kerakk, hozrg zamoda fudametal falar bla br qatorda ularg tatbqga bag shlaga masus kurslar ko proq o qtsh dolzarb masalalarda br bo lb qolad. «Еhtmollar azaryas va matematk statstka» masus kurs oly matematkag tatbqy bo lmlarda br bo lb, ug mavjud qouyatlar ma lum darajada blsh, tasodfy holatlar hsobga olga holda matqy ulosalar chqarsh va mavjud vazyat uchu optmal yechmlar topa olshga mko yaratad. O quv usluby qo llama oly matematkag ehtmollar azaryas va matematk statstka bo lm bo ycha mustaql sh masalalarda borat. Nazary ma lumotlar va amua uchu masalalar yechm ko rsatlga. Talabalar mustaql yechshlar uchu shahsy varatlar yetarlcha keltrlga Muallflar.

4 BOB. EHTIMOLLAR NAZARIYASI -. Na zary ma lumotlar Tasodfy hodsalar ustda amallar Ehtmollar azaryasda «tajrba» tushuchas bror shartlar majmuas aglatad. Bu shartlar bajarlgada (tajrba o tkazlgada) kuzatlsh mumk bo lga hodsalar-«tasodfy hodsalar» deylad. Shartlar majmu T br l bo lga kkta tajrba o zaro teg tajrbalar deylad. Buday holda T tajrba kk marta takrorlaad deymz. T tajrba atjasda albatta ro y beradga U T hodsa, bu tajrba uchu muqarrar hodsa deylad. Boshqacha aytgada, U T muqarrar hodsa shuday hodsak, T tajrba echa marta takrorlamas, u har gal ro y beraverad, T tajrba atjasda hech qacho ro y bermaydga hodsa, bu tajrba uchu mumk bo lmaga hodsa deylad, ya V T mumk bo lmaga hodsa shuday hodsak, T tajrba har qacha takrorlamas V T bror marta ham ro y bermayd. Tasodfy hodsalar lot harflar A,B,C, bla belglaymz. A hodsag ro y bersh B hodsa ro y bersh va akscha, B hodsag ro y bersh A hodsa ro y bersh ta mlasa, A va B hodsalar o zaro teg hodsalar deylad (A=B). Ikkala A va B hodsalarg br vaqtda ro y bersh fodalovch AB hodsa - A va B hodsalarg ko paytmas deylad. A va B hodsalarda hech bo lmagada bttasg ro y bersh fodalovch A+B hodsa - A va B hodsalarg yg ds deylad. A hodsa ro y berb, B hodsa ro y bermaslg fodalovch A\B hodsa - A va B hodsalarg ayrmas deylad. A hodsa ro y bermagalg fodalaydga A hodsa A ga teskar (qarama-qarsh) hodsa deylad.e shuday hodsa bo lsak, T tajrba atjasda ro y bersh mumk bo lga har qaday A hodsa uchu, E hodsa, yo A hodsa ro y bersh, yok A hodsa ro y bersh ta mlasa, E hodsa T tajrba uchu elemetar hodsa deylad. Elemetar hodsalar ω, =,, ko rshda kchk harflar bla, T tajrbag barcha elemetar hodsalar to plam Ω T yok bla belglaymz. T tajrba atjasda ro y bersh mumk bo lga har qaday A tasodfy hodsa, ma lum (A g ro y bersh ta mlaydga) elemetar hodsalarg yg ds shaklda, ya

5 A = εi ω ko rshda tasvrlaad. Agar qo shluvchlarg () yg ddag o r e tborga olmasa, () yg d A hodsa uchu yagoadr. Shu sababl har qaday A tasodfy hodsa A = ω \ εi ko rshda, ya () yg dga krga elemetar hodsalarg to plam ko rshda tasvrlash mumk.xususa, U T =Ω T, V T =. A to plamga kruvch ω, ϵi elemetar hodsalar A hodsaga mko yaratuvch elemetar hodsalar deylad. Ehtmollk fazos - bror T tajrba atjasda ro y bersh mumk bo lga barcha elemetar hodsalar to plam, A esa g qsm to plamlarda tuzlga, quydag shartlar qaoatlatruvch sstema bo ls: ) ε A ; ) agar A A bo lsa, A A, buda A = \A; )agar A, A ε A bo lsa, A A ε A u holda A -hodsalar algebras deylad, Agar hodsalar algebras ) A ε A, ε N A ε A = shart qaoatlatrsa, y σ-algebra deylad. A sstemada aqlaga va ) P (A) 0; A A; ) P()=. ) agar A, B ε A va A B= bo lsa, P (A+B)=P(A)+P(B) shartlar qaoatlaturvch P fuksya-ehtmollk fuksyas deylad. P(A) so esa A hodsag ehtmollg (yok ehtmol) deylad. Ehtmollk fuksyas quydag ossalarga ega; ) P( )=0; ) A B P(A) P(B), A,B ε A; ) 0 P(A) A A; ) P( A) = -P(A); ) Umuma aytgada () yg ddag qo shluvchlar so chekl yok cheksz (hatto saoqsz) bo lsh mumk.

6 ) P(A+B)=P(A)+P(B)-P(AB); ) P(A-B)=P(A)-P(AB). (, A,P) uchlk-ehtmollk fazos deylad Ehtmollkg klassk tarf Agar tajrba atjasda ro y bersh mumk bo lga elemetar hodsalar so chekl va ular teg mkoyatl bo lsa, A hodsa ehtmol, shu A hodsa ro y bershga mkoyat yaratuvch elemetar hodsalar so m g tajrba atjasda ro y beruvch barcha elemetar hodsalar so ga sbatga teg, ya P A = m. Geometrk ehtmollk Agar tajrba atjas ro y bersh mumk bo lga elemetar hodsalar cheksz ko p bo lb, teg mkoyatl bo lsa, A hodsag ehtmol hsoblash uchu barcha elemetar hodsalar to plam R fazog bror o lchovl qsmga teglashtramz. Agar bu qsmg o lchov S, A hodsaga mkoyat yaratuvch elemetar hodsalarga mos qsmg o lchov S A bo lsa, y holda A hodsag ehtmollg P A = S A S. formula orqal hsoblaad. Shartl ehtmollk. Ehtmollar ko paytmas haqdag teorema B hodsa ro y berd dega shart ostda A hodsag ro y bersh ehtmollg-shartl ehtmollk deylad va P(A/B) ko rshda yozlad. Shartl ehtmollk P(B) 0 bo lgada P(A/B) = P(AB) P(B) formula bla, P(B)=0 bo lgada P(A/B)=0 teglk bla hsoblaad. Yo qordag formulada, hodsalar ko paytmasg ehtmollg uchu P(AB) = P(A) P(B/A) = P(B) P(A/B) ayyat kelb chqad. Agar P(A/B)=P(A) teglk bajarlsa, A hodsa B hodsaga bog lq emas deylad. Bu holda P(B/A)=P(B) teglk ham bajarlad, ya

7 B hodsa ham A hodsaga bog lq bo lmayd. O zaro bog lq bo lmaga hodsalarg ko paytmas uchu. P AB = P A P(B) teglk o rl. Umuma ta hodsalar ko paytmas uchu quydag formula o rl P(A A. A ) = P(A ) P(A /A ) P(A /A A ) P(A /A.. A ) Agar A, A,A lar shuday hodsalar bo lsak, tyory m (m=,,) va k < k <... < k m tegszlklar qaoatlatruvch k j (j=,,,m) atural solar uchu P(A k, A km )=P(A k ) P(A km, ) teglk o rl bo lsa, y holda A, A,., A hodsalar o zaro bog lq bo lmaga hodsalar deylad, ya A, A,., A hodsalarg har qaday m tas (m ) ko paytmasg ehtmol, shu ko paytmaga krga hodsalar ehtmollarg ko paytmasga teg bo lsa, bu hodsalar o zaro bog lq bo lmaga hodsalar tashkl etad. Ehtmollar yg ds haqda teorema Ikk hodsa yg dsg ehtmol P(A+B)=P(A)+P(B)-P(A)P(B/A) yok P(A+B)=P(A)+P(B)-P(B)P(A/B) formula bla aqlaad. Agar A va B hodsalar o zaro bog lq bo lmaga hodsalar bo lsa u holda P(A+B)=P(A)+P(B) P(A)P(B), o zaro brgalkda bo lmaga hodsalar bo lsa P(A+B)=P(A)+P(B) teglklar o rly. Ityory hodsalar uchu P k= A k = P(A k ) P(A k A j ) + k= k= j =k+

8 + P(A k A j A ) + + P( k= j =k+ =j + j = Bog lq bo lmaga tajrbalar ketma- ketlg. Berull formulas Har br tajrbada A hodsa p ehtmollk bla ro y bers. ta bog lq bo lmaga tajrbalar ketma ketlgda A hodsag m marta ro y bersh ehtmol P m Berull formulas deb ataladga, m P m =C p m q m tehglk bla hsoblaad. Buda q=-p. ta k= teglk o rl, brgalkda bo lmaga hodsalar uchu esa teglk o rl. P k= A k = P(A k ) k= To la ehtmollk formulas. ) H k= Ω; ) H H J = Ø j,, j =,, k= shartlar qaoatlatruvch H, H,, H hodsalar - gpotezalar deylad. Yuqordag ) va ) shartlar bajarlgada H, H.., H lar, hodsalarg to la guruh tashkl qlad deymz. Tajrba atjasda ro y beradga har qaday A hodsag ehtmol topsh uchu P A = k= P (H k ) P (A/H k ) teglk o rl. Bu teglk to la ehtmollk formulas deylad. Gpotezalar ehtmol hsoblash ( Bayes formulas) A hodsa ro y bergada H k gpotezag ehtmol P(H k /A) = P H k P (A/H k ) P (A) formula bla hsoblaad, buda P(A) = P H j P (A/H k ) A k )

9 tajrbada A hodsa kam bla m marta ro y bersh ehtmol quydag formula bla hsoblaad: yok P ( k m) = k=m m P (k) P k m = P k. k=0 ta tajrbada A hodsag hech bo lmagada br marta ro y bersh ehtmol P m = q formula bla hsoblaad. ta tajrbada A hodsa hech bo lmagada br marta ro y bersh P ehtmollkda kam bo lmaslg tasdqlovch tajrbalar so l ( p) l ( p) formula bla hsoblaad. Buda p har br tajrbada A hodsag ro y bersh ehtmolg. ta tajrbada A hodsag ro y bershlar so m g eg katta ehtmolkka ershadga qymat µ P μ = ma P (m) 0 m (+)p sog butu qsmga teg. Agar (+)p butu so bo lsa, u holda P (m) eg katta qymatga kkta: μ =(+)p- va μ =(+) p solarda ershad. Agar bog lq bo lmaga tajrbalar ketma-ketlgg har br tajrbasda A hodsag ro y bersh ehtmollg har l bo lsa, ta tajrbada A hodsag m marta ro y bersh quydag g(u) = k= P k (u = q k ) = P (m)u m m=0 fodadag u m g koefftsyet orqal aqlaad, buda q k = p k, p k esa k tajrbada A hodsag ro y bersh ehtmollg. g (u) fuksyada olga hosla yordamda P (m) ehtmollk aqlash mumk:

10 P m = d m g u m! du m. Xususa, P (0)= q, q,, q. Tasodfy mqdorlar (, A,P) ehtmollk fazos bo lb, ξ mqdor to plamda aqlaga fuksya bo ls. Agar tyory εr uchu ξ < : = (ω/ξ ω < )ϵa shart bajarlsa, ξ fuksya tasodfy mqdor deylad. F ():= P (ξ < ) fuksya - ξ tasodfy mqdorg taqsmot fuksyas deylad. Taqsmot fuksyas quydag ossalarga ega:. P(a ξ < b) = F (b) - F (a);. Agar < bo lsa, F( ) F( );. lm F =, lm F = 0;. lm 0 F = F 0 < 0 Dskret tasodfy mqdorlar Agar tasodfy mqdorg qabul qlsh mumk bo lga qymatlar to plam chekl yok saoql bo lsa, bu tasodfy mqdor dskret tasodfy mqdor deylad. Dskret tasodfy mqdor taqsmot qou yok jadval ko rshda berlsh mumk. ξ tasodfy mqdorg qymatlar va bu qymatlarga mos ehtmollar p = P(ξ = ) lar yordamda tuzlga.. p p p.. p jadval taqsmot jadval deylad. Bu jadvalda p > 0, =,, = ( - chekl yok saoql bo lsh mumk) shartlar bajarlad. Argumetg,,, qymatlar uchu aqlaga 0 p =

11 P(ξ = ) = f, =,, fuksya - ξ dskret tasodfy mqdorg taqsmot qou deylad. Buda f > 0, =,, ; f( ) = shartlar ta mlash lozm. Amalyotda ko p uchraydga ba z dskret tasodfy mqdorlar :. Qymatlar a ga to plaga ayga taqsmot: P(ξ = a) =,. p (0<p<) parametrl Berull taqsmot: P ( ξ = ) = p, P ( ξ = 0) = p;. (,p) (0< p < ) parametrly bomal taqsmot : m P(ξ = m)=c p m ( p) m, m=0..,. (N, M, ) (N, M, atural solar, M N, N) parametrl gpergeometrk taqsmot = m P ξ = m = C m C N M. C N m=0,,,m (M,). p (0< p < ) parametrly geometrk taqsmot : P(ξ = m)= ( p) m p, m=,. γ ( γ > 0 ) parametrl Puasso taqsmot: γ m P( ξ = m )= m! e γ, m=0,,,. p (0< p < ) parametrl logarfmk taqsmot ( P )m P( ξ = m )=, m=, m Lp Absolut uzluksz tasodfy mqdor Agar tasodfy mqdorg F( ) taqsmot fuksyas uchu F = f d teglk qaoatlatruvch f fuksya mavjud bo lsa, tasodfy mqdor - absolut uzluksz tasodfy mqdor deylad. f fuksya esa absolut uzluksz tasodfy mqdorg zchlk fuksyas deylad. Zchlk fuksyas quydag ossalarga ega :. f - uzluksz bo lga uqtalarda f = F ();. f 0 ;

12 +. f d = ; b. P a ξ < b = f d a F ( p ) = p teglk bla aqlaga p qymat p - tartbl kvatl deylad. 0, kvatl medyaa deylad. Zchlk fuksya maksmmumga ershadga g qymat moda deylad. Absolut uzluksz tasodfy mqdorlarga msollar.( a, b ) ( a< b) kesmada teks taqsmlaga tasodfy mqdor:, ε a, b, f () = b a 0, a, b.. γ ( γ > 0 ) parametrl ko rsatkchl taqsmot: f () = γ e γ, 0, 0, < 0.. ( a,b ) (b> a) parametrl ormal (Gauss) taqsmot: f = ( a) σ π e σ. Normal taqsmlaga ξ tasodfy mqdorg (, ) oralqqa tushsh ehtmollg quydag formula bla aqlaad : P( < ξ < ) = Ф a Ф a, b b buda Ф() = π e t dt 0 Laplas fuksyas. Laplas fuksyasg qymatlar jadval yordamda toplad. Tasodfy mqdorg sol arakterstkalar p = P ξ =, =, taqsmot qou bla berlga dskret tasodfy mqdorg matematk kutlmas deb, M (ξ) = =. p

13 teglk bla aqlaga soga aytlad. f () zchlk fuksyas bla berlga absolut uzlksz tasodfy mqdorg matematek kutlmas deb, M ( ξ ) = + f d teglk bla aqlaga soga aytlad. tasodfy mqdorg matematek kutlmas ba za ko rshda ham belglaad: m k = M ξ k, μ k = M (ξ ) k formulalar bla aqlaga m k va μ k solar mos ravshda tasodfy mqdorg k tartbl boshlag ch va markazy mometlar deylad. Xususa, -tartbl boshlag ch momet matematk kutlmadr. tasodfy mqdorg tartbl markazy momet - ug dspersyas deylad va D(ξ) ko rshda belglaad, ya D ( ξ ) = M(ξ ) = Mξ (Mξ) ξ tasodfy mqdorg dspersyasda olga kvadrat ldz σ(ξ) = D (ξ) ξ tasodfy mqdorg o rta kvadratk chetlash deylad. Tasodfy mqdorg k-tartbl mometlar ug soly arakterstkalar deylad. Matematek kutlma va dspersyag ossalar X, Y tasodfy mqdor va C o zgarmas so uchu quydag teglklar o rl: ) M (C) =C; ) M (CX)=CM(X); ) M (X) M X ; ) M X + Y = M X + M Y ; ) agar X va Y o zaro bog lq bo lmasa, u holda M X Y = M X M Y ; ) D (X)> 0; ) D (C)=0; ) D (CX)=C D X ; ) agar X va Y o zaro bog lq bo lmasa, u holda D (X+Y)= D (X)+D (Y).

14 -. Namuavy msol va masalalar yechm -msol. Tasodfy solar jadvalda tyory ravshda kkta so talab old. A hodsa ularda hech bo lmagada bttas tub so, B hodsa esa, ularda hech bo lmagada bttas juft so eka bldrsa, AB va A+B hodsalar ma aglatad? Yechm. AB hodsa kkala hodsag br vaqtda ro y bersh, ya talaga solarda br tub, kkchs juft ekalg yok solarda br, kkchs esa htyory tasodfy so eka aglatad. A B hodsa, yo talaga solar tub eka, yo talaga solar juft eka, yok talaga solarg br tub, boshqas esa juft ekalg aglatad. -msol. Ityory talab olga N atural sog uchch darajasg org kkta raqam bo lsh ehtmol topls. Buda N atural so deylgada 0 da gacha bo lga, teg mkoyatl hd-arab raqamlar yordamda yozlga so azarga tutlga. Yechm. N so N a 0b... ko rshda fodalaymz, buda a, b, lar tyory raqamlar. U holda N = a + a 0b + 00c, c atural so, buda ko radk, N g org kkta raqam a va b raqamlarg qymatga bog lq. U holda ro y bersh mumk bo lga mkoyatlar so 00 ta. N sog org raqam a g faqat btta qymatda, ya a da bo lad. Orda kkch raqam ham bo lsh uchu N 0 g org raqam ga teg bo lsh kerak, ya b so raqam bla tugash kerak. Bu faqat b bo lgada amalga oshad. Shuday qlb, ro y bersh mumk bo lga mkoyat btta: a, b. Bu holda zlaayotga ehtmollk p 0, 0ga teg bo lad. -msol. Uzlg 00m bo lga magtafo tasmasg - qatorg 0m ga ma lumotoma yozlga. -qatorg ham 0m ga ma `lumotoma shuday yozlgak, tasma magtafo kallagda o tgada avval -qatordag ma lumotoma kelad. Agar ma lumotoma yozlsh tasmag hamma qsm uchu teg

15 mkoyatl bo lsa, tasmag 0- metrda 0- metrgacha bo lga qsmga ma lumotoma yozlmaga bo lsh ehtmol topls. Yechm. Magtafo tasmas chapda o gga tarag tortlga deb faraz qlb, ug -qatordag ma lumotoma yozlga qsmg chap chegaras bla -qatordag ma lumotoma yozlga qsmg chap chegaras y bla belglaymz. U holda o 0, 0 y 0, y (- rasm). Bu tajrba uchu elemetlar hodsalar to plam y tekslkdag = (, y)/0 0, y uchburchakda borat bo lad. Agar 0 0 Y A 0 0 A 0 -rasm A orqal tasmag 0- metrda 0- metrgacha bo lga qsmga ma lumotoma yozlmaga bo lsh hodsas belglasak, u holda A ga g A {(,y )/ 0 y 0} va A {(, y ) / 0 0, y 0} qsmlar mos kelad. uchburchakg yuzas S( ) 0 ga, A qsmg yuzas esa S ( A) S( A ) S( A ) ga teg bo lad. Ehtmollkg geometrk tarfga asosa 0 0

16 0 0 P ( A ) msol. Umumy mahsulotg % yaroqsz bo lb, yaroql mahsulotlarg % brch avl bo lsa, tyory olga mahsulot brch avl bo lsh ehtmol topls. Yechm. A - olga mahsulot yaroql bo lsh hodsas, B - mahsulot brch avl bo lsh hodsas, C -olga mahsulot brch avl bo lsh hodsas bo ls. U holda C AB ; P( A ) 0,0 ; P( A) P( A ) 0, 0 0, ; P B 0,. A Ehtmollk ko paytrsh formulasga asosa -msol. Koroaga favqulodda hodsa ro y bergada abar beradga, br-brda mustaql shlovch ta asbob o ratlga. Favqulodda hodsa ro y bergada brch asbobg shlash ehtmol 0, ga, kkchs uchu bu ehtmollk 0, ga teg. Favqulodda hodsa ro y bergada faqat btta asbobg shlash ehtmol topls. 0,0, 0, P( C) P( A B) P( A) P B. A -msol. Ichda tas yaroqsz bo lga 00 ta mahsulotda tavakkalga 0 tas old. Shu 0 ta mahsulotda ko p bla tas yaroqsz bo lsh ehtmol topls. Yechm. A - tavakkalga olga 0 ta mahsulotda bttas ham yaroqsz bo lmaslk hodsas, B - olga 0 ta mahsulotda bttas yaroqsz bo lsh hodsas bo ls. U holda A va B lar brgalkda bo lmaga hodsalar bo lb, ularg ehtmollklar gpergeometrk taqsmot formulas bla, bu hodsalarda hech bo lmagada bttasg ro y bersh ehtmol esa P P( A B) P( A) P( B) formula bla hsoblaad: 0 0 C C C C P( A B) P( A) P( B) 0 0 C C ,.

17 Yechm. A - brch asbobg shlash hodsas, A - kkch asbobg shlash hodsas bo ls. U holda P ( A ) 0,,,, B A A A A A A va A A lar brgalkda bo lmaga hodsalar bo lga uchu P( B ) P( A A ) P( A A ) P( A ) P( A ) P( A ) P( A ) P( A ) 0, P( A ) P( A ) 0, P( A ) P( A ) 0,. - hodsa faqat btta asbobg shlash fodalayd. teglk o rl bo lad. P( B ) 0, 0, 0, 0, 0, 0, 0,. -msol. Samolyot bortga o ratlga qurlma kk l rejmda samolyotg teks parvoz vaqtdag ormal rejmda hamda samolyotg ko tarlsh va qo sh vaqtdag zo rqsh rejmda shlash mumk. Samolyotg teks parvoz, ug umumy parvozg 0 %, ko tarlsh va qo sh vaqtdag parvoz esa 0% tashkl etad. Qurlmag ormal rejmda shlash vaqtda shda chqsh ehtmol 0,, zo rqsh rejmda shlash vaqtda shda chqsh ehtmol 0, ga teg. a)qurlmag parvoz vaqtdag shochllg topls; b)agar qurlma shda chqqa bo lsa, bu hodsa teks parvoz vaqtda sodr bo lsh ehtmol topls. Yechm. A -parvoz vaqtda qurlmag buzlmay shlash hodsas, H -qurlmag ormal rejmda shlash hodsas, H - qurlmag zo rqsh rejmda shlash hodsas bo ls. U holda, P( H ) 0, ; P( H ) 0, ; P A 0, 0, 0 H ; P A 0, H. a)to la ehtmollk formulasga ko ra P( A) P( H) P ( ) A P H P A H H 0, 0, 0, 0, 0,

18 b)bayes formulasga ko ra P( H) P A H 0, 0, H P 0,. A P( A) 0, -msol. Teg kuchl raqblar orasdag o yda qays hodsag ehtmol katta: a)br o ychg to rtta o yda uchtasda g olb bo lshm yok sakkzta o yda beshtasda g olb bo lshm? b)br o ychg to rtta o yda kamda uchtasda g olb bo lshm yok sakkzta o yda kamda beshtasda g olb bo lshm? O yda durag holat ro y bermayd deb hsoblaad. Yechm. O ychlar teg mkoyatl bo lga uchu ularg yutsh yok yutqazsh ehtmollklar tegdr: p q. a)br o ychg to rtta o yda uchtasda yutsh ehtmol P ( ) C p q. Sakkzta o yda beshtasda yutsh ehtmol P ( ) C pq Demak,. bo lgalg uchu, to rtta o yda uch marta yutsh ehtmol ortq. b)br o ychg to rtta o yda kamda uch marta yutsh ehtmol ( K ) P ( ) P ( ) P. Sakkzta o yda kamda besh marta yutsh ehtmol

19 P ( K ) P ( ) P ( ) P ( ) P ( ) Demak, bo lga uchu, sakkzta o yda kamda besh marta yutsh ehtmol ortq. -msol. Partyadag 00 ta mahsulotda 0 tas yaroqsz. Mahsulot sfat tekshrsh uchu shu partyada tyory tas talab old. tasodfy mqdor talab olga mahsulotlar chdag yaroqsz mahsulotlar so bo lsa, shu tasodfy mqdorg taqsmot jadval tuzls. Yechm. Talab olga ta mahsulot chdag yaroqszlar so 0 da gacha bo lsh mumk. Shu sababl tasodfy mqdor X 0, X, X, X, X, X qymatlar gpergeometrk taqsmot qou bo ycha, K K C0 C0 P( K ), K 0,,..., C 00 ehtmollklar bla qabul qlad. Bu ehtmollklar 0,00 aqlkda taqrba hsoblaymz: P P( 0) 0,, P P( ) 0,, P P( ) 0, 0, P P( ) 0, 00, 0-msol. a radusl aylaa tyory uqtas radus vektorg shu aylaa dametrga proyeksyas quydag taqsmot fuksyas (arksus qou) ga ega. P P( ) 0, P P( ) 0. Ed bu tasodfy mqdorg taqsmot jadval tuzshmz mumk X 0 P 0, 0, 0,00 0,00 0 0

20 , a, F( ) arcs, a a, a 0, a. tasodfy mqdorg a a) Qymatlar, a oralqqa tushsh ehtmol; b) f zchlk fuksyas; c)taqsmotg moda va medaas aqlas. a Yechm. -tasodfy mqdorg qymatlar, a oralqqa tushsh ehtmol a a a a P F F arcs arcs ; b) Zchlk fuksyas ( a, a) oralqda df( ) f ( ) d teglk bla, to plamda d d arcs a (, a ) ( a, ) f ( ) 0 a teglk bla aqlaad. c) Zchlk fuksyasg modas mavjud emas, chuk f ( ) a fuksyasg maksmum qymat yo q. Quydag arcs 0 a

21 teglama yechb, medaa olga teg ekalgga shoch hosl qlamz. -msol. 00 ta mahsulotda 0 tas sfatsz bo lb, mahsulot sfat tekshrsh maqsadda tas talab old. Agar - tasodfy mqdor - talab olga mahsulotlar chdag sfatsz mahsulotlar so bo lsa, shu tasodfy mqdorg matematk kutlmas topls. Yechm. tasodfy mqdor 0,,,,, qymatlar qabul qlsh mumk. Bu qymatlar qabul qlsh ehtmollklar k k C0 C0 P( k ) ( k 0,,..., ) C00 formula bla hsoblaad. tasodfy mqdorg matematk kutlmas MX = k=0 k = 0 k C C 00 ga teg. Bu fodadag C 0 k k C 0 = k k C C 00 C 0 00 k= yg d hsoblash uchu k= C k 0 = 0 j C j =0 C j C 00 j C 0 j =0 0 ( u) ( u) ( u ) C k 0 = j C 0 ayyatda foydalaamz. Bu ayyatg chap tomodag ko phadda u g koefftset j =0 C j j C 0 ga teg. Bu koefftset o g tomodag ko phad uchu teg. Demak, C ga

22 va j =0 C j 0C M C C j 0 = C 00 -msol. Kemag yoboshga tebrash ampltudas zchlk fuktsyas a e, f ( ) 0 a (Reley qou) da borat tasodfy mqdordr. Shu tasodfy mqdorg a) M matematk kutlmas; b) D dspersyas va o rta kvadratk chetlash; c) va - mos ravshda uchch va to rtch tartbl markazy mometlar topls. Yechm. Quydag J t e t dt 0 (-atural so) tegral yordamda tyory tartbl mometlar hsoblash mumk. - juft bo lgada ( k )!! J k Ak k buda ( k )!! ( k )( k )..., -toq bo lgada J k! A( k ). k a)matematk kutlma M 0 f ( )d a e 0, a d

23 t a almashtrsh bajarsak, 0 a a J a dt e t a M : t ga teg bo lad. b), ) ( ) ( ) ( M M D buda. a a J a d e a ) X M( : m a 0 Shu sababl, ) ( a a a D buda. ) ( a D c), ) ( m m ] ) M [( buda, a a J a m U holda, ) ( a a a a a, ) ) (( m m m M buda, a J a m shug uchu

24 a. -msol. Radoapparat 000 ta elektroelemetga ega. Yl davomda btta elemetg shda chqsh ehtmollg 0,000 ga teg. a)ikkta elemetg; b)kamda kkta elemetg shda chqsh ehtmol topls. Yechm. Ishda chqqa elemetlar so tasodfy mqdor bo lb, u p 000 0,00 parametrl Puasso taqsmot qouga bo ysuad deb hsoblash mumk. U holda a) kkta elemetg shda chqsh ehtmollg P( ) P000( ) e 0,! e ga, b) kamda kkta elemetg shda chqsh ehtmollg ( ) P( ) P (0 ) P () P e ( ) e 0, ga teg. -msol. Ma lum ob yektgacha bo lga uzoqlk o lchash sstematk va tasodfy atolar keltrb chqarad. Uzoqlkg o lchashdag sstematk ato - 0 m ga teg. Tasodfy atolar esa o rta kvadratk chetlash 00 m bo lga, ormal taqsmot qouga bo ysuad. a)uzoqlk o lchashdag atolk, absolut qymat bo ycha, 0 m. da oshmaslk ehtmol; b)uzoqlk o lchashdag atolk, shu uzoqlkg haqqy qymatda oshmaslk ehtmol toplls. Yechm. Uzoqlk o lchashdag atolk deb olsak, masala shartga ko ra - parametrlar (-0, 00) bo lga ormal taqsmot qouga bo ysuvch tasodfy mqdor bo lad. Shu sababl

25 a) P( 0 ) P( 0 0 ) Ф Ф Ф( ) Ф( ) Ф( ) Ф() 0, 0, 0, buda Ф ( ) va Ф () u Ф( ) e du 0 Laplas fuksyasg va uqtadag qymatlar bo lb, ular jadval yordamda topld. b) Uzulk o lchashdag atolk, shu uzulkg haqqy qymatda oshmaslg ehtmol P( 0 ) Ф Ф Ф(0, ) Ф( ) 0, 0, 0,. Mustaql yechsh uchu msol va masalalar -varat. A+A va A-A hodsalar ma aglatad?.talaba dasturdag 0 savolda tas blad. Har br mtho blet uchta savolda tashkl topga. Quydag hodsalarg еhtmol topg: Talaba tushga bletg: a) barcha uchta savol blad; b) faqat kkta savol blad; d) faqat btta savol blad..uzulg L ga teg bo lga AB telefo lyasg C uqtasda uzlsh sodr bo ld. (Buday uzulsh telfo lyasg har br uqtasda teg mkoyat bla sodr bo lsh mumk) Shu C uqta A uqtada l da kam bo lmaga masofada joylashsh ehtmol topls.. Brgalkda bo lmaga ta hodsa ro y bersh ehtmollklar mos ravshda 0,0, 0,00, 0,00 va 0,00 ga teg. Tajrba atjas-

26 da shu hodsalarda hech bo lmagada tasg ro y bersh ehtmol topls.. 0 ta br l dshg tasda tada qora va tada oq shar, tasda esa ta oq va ta qora shar bor. Idshlarg brda oq shar old. Bu sharg ta oq shar bor dshda olga bo lsh ehtmol topls.. Basketbolch to p korzaga otad. Agar to p korzaga tushsa, basketbolch to p yaa korzaga otad. Shu jarayo, to to p korzaga brch bor tushmay qolguga qadar davom ettrlad. To pg ta otshda korzaga tushsh ehtmol p=0, ga teg. Korzaga tushga to plarg tasodfy so uchu taqsmot qator va taqsmot fuksyas tuzls.. Btta asbob shochllg tekshrlmoqda. Ug tekshruv vaqtda shda chqsh ehtmol p ga teg. Agar X-tekshruv vaqtda shda chqqa asboblar so bo lsa, shu tasodfy mqdorg matematk kutlmas topls..radoapparatg 0000 soat chda shda chqshlar sog matematk kutulmas 0 ga teg. Radoapparatg 00 soat chda shda chqsh ehtmol topls.. X tasodfy mqdor o zg taqsmot fuksyas F() bla berlga. Ug zchlk fuksyas, matematk kutlmas va dspersyas topls. Taqsmot va zchlk fuksyalarg grafg chzls. F = 0, agar 0 bo lsa,, agar 0 < bo lsa,, agar > bo lsa. -varat. Tavakkalga olga domo doas dubl emas. Tavakkalga olga - domo doas ham dubl bo lmaslk ehtmol topls.. Ikkta yashkg brda ta oq va kkchsda 0 ta qora shar bor. Brch yashkda kkchsga tavakkalga br shar old, so gra kkch yashkda tavakkalga br shar old. Olga shar qora bo lshlg еhtmol topg.. t vaqt davomda hsoblash mashasg k - blokg shda chqsh ehtmol p k (k=,,,) ga teg. Agar masha bloklarg har br, boshqalarga bog lq bo lmaga holda shlasa, shu t vaqt da-

27 vomda ta blokda kamda tasg shda chqsh ehtmol maga teg?. Domo doalarg to la aborda tavakkalga tas old. - doag - doa ketda qo ylsh mumk bo lsh ehtmol topls.. Olada 0 ta farzad bor. O g l farzad tug lsh ehtmol 0, ga teg bo lsa: a) olada ta o g l bo lsh; b) olada kamda ta, ko p bla ta o g l farzad bo lsh ehtmol topls.. Tasodfy mqdorg F() = π e t taqsmot fuksyas berlga. Shu tasodfy mqdorg zchlk fuksyas topls.. X tasodfy mqdor o zg taqsmot fuksyas F() bla berlga. Ug zchlk fuksyas, matematk kutlmas va dspersyas topls.taqsmot va zchlk fuksyalarg grafg chzls. dt. X tasodfy mqdorg taqsmot fuksyas F() = 0, agar bo lsa, a + b arcs, agar < bo lsa,, agar > bo lsa. ko rshga ega (arksus qou). a va b o zgarmaslarg qymat aqllas hamda M(X) va D(X) lar topls.. Samolyotg ma lum baladlk saqlaga holda uchshda yo l qo yladga sstematk atolk 0 metr. Baladlk saqlashdag tasodfy atolk metr ga teg o rta kvadratk chetlashga ega. Samolyot uchsh uchu baladlg 00 metr bo lga yo lak ajratlga. Agar samolyot uchu shu yo lakg o rtas ko rsatlga bo lsa, samolyotg yo lakda pastda, yo lakg chda va yo lakda yo qorda uchsh ehtmollar topls

28 0, agar bo lsa, F = 0 ( + ), agar < bo lsa,, agar > bo lsa. -varat.nsho 0 ta kosetrk aylaalarda borat bo lb, bu aylaalarg radyuslar r k (k =,,,0) r < r < < r 0 shart qaoatlatrad. Agar A k - uqtag r k radyusl aylaaga tushsh hodsas bo lsa, B = k= A k,j = 0, N = A k hodsalar ma aglatad?.uchta merga br l va bog lqsz sharotda btta mo ljalga qarab br martada o q uzshd. Brch mergag mo ljalga o q tekkzsh еhtmol 0, ga, kkchsk 0, ga, uchchsk еsa 0, ga teg. Quydag hodsalarg еhtmol topg: a) faqat br merga mo ljalga o q tekkzd; b) faqat kkta merga mo ljalga o q tekkzd; d) uchta merga ham mo ljalga o q tekkzd. Radyus R ga teg bo lga dorada, berlga yo alshga parallel bo lga vatarlar o tkazlga. Agar vatarlar berlga yo alshga perpedekular bo lga dametr teg mkoyat bla kesb o tsa, shu vatarlarda tavakkalga olga tasg uzulg R da oshmaslg ehtmol maga teg?. ta br l katakchaga bo lga kvadratga sharcha tashlad. Sharchag - qator, j ustudag katakchaga tushush ehtmol p j ga teg u p j =. k=

29 Sharchag k-qatorga tushush ehtmol topls.. Ishlab chqarlayotga mahsulotg % stadartga to g r kelad. Nazoratg soddalashtrlga semasda, tekshrlayotga mahsulot - 0, ehtmollk bla stadartga to g r kelad deb, 0,0 ehtmollk bla ostadart deb toplad. Soddalashtrlga azoratda o tga mahsulotg stadartga to g r kelsh ehtmol tops.. ta asbobg shochllg br-ket shuday tekshrlmoqdak, agar tekshrlga asbob shochl bo lsa, keygs tekshrlad,aks holda tekshrsh to tatlad. Har br asbob tekshrsh atjasda 0, ehtmollk bla shochl deb toplad.tekshrlga asboblarg tasodfy so uchu taqsmot qator (jadval) tuzls..tekshralyotga asbob ta elemetda tuzlga. - elemetg shda chqsh ehtmol p = 0, + 0,( ) ga teg. Agar elemetlarg shda chqsh o zaro bog lq bo lmasa, shda chqqa elemetlar tasodfy sog matematk kutlmas va dspersyas topls.. Apparat 000 ta, shochllklar br l bo lga elemetlarda borat. Har br elemetg shda chqsh ehtmol p=0,000 ga teg. Shu elemetlarda hech bo lmagada tasg shda chqsh atjasda, apparatg shlamay qolsh ehtmol maga teg?. X tasodfy mqdor o zg taqsmot fuksyas F() bla berlga. Ug zchlk fuksyas, matematk kutlmas va dspersyas topls. Taqsmot va zchlk fuksyalarg grafg chzls. F = 0, agar 0, bo lsa, +, agar 0, < 0 bo lsa,, agar > 0 bo lsa. -varat.harfl qulf umumy o qqa joylashtrlga ta dsska ega. Dsklarg har br ta sektorga bo lga va sectorlarga turl harflar yozlga. Dsklar o q atrofda aylatrlb, ulardag harflarg ta ma lum kombatsyas tuzlgada va faqat shu holdaga qulf ochlad. Dsklardag harflarg tavakkalga olga kombatsyasda qulfg ochlsh ehtmol topls.

30 .Br l va bog lqsz tajrbalarg har brda hodsag ro y bersh еhtmol 0, ga teg. 00 tajrbada hodsa 00 marta ro y bersh еhtmol topg. Detall -dastgohda tayyorlagada, ug -avl bo lsh ehtmol - 0, ga teg. Shu detal -dastgohda tayyorlagada, bu ehtmollk 0, ga teg. - dastgohda ta, -dastgohda ta detall tayyorlad.tayyorlaga hamma detalg -avl bo lsh ehtmol topls.. ta dshg har brda m tada oq va k tada qora shar bor.- dshda tavakkalga ta shar olb, -sga sold. Shuda so g - dshda ta shar olb, -sga sold, va hokazo. Org dshda oq shar olsh ehtmol topls.. Kutuboada faqat teka va matematkaga od ktoblar bor. Ityory o quvchg tekaga od ktob olsh ehtmol 0, va matematkaga od ktob olsh ehtmol 0, ga teg. Ketma- ket kelga o quvchg, yok faqat tekaga od ktob, yok faqat matematkaga od ktob olga bo lsh ehtmol topls.. Radoapparaturag uzluksz buzulmay shlash vaqtg taqsmot fuksyas F t = e t (t 0) ga teg (ekspoesal taqsmot). a) radoapparaturag t vaqt mobayda uzluksz buzulmay shlash ehtmol; b) shu tasodfy mqdorg zchlk fuksyas f (t) topls.. X tasodfy mqdorg zchlk fuksyas f = π a ( a < < a) ko rshga (arksus qou) ega bo lsa, shu tasodfy mqdorg dspersyas va o rta kvadratk chetlash topls.. Detal ma lum dastgohda tayyorlagada ug o lchamg omal o lchamda tasodfy chetlash matematk kutlmas 0 ga, o rta kvadratk chetlash mk. ga teg bo lga tasodfy mqdordr. Agar o lchamg omal o lchamda chetlash mk. da kchk bo lga detallar yaroql detal deb qaralsa, mazkur dastgohda 0

31 echta detal tayyorlagada, shu detallarda hech bo lmagada tas yaroql bo lsh hodsasg ehtmol 0, da kam bo lmayd.. X tasodfy mqdor o zg taqsmot fuksyas F() bla berlga. Ug zchlk fuksyas, matematk kutlmas va dspersyas topls. Taqsmot va zchlk fuksyalarg grafg chzls. 0, agar π bo lsa, F = cos, agar π < π bo lsa,, agar > π bo lsa. -varat.har brda kamda ta tom bo lga ta turl asarlar to plam mavjud. A, B va C hodsalar mos ravshda -- - to plamda kamda ta ktob olga aglatad. A s va B k hodsalar esa mos ravshda -asarlar to plamda s ta, - asarlar to plamda k ta tom olga aglatad. a) A+B+C; b) ABC; d) A + B ; e) A B ; f) A B + B A C hodsalar ma aglatad?.avarya ro y bersh bldrsh uchu uchta br-brga bog lq bo lmaga holda shlovch qurlma o ratlga. Avarya vaqtda brch qurlma shga tushshg еhtmol 0, ga, kkch qurlma shga tushshg еhtmol 0, ga va uchchs shga tushshg еhtmol 0, ga teg. Quydag hodsalarg еhtmol topls: avarya vaqtda: a) faqat btta qurlma shga tushsh; b) faqat kkta qurlma shga tushsh; d) barcha qurlmalar shga tushsh..to g r to rtburchak shakldag pajara ko dalag qsmg radus r ga teg bo lga, sldr shakldag prutlarda (hvchlarda) yasalga. Ikkta yoma-yo joylashga (o zaro parallel holda) prutlarg o qlar orasdag masofa mos ravshda a va b ga teg. Shu pajaraga d dametrl sharcha tashlad. Agar sharcha trayektoryas pajara tekslgga perpedkular bo lsa, sharchag pajarada (prutlarga urlmasda) o tb ketsh ehtmol topls.

32 . O y kartasg talk to la to plamda suratl karta yok qarg a mastl karta sug urb olsh ehtmol maga teg? (suratl karta deylgada valet, dama va karol azarda tutlad) ta lampochkag chdag yaroqszlarg so teg ehtmol bla 0 tada tagacha bo lsh mumk. Agar shu 000 ta lampochkada tavakkalga olga 00 tasg chda brorta ham yaroqsz bo lmasa, 000 ta lampochkag hammas yaroql bo lsh ehtmol maga teg?. Ikk basketbolch, to ularda br to p korzaga tushura olmay qolguga qadar, galma-gal to p korzaga tashlamoqdalar. Agar brch basketbolch 0, ehtmollk bla, kkch basketbolch esa 0, ehtmollk bla to p korzaga tushura olsa, har br o ychg to p korzaga tashlashlar so uchu taqsmot jadval tuzls.. Agar ta asbobg har br t vaqt mobayda p ehtmollk bla shda chqsh mumk bo lsa, shu vaqt mobayda shda chqqa asboblar tasodfy sog matematk kutulmas topls.. Mahsulotg sovda o ta olmaslk ehtmol 0,00 ga teg. 000 ta mahsulotda kamda tasg sovda o ta olmaslk ehtmol topls. Bomal taqsmot va Puasso taqsmot orqal toplga atjalar taqqoslas. Hsoblashlar logarfmg oal jadval yordamda amalga oshrls.. X tasodfy mqdor o zg taqsmot fuksyas F() bla berlga. Ug zchlk fuksyas, matematk kutlmas va dspersyas topls. Taqsmot va zchlk fuksyalarg grafg chzls. 0, agar 0 bo lsa, F =, agar 0 < bo lsa,, agar > bo lsa. -varat.hamyoda ta 000 so mlk va ta 00 so mlk pul qog ozlar bor. Shu hamyoda ta pul qog oz old. Shuda so g yaa ta pul qog oz old. Keyg olga pul qog oz 000 so mlk bo lb chqd. Brch olga pul qog ozg ham 000 so mlk eka ehtmol topls.

33 .Br l va bog lqsz tajrbalarg har brda hodsag ro y bersh еhtmol 0,0 ga teg. 0 ta tajrba o tkazlgada hodsa marta ro y bersh еhtmol topg.. N ta lampaga ega bo lga asbob btta lampas shda chqsh bla shlashda to tayd. Shu shda chqqa lampa aqlash uchu, asbobdag lampalar, to shu lampaga yetb kelgucha ketmaket tekshrlad. Agar har br lampag shda chqsh ehtmol br l bo lsa, ta lampag tekshruvda o tsh ehtmol topls..brch partyadag detallarg qsm yaroqsz. Qolga partyadag detallarg hammas sfatl. Shu ta partyag brda tavakkalga btta detal old. Olga detalg yaroqsz bo lsh ehtmol topls.. ta A tpdag, ta B tpdag, ta C tpdag blokka ega bo lga elektr qurlma -rasmda ko rsatlga semada tuzlga. A A B K C C C C -rasm Agar A tpdag blokg shda chqsh ehtmol 0., B tpdagsk 0., C tpdagsk esa 0. bo lsa, elektr zajrg K kalt yordamda tklab bo lmaydga uzlsh ehtmol topls..veybulag m F() = e 0 ( 0) taqsmot fuksyas berlga. Shu taqsmotg: a) f () zchlk fuksyas; b) p-tartbly kvatl; d) modas topls..gaz molekulalar tezlg zchlk fuksyas f v = A v e v v 0 (Makcvell qou) bo lga tasodfy mqdordr. A va h kattalklar hamda M(v) matematek kutlma va D(v) dspersya topls.

34 .Normal taqsmlaga X tasodfy mqdor σ=0 m o rta kvadratk chetlashga ega. Argumetg, qadam 0 m ga teg dskret qymatlar uchu, taqsmot fuksyasg qymatlar jadval tuzls va grafg chzls.. X tasodfy mqdor o zg F() taqsmot fuksyas bla berlga. Ug zchlk fuksyas, matematk kutlmas va dspersyas topls. Taqsmot va zchlk fuksyalarg grafg chzls. 0, agar bo lsa, F = L, agar < e bo lsa,, agar > e bo lsa. -varat.agar A olga ta mahsulotda kamda tas yaroqszlg hodsas, B esa shu ta mahsulotda kamda tas yaroqsz bo lsh hodsas aglatsa, A va B hodsalar ma bldrad?.000 doa tovarda 0 ta yaroqszmahsulotuchrayd. Shu 000 doa tovarda tavakkalga 0 doa olgada ularg rosa doas yaroqsz bo lshlg еhtmol topg..raduslar k r (k=,,,,) ga teg bo lga ta kosetrk aylaalar chzlga. r radusl dora va tashq raduslar r va r bo lga halqalar bo yalga. Radus r ga teg bo lga doraga uqta tashlad. Nuqtag: a) r radusl doraga tushush; b) Bo yalga sohaga tushush ehtmol topls;. A, B va AB hodsalarg ehtmol ma lum. AB hodsag ehtmol va P( A B ) shartly ehtmolk topls.. ta mergada tas 0, ehtmollk bla, tas 0, ehtmollk bla, tas 0, ehtmollk bla va tas 0, ehtmollk bla o q shoga tekkza olad. Mergalarda br shoga o q otb, tekkza olmad. Shu mergag qays guruhga tegshl bo lsh ehtmol kattaroq.

35 .Asboblarg shga tushursh uchu har sekudda sgal yuborlad.sgal yuborlga vaqtda asbob shga tushga vaqtgacha sekud vaqt o tad. ta asbob shga tushsh bla sgal yuborsh to tayd. Agar har br asbob sgal yuborlgada ehtmollk bla shga tushsa, yuborlga sgallarg tasodfy so uchu taqsmot qator tuzuls..brch o ych ta, - o ych esa ta br l taga tashlamoqda. Qays o ychga gerb ko proq tushsa o sha o ych yutad va tashlaga ta tagag hammas olad. Agar gerblar so teg bo lb qolsa, o y aq atjaga ershlgucha qaytada o yalaverad. Har br o ychg yutb olga tagalarg tasodfy solarg matematk kutulmas topls..agar yaroqsz mahsulotlar mqdor, mahsulotlar umumy mqdorg % tashkl qlsa, olga 00 ta mahsulotg tada ortg yaroqsz bo lsh ehtmol topls.. X tasodfy mqdor o zg taqsmot fuksyas F() bla berlga. Ug zchlk fuksyas, matematk kutlmas va dspersyas topls. Taqsmot va zchlk fuksyalarg grafg chzls. 0, agar 0 bo lsa, F = ( х ), agar 0 < bo lsa,, agar > bo lsa. -varat. Tavakkalga olga atural sog a) kvadrat; b) -darajas; d) tyory atural soga ko paytmas raqam bla tugash ehtmol topls..br l va bog lqsz tajrbalarg har brda A hodsag ro y bersh еhtmol 0, ga teg. ta tajrba o tkazlgada A hodsa da kam bo lmaga va 0 da ko p bo lmaga marta ro y bersh еhtmol topg.. ta bog lq bo lmaga tajrbada A hodsag kamda marta ro y bersh ehtmol 0, ga teg. Agar A hodsa hamma tajrbada br

36 l ehtmollk bla ro y bersa, ug ta tajrbaga ro y bersh ehtmol topls.. Tasodfy uqta faqatga rombg B j (j=,,,) uchlarda brda bo lsh mumk. Buda, br qadamda p k k =,,, ehtmollk bla B k da B k+ ga, q j = p j (j=,) ehtmollk bla B j da B j + ga, q = P ehtmollk bla B da B ga o ta olad. Tasodfy uqtag a) s da ortq bo lmaga qadamda (s=,); b) qachodr B da B ga o tsh ehtmol topls.. Bog lq bo lmaga ta tajrbag har brda A hodsag ro y bersh ehtmol 0, ga teg. Agar A hodsa kamda marta ro y bersa, B hodsa ehtmollk bla ro y berad; A hodsa ro y bermasa, B hodsa ham ro y bermayd, agar A hodsa br marta ro y bersa, u holda B 0, ehtmollk bla ro y berad. B hodsag ro y bersh ehtmol topls. ( < < + ) taq-. X tasodfy mqdorg F(X)=c+b arctg a smot fuksyas (Kosh qou) berlga. a) c va b o zgarmaslar; b) P (γ < X < β) ehtmollk; d) zchlk fuksyas topls..x tasodfy mqdorg zchlk fuksyas f = 0, agar 0 bo lsa; 0, agar < 0 bo lsa; ( 0 > 0) ko rshga ega. M(X) va D(X) topls..normal taqsmlaga X tasodfy mqdorg E = M( X X ) o rta arfmetk chetlash bla ug o rta kvadratk chetlash orasdag muosabat aqlas.. X tasodfy mqdor o zg F() taqsmot fuksyas bla berlga. Ug zchlk fuksyas, matematk kutlmas va dspersyas topls. Taqsmot va zchlk fuksyalarg grafg chzls.

37 0, agar 0 bo lsa, F = ( х ), agar 0 < bo lsa,, agar > bo lsa. -varat. Qays hollarda a) A+B=A ; b) AB=A ; d) A+B=AB teglklar o rl bo lad?.br kkchsga bog lq bo lmaga holda shlaydga uchta dastgohda br turdag detal tayyorlaad.tayyorlaga barcha detallarg 0 foz brch dastgohda, 0 foz kkchsda, 0 foz uchchsda shlab chqlga.detalg yaroql bo lb tayyorlash ehtmol brch dastgoh uchu 0,, kkch dastgoh uchu 0, va uchch dastgoh uchu 0, ga teg. Barcha tayyorlaga detallarda tavakkalga olga btta detalg yaroql bo lsh ehtmol topls.. L uzulkdag kesmada htyory ta uqta olga. Bu uqtalar orasdag masofa k da kchk bo lsh ehtmol topls, buda 0 k L.. Agar A hodsa B hodsag qsm bo lsa, P(A) P(B) tegszlkg to g rlg sbotlag.. ta ovch brdaga to g zga qarata o q uzshd va u ta o qda halok bo ld. Agar brch, kkch va uchch ovchg o q shoga tekkaza olsh ehtmollar mos ravshda 0,; 0, va 0, ga teg bo lsa, to g zg a) brch; b) kkch; d) uchch ovch o qoda halok bo lgalg ehtmol topls.. ta mahsulotg shochllg tekshrmoqda har br mahsulot tekshruvda so g p ehtmollk bla shochl deb toplad. Tekshruv atjasda shochl deb toplga mahsulotlar tasodfy sog taqsmot qou topls.. Uchta A,B,C o ychlar quydaga shartlar asosda o y o yashmoqda; har partyada tada o ych shtrok etad. Yutqazga o ych o r o ychga bo shatb berad. Har partyada o yayotga o ychlarg mkoyatlar teg. O y, o ychlarda br to ta partya ketma ket yutmagucha davom ettrlad va yutga o ych barcha o yalga partyalar soga teg bo lga yutuq olad. Agar brch partya A va B o ychlar o yaga bo lsa,

38 A va C o ychlarg yutuqlar sog matematk kutulmas maga teg.. Rezerford va Xeygerg kuzatuvlarga ko ra radoaktv modda, sekud davomda o rtacha, ta α-zarracha chqarad. Shu moddag br sekud davomda hech bo lmagada ta α -zarracha chqarsh ehtmol topls.. X tasodfy mqdor o zg taqsmot fuksyas F() bla berlga. Ug zchlk fuksyas, matematk kutlmas va dspersyas topls. Taqsmot va zchlk fuksyalarg grafg chzls. 0, agar 0 bo lsa, F = х х, agar 0 < bo lsa,, agar > bo lsa. 0- varat.brch uchraga avto mashag omer a) turl raqamlarda tashkl topsh ; b) kkta br l raqamga ega bo lsh ; d) uchta br l raqamga ega bo lsh ; e) kk juft br l raqamga ega bo lsh ; f) br l raqamlarda tashkl topsh ehtmol tops. Buda, omerlar oal, 000da boshlaad va takrorlamayd hamda barcha omerlar teg mkoyatl deb hsoblaad..aka-uka har br kshda borat kkta sport komadasga qatashadlar. Ikk yashkda da gacha omerlaga ta blet bor. Har br komada a zolar tavakkalga bttada blet aq br yashkda olshad. Olga blet yashkka qaytarlmayd. Ikkala akaukag - omerl blet olshlg еhtmol topls..tavakalga yozlga oddy kasrg qsqarmaydga kasr bo lsh ehtmol topls (Chebshev masalas )..Chorrahaga o ratlga aftomat svetoforg chroqlar yashl sarq qzl-sarq yashl - shemada avbatma- avbat yob uchb turad. Yashl chroqg o chmasda uzluksz yob tursh davr - mut. Bu davr qzl chroq uchu - 0, mut, sarq chroq uchu esa sekud. Agar vaqtg tyor mometda svetoforg ta chrog da br (avbat kelga) yob turad deb faraz qlsa, shu

39 chorrahaga tasodfa kelb qolga yegl mashag to tamay o tb ketsh ehtmol maga teg?. ta oq, ta qora shar solga dshda to qora shar chqquga qadar ta tada shar olmoqda. Agar a) olga shar oq bo lsa, u qaytada dshga solb, so g keyg shar olayotga bo lsa; b) olga sharlar qayta dshga solmayotga bo lsa, dshda marta shar olgalg ehtmol topls.. f = a ( < < + ) + fuksya, X tasodfy mqdorg zchlk fuksyas. a) a parametr; b) X tasodfy mqdorg taqsmot fuksyas; d) Tasodfy mqdorg (-;) oralqqa tushsh ehtmol topls.. X tasodfy mqdor f = A e β, agar 0 bo lsa, ( > ; β > 0) 0, agar < 0 bo lsa, zchlk fuksyasga ega. A parametr, X tasodfy mqdorg matematek kutulmas va dspersyas topls..ov mltg o qg poro zaryad, og rlk o lchashdag o rta kvadratk atolg 0 mg. bo lga tarozda tortlad. Poro zaryadg omal og rlg, gr. Agar yo l qo yladga maksmal poro zaryad, gr bo lsa, mltqg shda chqsh ehtmol topls.. X tasodfy mqdor o zg F() taqsmot fuksyas bla berlga. Ug zchlk fuksyas, matematk kutlmas va dspersyas topls. Taqsmot va zchlk fuksyalarg grafg chzls. 0, agar 0 bo lsa, F = s, agar 0 < π bo lsa,, agar > π bo lsa. -varat.btta tajrba atjasda sodr bo lsh mumk bo lga tyory A va B hodsalar uchu A B + A B + A B = AB teglkg to g r eka sbotlas.

40 .Uchta qurolda br vaqtda mo ljalga qarab o q uzshd. Br otshda mo ljalga tekkzsh еhtmol brch qurol uchu 0, ga, kkch qurol uchu 0, ga va uchch qurol uchu еsa 0, ga teg. Quydag hodsalarg еhtmol topg: a) faqat br o q mo ljalga tegsh; b) faqat kkta o q mo ljalga tegsh; d) barcha uchta o q mo ljalga tegsh; e) hech bo lmagada br o q mo ljalga tegsh. L uzulkdag kesma tavakkalga olga uqta yordamda uch qsmga bo ld. Shu uch kesmada uchburchak yasash mumklg ehtmol topls..yo l chetdag chzq bo ylab br-brda teg uzoqlkda o smlk ko chatlar o tqazlga. Yo l belglamaga joyda kesb o tayotga yo lovch p ehtmollk bla ko chatlarda br bosb o tsh mumk. Agar yo lovchlar yo l ketma-ket, br-brlarga bog lq bo lmaga holda kesb o tshayotga bo lsa, belglamaga joyda o tayotga t - yo lovchg ko chatlarda br bosb o tsh ehtmol topls..idshdag ta sharg har brg rag, teg ehtmollk bla, oq va qora bo lsh mumk. Idshda k marta shar old. Bu shuday amalga oshrldk, har gal olga sharg rag aqlaad va qaytarb dshga solad, shuda so g keyg shar olad. Agar dshda olga sharlar chda brorta ham qora shar bo lmasa, dshdag sharlarg hammas oq eka ehtmol maga teg?.btta otshda o qg shoga tegsh ehtmol p ga teg. Nshog shu shoga kelb tekka m ta o q tufayl shda chqsh ehtmol P(m)=-(- ω )m, m 0, ω >. Nsho shda chqquga qadar otlga o qlar so uchu taqsmot qator tuzls..br radostasya jo atga chaqrq sgal boshqa radostasya tomoda qabul qlsh ehtmol 0, ga teg. Radostasya, o z muqarrar tarzda qabul qladga javob sgal olmagucha, har sekudda chaqrq sgal yuboraverad. Chaqrq sgal borb, uga javob sgal kelgucha sekud o tad. tommolama aloqa o ratlguga qadar yuborlga chaqrq sgallarg o rtacha so topls. 0

41 .Uchsh vaqtda raketag apparatlar bo lmga P r, γ = r! e γ ehtmollk bla r ta elemetar zarracha tushsh mumk. Shu elemetar zarrachalarg har br p ehtmollk bla apparatlar bo lmg ozk blokga krb qolsh mumk. Nozk blokka a) roppa-rosa k ta; b) kamda ta elemetar zarracha krb qolsh ehtmol topls.. X tasodfy mqdor o zg F() taqsmot fuksyas bla berlga. Ug zchlk fuksyas, matematk kutlmas va dspersyas topls. Taqsmot va zchlk fuksyalarg grafg chzls. 0, agar 0 bo lsa, F = ( х ), agar 0 < bo lsa,, agar > bo lsa. -varat.sakkzta br l kartochkaga mos ravshda,,,,,, va solar yozlga. Shu kartochkalarda tavakkalga tas old. Olga kartochkalardag solarda qsqaradga kasr tuzsh ehtmol maga teg?.uch merga br vaqtda shoga o q uzshd. Nshoga o q tekkzsh еhtmol brch merga uchu 0, ga, kkch merga uchu 0, ga, uchch merga uchu еsa 0, ga teg. Quydag hodsalarg еhtmol topg: a) faqat br merga shoga o q tekkzsh; b) faqat kk merga shoga o q tekkzsh; d) barcha uchta merga shoga o q tekkzsh; e) hech bo lmagada btta merga shoga o q tekkzsh еhtmol topg..elektr zajrdag kuchlash omal (me yory) qymatda oshsh ehtmol p ga teg. Elektr tok ste mol qluvch asbobg kuchlash oshsh atjasda shda chqsh ehtmol p ga teg. Shu asbobg shda chqsh ehtmol topls..000 ta lampag chda yaroqszlar, teg ehtmol bla, 0 tada tagacha bo lsh mumk. Shu 000 ta lampada tavakkalga olga 00 tasg hammas yaroql bo lsh ehtmol topls. γ r

42 .A hodsa o zaro bog lq bo lmaga ta tajrbag har brda 0, ehtmollk bla ro y berad. Shu hodsag kamda marta ro y bergalg ehtmol topls..azmut burchag o lchash asbob shkalas 0 da bo lga. Agar o lchash atjas, eg yaq butu sogacha yaltlaga bo lsa, azmut burchag o lchashdag atolk ±0 chegarada bo lsh ehtmol topls.. X tasodfy mqdorg zchlk fuksyas f() = A( + ) + ga teg. > butu so, A o zgarmas so, M(X) matematk kutulma va D(X) dspersya topls..tekslkda br-brda L masofada joylashga ta parallel to g r chzq o tkazlga. Shu tekslkka R radusl dora tashlad. Dora tushsh mumk bo lga sohag o rtas, to g r chzqlarg brda b masofaga tashq tomoda joylashga. Dora markazg tog r chzqlarga perpedkular yo alshda ortacha chetlashsh E ga teg. Agar L=0m, R=m, b=m, E=0m. bo lsa, dorag a) kamda ta to g r chzqg ustga tushsh; b) kkala chzqg ustga tushsh ehtmol topls.. X tasodfy mqdor o zg taqsmot fuksyas F() bla berlga. Ug zchlk fuksyas, matematk kutlmas va dspersyas topls. Taqsmot va zchlk fuksyalarg grafg chzls. 0, agar bo lsa, F = ( ), agar < bo lsa,, agar > bo lsa. -varat. A va A + B hodsalar brgalkdam?.talaba dasturg 0 ta savolda 0 tas blad. Imtho blet ta savolda borat. Quydag hodsalarg еhtmol topg: Talaba a) faqat kkta savol blad; b) uchta savol blad; d) hech bo lmagada btta savol blad.

43 .Uzulg l da katta bo lmaga ta kesmada uchburchak yasash mumk bo lsh ehtmol topls..bletg omer ta raqaml soda borat bo lb, brch ta raqamg yg ds, keyg ta raqam yg dsga teg bo lsh ehtmol 0,0 ga teg. Tavvakalga ta blet old. Agar a) Olga bletlarg raqamlar ketma-ket bo lsa; b) Bletlar htyory olga bo lsa, bu bletdqa brg dastlabk ta raqamg yg ds, keyg ta raqam yg dsga teg bo lsh ehtmol topls..br l jsl egzaklarg tug lsh ehtmol, turl jsly egzaklarg tug lsh ehtmolda marta ko p. Turl jsl egzaklarg tug lsh kkala (o g l-qz, qz-o g l) tartb uchu br l mkoyatl. Egzaklarg brchs o g l bo lsh ehtmol 0, ga teg. Agar egzaklarg brchs o g l bo lsa, kkchsg ham o g l bo lsh ehtmol topls..taga br marta tashlagada, ug gerb tomo bla tushsh ehtmol 0, ga teg. Taga marta tashlad. Gerblar sog raqamlar soga sbat uchu taqsmot jadval tuzuls.. m ta elemet sh qoblyat yo qotgada asboblarg shlamay qolsh ehtmollklar a) A asbob uchu P(m)= - e γm, γ > 0, m =, ga teg, b) B asbob uchu 0, agar m = bo lsa; P (m) = e γ(m ), agar m =, bo lsa. A va B asboblarg shda chqshga sabab bo lga sh qoblyatlar yo qotga elemetlar sog matematk kutlmas topls..agar radoaktv moddag massas M=0,g., atom og rlg A=, yarm parchalash davr T =, 0 yl bo lsa, shu moddada r=sm uzoqlkda, zarrachalar yo alshga perpedkular holatda joylashtrlga, S=0, sm yuzal ekraga sekud davomda a) Roppa-rosa 0 doa α-zarracha tushsh; b) Ikktada kam bo lmaga α -zarracha tushsh ehtmol topls.. X tasodfy mqdor o zg taqsmot fuksyas F() bla berlga. Ug zchlk fuksyas, matematk kutlmas va dspersyas topls. Taqsmot va zchlk fuksyalarg grafg chzls.

44 F = 0, agar 0 bo lsa,, agar 0 < bo lsa,, agar > bo lsa. -varat. 0 ta lotereya blet chda tas yutql. Shu bletlarda tavvakalga tas old. Olga bletlar chda: a) ta yutuql blet bo lsh; b) kamda ta yutuql blet bo lsh ehtmol topls..har br 0 sportchda borat kk komada musobaqa qatashchlarga omer bersh uchu qur a tashlashmoqda. Ikk aka-uka turl komadalarg a zosdr. Aka-ukag kkalas ham musobaqada - omer bla qatashsh еhtmol topg..uch o ych quydag shart asosda o y o yashmoqda. Avval -ch o ych, -ch va -ch o ychlarga qarsh o yayd. Agar bu o ylarda - o ych yutmasa, u holda - va - o ychlarg yutsh ehtmol br l va 0, ga teg. Agar - o ych yutqazmasa, u holda u - va - o ychlar bla yaa br martada o yayd va bu o ylarg har brda 0, ehtmollk bla yuta olad. Shu bla o y tugayd. - o ychg kamda ta raqb umumy hsobda yutsh ehtmol topls..yashkdag ta tes koptogda tas yag. - o y uchu shu koptoklarda tavakkalga tas old va o y tugagach yaa yashkka qaytarb sold. - o y uch yaa tavakkalga ta koptok old. - o y uchu olga uchchala koptokg yag bo lsh ehtmol topls..agar lotereya blet yutuql bo lsa, bu yutuqg velosped yok kr yuvsh mashas eka ehtmol, mos ravshda 0,0 va 0,0 ga teg. Turl seryada tavakkalga olga 0 ta yutuql blet chda yuqordag buyumlarda hech bo lmagada tasg borlg ehtmol topls..o zaro L masofada joylashga pukt orasda avtobus qatayd. Avtobusga qadaydr br yo lovch chqd va ma lum yo l bosb o tb, yaa tushb qold. Yo lovchg (0 L) uqtada avtobusga chqsh ehtmol zchlg (L ) ga proportsoal,

45 y ( y L) uqtada tushb qolsh ehtmol zchlg (y), 0 ga proporsoal. a) yo lovchg z puktda avval avtobusga chqsh; b) uqtada avtobusga chqqa yo lovchg z puktda so g aftobusda tushb qolsh ehtmol topls.. lm [F ] = 0 lm + {[ F ]} = 0 shartlar bajarlgada, matematk kutulma uchu M X = 0 F d 0 F d teglk to g rlg sbotlas..agar detal o lchamg, shu detal o lcham uchu yo l qo yladga oralq o rtasda og sh - =0, σ=mk. parametrl ormal taqsmlaga tasodfy mqdor bo lsa, htyory olga detal o lcham 0,00 ehtmollk bla yo l qo yladga oralqda tashqarga tushsh uchu yo l qo yladga oralqg keglg qaday bo lsh kerak?. X tasodfy mqdor o zg taqsmot fuksyas F() bla berlga. Ug zchlk fuksyas, matematk kutlmas va dspersyas topls. Taqsmot va zchlk fuksyalarg grafg chzls. 0, agar bo lsa, F =, agar < 0 bo lsa,, agar > 0 bo lsa. -varat.ikk shamatch br partya o y o yayd. A brch shamatch yutsh hodsas, B kkch shamatch yutsh hodsas. Bu kk hodsaga yaa qaday hodsa qo shlgada, ular brgalkda hodsalarg to lq gruppas tashkl qlad..ikk merga mo ljalga bttada o q uzshd. Har br mergag mo ljalga o q tekkzsh еhtmol 0, ga teg. Quydag hodsalarg еhtmol topg: a) kkala merga mo ljalga o q tekkzshd; b) kkala merga mo ljalga o q tekkzshmad; d) hech bo lmagada br merga mo ljalga o q tekkzd

46 .Bekatga A yualshdag avtobus har mutda, B yo alshdag avtobus esa har mutda kelad. A va B yo alsh avtobuslarg bekatga kelb to tashlar teg mkoyatl. a) Bekatga kelb to taga avtobus A yo alshda bo lsh; b) kk mut chda kkala yo alsh avtobuslarg bttasg kelb to tash ehtmollar topls..p(x 0) = 0,, P( Y ) = 0, eka ma lum. X va Y g o zaro bog lq va bog lq emaslgda qat y azar Z=X+Y tasodfy mqdor uchu P(Z ) 0,, P(Z ) 0,, teglklar o rl eka sbotlas..uchta o zaro bog lq bo lmaga tajrbag har brda A hodsag ro y bersh ehtmol 0, ga teg. Boshqa B hodsag ro y bersh ehtmol A hodsag ro y bershlar soga bog lq: A hodsa marta ro y bergada bu ehtmollk 0, ga, marta ro y bergada esa 0, ga teg. A hodsa ro y bermagada B ham ro y bermayd. Agar B hodsa ro y berga bo lsa, A hodsag eg katta ehtmollk bla ro y beradga so aqlas.. a g qaday qymatda f = a ( < < ) + fuksya X tasodfy mqdorg zchlk fuksyas bo lad. a) X tasodfy mqdorg taqsmot fuksyas, b) X tasodfy mqdorg (-;) tervalga tushsh ehtmol topls..asbob kosmk urlashga sezgr va brga zarracha tushsh bla shda chqadga ta A,B va C elemetlarga ega. Asbob yo A elemet shda chqqada, yok B va C elemetlar baravarga shda chqqada shlamay qolad. Asbobga krga zarra, mos ravshda 0,, 0, va 0, ehtmollk bla A, B va C elemetlarga tushsh mumk. Agar asbob uga krga zarra tufayl shlamay qolga bo lsa, asbobga krga zarralarg tasodfy so va matematk kutlmas topls.. Normal taqsmlaga X tasodfy mqdorg E = M( X X ) o rta arfmetk chetlash va o rta kvadratk chetlash orasdag bog lash aqlas.