Giáo trình xác suất thống kê pgs ts đinh ngọc thanh, ths nguyễn quốc huy, ths trần thị ngọc tuyết

BQ GIAO DUG VA DAO TAO TRUdNG DAI HOC DAN LAP Clftr LONG Nguyen Quoc Huy - Dinh Ngoc Thanh Tran Thi Ngoc Tuyet Gido Trinh XAC SUAT THONG KE thU viM | — Vmh Long - 2007 i • J Ldl DE TUA Trifcfng Dai hoc Dan lap Ctfu Long muon khang dinh la trung tam dao tao nguon nhan life co trinh cao cho khu vile dong bang song Cdu Long de phuc vu cho cong cuoc cong nghiep ho a, hien dai ho a dat nifdc thi phai nang cao chat lifo'ng dao tao toan dien Mot nhifrig yeu to' yeu la doi ngu thay co, he thong cac giao trinh va trang thiet bi day hoc Gung vdi cac nganh cac cap toan quoc dang day nhanh tien xay difng va cung cd vi the xu the hoi nhap, triTcfng Dai hoc Dan lap Cdu Long dan tiTng birdc hoan thien de diTOng dau vdi nhtfng thach thde do, gop phan dtfa nen giao due dai hoc Viet Nam ddng vifrig, ngang tam khu vifc va the gio’i Ke hoach xay ddng ve mot Bo giao trinh sd dung cho cac giang vien lam tai lieu ccf ban de giang day, cho sinh vien nghien cdu tham khao hoc tap la nhu cau cap bach cua nha trddng An pham la giao trinh Xac Suat Thong Ke cua Khoa Khoa Hoc Cd Ban, Trdd’ng Dai hoc Dan lap Cdu Long, PGS TS Dinh Ngoc Thanh, ThS Nguyen Quoc Huy va ThS Tran Thi Ngoc Tuyet bien soan Lanh dao trddng Dai hoc Dan lap Cdu Long tran nhdng cong hien, dong gop cua Quy thay co mdi giang cung nhu" cac thay co can bo - giang vien co hdu cua nha trddng Hy vong rang, mot khong xa Trufrng Dai hoc Dan lap Chu Long se co day du mot Bo giao trinh hoan chinh dung nha trildng Tran gid'i thieu giao trinh den Quy thay cd va cac em sinh vien trufrng Dai hoc Dan lap Cufu Long Vinh Long, thang 09 nam 2007 Q HIEU TRUdNG ThS Nguyen Cao Dat h DAI CVONG Vfi GIAI TICH TO HOP TAP H0P Tap hop la mot khai niem nguyen thuy, khong co dinh nghia Sir gom gop mot so ddi toong lai vdi cho ta hinh anh cua tap hop va cac ddi toong dooc gom gop trd phan tit, cua tap hop NgObi ta thifdng ky hieu tap hop bang cac ky tii in nho A, B, C, , X, Y, va phan tuf bdng cac ky to thobng nho a, b, c, , x, y Neu x la mot phan td cua A, ta viet x g A Ngooc lai, neu x khong la phan td cua A, ta viet x g A 1.1 Cac phtfcTng phap xac dinh tap hofp Co ba phoong phap xac dinh tap hop : Phoong phap liet ke, phoong phap trOng tinh va phoong phap dung gian Venn Phiiong phdp liet ke : Cac phan tu* cua tap hop duttc viet xuong gitfa hai ngoac nhon, va phan tur khac ditoc phan each bdi dau phay va thoa hai dieu kien : - Khong chu y thdf tit liet ke, - Moi phan tu’ chi dooc liet ke mot lan, khong lap lai Chang han, cac tap hop A = {a, b,c} va B = {c,b,a} la nho chung chi khac thd to liet ke cac phan tu”; C = {1,0,1) khong hop le vi phan td difoc liet ke hai lan Phuong phdp tri£ng tinh : Dtfa mot tinh chat ma chi co phan td cua tap hop tOOng ting duOc thoa Chang han, gpi A la tap cac so nguyen chSn Tinh chat so nguyen chdn la tinh chat d&c tritog cho tap A Khi do, V2,|gA vi chung khong la so' nguyen; 3,5 g A vi chung la so nguyen nhifng khong la so chan; 10,100 g A vi chung la so nguyen va la so' chin Tong quat, ngifbi ta dung mot ham menh de p(x) theo mot bien x g X, nghia la dng vdi moi x g X, ta cd mot menh de p(x) Tap A cac phan tuf x g X cho menh de p(x) cd chan tri dung dope ky hieu la A = (x e X|p(x)} Khi do, ta co VxgXjg Ao p(x) Noi khac di, x thuoc ve A neu va chi neu p(x) la menh de dung Chang han, tap A cac so' nguyen chSn dooc viet lai la A = |x e N |x chia het cho 2} Phuong phdp dung gidn Venn : Ngufti ta dung mot dirbng cong don khep kin, chia mat phang lam hai mien, mien phia dodng cong danh de liet ke cac phan tuf cua tap hop, mien phia ngoai duftng cong danh de liet ke cac phan tuf khong nSm tap hop Ching han, ta cd x,a e A , y,b g A • y a CAC PHEP TOAN TAP HOP Vdi hai tap hop A va B, ta noi A la mot tap cua B, ky hieu A cB, moi phan td cua A deu la phan tuf cua B, Vx,x g A => x e B Tap tat ca cac tap cua mot tap X cho trade daoc ky hieu la ^(X), ^>(X) = {A|AcX{ Hien nhien X c X va do X e ^(X) Ngoai ra, tap hnp khong cd phan td nao ca dude goi la tap hap rdng, ky hieu va ta quy ddc tap hdp rdng la tap cua moi tap hdp, nghia la c: X hay e ^(X), vdi moi tap hop X Vi du, vdi A = (a, b, c,d}, B = {a,b,c} va C = {b, c, d), ta cd B,CcA vi moi phan td cua B cung nhd cua C deu la phan td cua A Tuy nhien B C) = P (tCr choi H | H dung) va ta ket loan |T| > C : tcf choi H, |T| < C : chap nhan H Vi du Goi X, Y la khoi ItTOng tre so sinh trai va gai Quan sat ngau nhien 20 be trai, ta tinh di/Oc X = 3200g; Sx - 400g va quan sat 17 be gai, ta tinh difOc Y = 3000g; SY = 380g Hoi rang khoi hrtfng cua tre so sinh trai va gai co gio'ng khong ? Ket loan vdi a = 0.05 Ta co bai toan kiem dinh [H: nx =llY * |iY (H : Ngoai ra, 19 ■ 4002 + 16 ■ 380i2: — = 152686, 35 (n - 1) S^ + (m -1) S^ n+m-2 S = 390.98g, T= X-Y ~ St (20 + 17 - 2) = St (35) * N (0; 1) S Ji + i Vn m 3200-3000 - , _ ■ — = 1.55 390.98+ i Vcri a = 0.05, ta co C = z 1-0.05 = 1-96 147 Ket ludn : |T| < C nen ta chap nhan H, nghia la khoi lifo'ng tre scf sinh trai va gai nhil 3.5 Trifofng hofp so li^u ttfiig cap Gia suf ta dung thuoc A cho 10 ngilcri Bang sau ghi lai nhip tim/phut cua tuTng ngildi trade va sau dung thuoc Ngudi 10 Trude : X 70 77 78 72 81 78 73 74 79 80 Sau : Y 76 75 78 77 85 81 76 74 85 80 Hoi rang thuoc A cd lam thay doi nhip tim hay khdng ? Ket luan vdi a = 0.05 Day la trddng hop day so lieu tufng cap Ta khdng the can cd tren nhip tim trung binh cua tdng day so de so sanh ma ta phai quan tarn den sd thay doi cua tdng ca the Dat d = Y-X de chi sb Ido’ng gia tang nhip tim Ta cd bai toan kiem dinh fH:pd =0 [H : pd Vi T= va neu H (d-O)Vn dung thi (2.5-0)710 ” 2^84 T~St(9) Ma = 2.79 vdi a = 0.05 thi C - t-0.05 = 2.262 Ket luan : Do |T| > C, ta bac bo gia thuyet H va ket luan rang thuoc A cd lam thay doi nhip tim 148 3.6 So sanh hai ty 1? Gia str quan sat n = 100 be trai thay cd 15 chau mac benh B; cung quan sat m = 150 be gai, thay co 18 chau bi benh B Hoi rang ty le nhidm benh B doi vdi be trai va be gai co gidng khong ? Ket luan vdi a = 0.05 Goi Pj = P(B| trai) = ty le benh cua be trai, va p2 = P(B| gai) = ty le benh cua be gai Ta co bai toan kiem dinh H : Pi = p2 H : Pi * p2 Ta co fx = tan suat bi benh cua be trai = 15 = 0.15, 18 f2 = tan suat bi benh cua be gai = —— = 0.12 J.OU Dat P= T= nfj + mf2 n+m fl-f2 va vdi m^u tren, ta co T= 15 + 18 = 0.312, 100 + 150 ■ ~ St(n + m - 2) = St(248) « N(0;l) 0-15-0.12 = 6g65 V0^132-0-868^^) va vdi a = 0.05, thi C = Zi-q.qs = 1.96 Ket luan : |T| < C nen ta chap nhan H, nghia la ty le nhidm benh B cua be trai va be gai nhif 149 3.7 So sanh hai phtfdng sai Ta quan sat hai m^u doc lap Xn ; Xi ~ N(px; Xj, X2, j; Yj, Y2, , Yn; Yi-N( py;cty)- Ta muon so sanh hai phifong sai Ox gy bang each difa vao cac gia tri quan sat d hai mau tren Gia sd Sx > SY (trong trufrng hop nguOc lai thi ta thay doi vai tro cua X va Y) Ta cd bai toan kiem dinh H: oY Ta co (n-l)Sl ~X2(n- 1) (m -1) Sy -X2(m- !)• C) = P (til chdi Hjkhi H dung) va ta ket luan F > C : tit choi H, F < C : chap nhan H Vi du Quan sat lifting X cua 25 tre so sinh trai va trpng lifting cua 19 tre sd sinh gai Ta tinh diftlc Sx = 450g2 va S2 = 360g2 Hay so sanh hai phiftlng sai Ox ■ Ket luan vori a = 0.05 Ta co bai toan kiem dinh H: = oy H:ox > °y F = < = 150=1.25 360 Neu H dung thi F~F(24,18) nen vdi a = 0.05, ta diftlc fo.os = 2-15 Ket luan : F < C , chap nhan H, nghia la ox = oY 151 Bai tap Cd mot 16 hang ma ngiTdi giao hang cho biet ty le hong 0.10; thuf pham 0.30; dat 0.40; tot 0.20 Ta kiem tra mot so' tritbng hpp thay co 25 san pham hong; 50 thtf pham; 50 san pham dat; 25 san pham tot Hoi rang led ngufti giao hang noi co dung khong ? (ket luan vdi a = 5%) Do cholesterol (defn vi mg%) cho mot nhom nghcfi, ta ghi nhan lai difo'c Choi 150-160 160-170 170-180 180-190 190-200 200-210 So ngirdi 11 a) Tinh trung binh X va phifcrng sai Sx , b) Tim khoang irdc liTPng cho trung binh cholesterol dan so', px , d tin cay y = 0.95, c) Co tai lieu cho biet Itfatag cholesterol trung binh la p0 = 175mg% Gia tri co phu htfp vdi m6u quan sat khong ? (ket luan vdi ct = 0.05) Quan sat stfc nang cua be trai (X) va be gai (Y) luc sP sinh (defn vi gam), ta co ket qua Khoi Itfgng 3000-3200 3200-3400 3400-3600 3600-3800 3800-4000 So be trai 10 So be gai 10 10 a) Tinh X, Y, , S^, b) So sanh cac phiro'ng sai Ox > gy luan vdi a = 5% ), c) So sanh cac trung binh px » I^Y d) Nhap hai m£u lai Tinh trung binh va lech chuan cua m^u nhap Dung mau nhap de tfefe litdng sdc nang trung binh cua tre so' sinh tin cay 95% 152 Mot may dong gdi cac san pham cd khbi limng 2kgngd may boat dong khong binh thifdng, ngildi ta chon mot mdu ngdu nhien gdm 100 san pham thi thay nhuf sau : Khbi kmng 0.95 0.97 0.99 1.01 1.03 1.05 Sb gdi 31 40 15 Vdi mtfc y nghia 0.05, hay ket luan ve nghi ngd tren Quan sat so hoa hong bani mot cua mot cufa hang ban hoa sau jmot thcri gian, ngi/cri ta ghi di/oc so lieu sau So hoa hong (doa) 16 15 13 12 17 18 19 7 Sb a) Tim Ode lifOng khong chech cua sb hoa hong trung binh ban dime mot b) Sau tinh toan, ong chu eda hang noi rang neu trung binh mot khong ban dime 15 doa hoa thi chang tha dong eda hem Diia vao sb lieu tren, anh (chi) hay ket luan giup ong chu eda hang xem co nen tiep tuc ban hay khong d mdc y nghia a = 0.05 c) Gia sd nhtfng ban dime td 13 den 17 doa hong la nhufng "binh thitdng" Hay tide limng ty le nhtfng binh thircmg cua eda hang d tin cay 90% (Gia thiet rang sb hoa ban co phan phbi chuan) Mot xl nghiep due mot sb rat Idn cac san pham bang thep vdi sb khuyet tat trung binh d moi san pham la Ngufdi ta cai tien each san xuat va kiem tra 36 san pham Ket qua nhii sau : Sb khuyet tat tren san pham 6 7 Sd san pham tJdng ting Gia sd sb khuyet tat cua cac san pham co phan phbi chuan a) Hay Ode limng sb khuyet tat trung binh d mdi san pham sau cai tien, vdi tin cay 90% 153 b) Hay cho ket loan ve hieu qua cua viec cai tien san xuat d mile y nghia 0.05 Quan sat ngau nhien mot sb trUbng hop 16 thubc (rat nhieu), ta ghi nhan diTOc Tot Tam dung Hong L6 A 125 52 23 L6 B 117 61 22 L6 C 178 97 25 Hoi rang chat ItTo’ng cua 16 thubc co nhil khong ? (ket luan vdi a = 0.05) Trong mot cong ty, ngiTd’i ta chon ngAu nhien 1000 cong nhan va theo doi sb nghi cua hp mot nam Ket qua thu dupe : Nu” Nam 0-5 300 500 5-20 80 70 >20 20 30 Gio’i tinh So' nghi Vdi mile y nghia 0.01, hay kiem dinh gia thiet cho rang sif nghi viec khong phu thuoc vao gidi tinh Nghien cilu anh hildng cua hoan canh gia dinh dbi vhi tinh hinh pham toi cua tre em vi nien, ngilcfi ta thu duqc Bd hose me da chet Bd me ly hon Con ca bd me Khong pham toi 20 25 13 Pham toi 29 43 18 Hoan canh gia dlnh Tinh trang pham toi 154 Vdi mufc y nghia a = 0.05, co the ket loan la hoan canh gia dinh cua tre em doc lap vdi tinh trang pham toi hay khong ? 10 Co 90 ngtfdi dung DDT de tri benh ngoai da thi co 10 ngiidi nhidm benh; Co 100 ngtibi khong dung DDT thi co 26 ngPdi mac benh Hoi rang DDT co tac dung ngha benh ngoai da khong ? (ket luan vdri a = 0.05) 11 Trong mot vung dan ci! co 18 be trai va 28 be gai mac benh B Hoi rang ty le nhmm benh cua be trai va be gai co nhtf khong ? (ket luan vdi a = 0.05 va gia sd rang so luVng be trai va be gai vung tiio’ng dtfo'ng nhau, va rat nhieu) 12 Thong ke so tai nan lao dong tai xi nghiep, ta co cac so lieu sau : Xi nghiep So cong nhan So tai nan lao dong A 200 20 B 120 800 Vdi mdc y nghia a = 0.05, hay ket luan xem chat Itftfng cong tac bao ve an toan lao dong tai xi nghiep tren co khac khong ? 13 Doi vdi ngirbi micfc ngoai, litong huyet sac to trung binh la 138.3g/l Kham cho 80 cong nhan d nha may co tiep xuc hoa chat, thay huyet sSc to trung binh la 120g/l; S = 15g/l Td ket qua tren, co the ket luan ItfOng huyet sac to trung binh cua co ng nhan nha may hoa chat thap hon mdc chung hay khong ? Ket luan vdri a = 0.05 14 Ham litong difcfng mau cua cong nhan sau gib lam viec vdi may sieu cao tan da dude d hai thbi diem trddre va sau gib lam viec Ta co ket qua sau : Trade = 50; X = 60mg%; Sx = ; Sau n2 = 40 ; Y = 52mg%; Sx = 9.2 155 V6i mile y nghia a = 0.05, co the khing dinh ham liTtfng dufrng mau sau gid lam viec da giam di hay khong ? 15 Danh gia tac dung cua mot che an hoi diYdng ma dau hieu quan sat la so hong cau Ngufri ta dem so hong cau cua 20 ngudi trade va sau an bdi dirdng : xi 32 40 38 42 41 35 36 47 50 30 Yi 40 45 42 50 52 43 48 45 55 34 xi 38 45 43 36 50 38 42 41 45 44 Yi 32 54 58 30 60 35 50 48 40 50 Vdi mure y nghia a = 0.05, co the ket luan gi ve tac dung cua che an hoi dadng ? 16 Trong dot thi dua, phan xirdng A bao cao chat laqng san pham lam nha sau : cd 85% loai 1; 10% loai va 5% loai Ban thi dua da lay ng^u nhien tif 16 san pham chifa phan loai cua phan xifdng A 100 san pham, thay cd 80 loai 1, 13 loai va loai Vdi mdc y nghia a = 0.01, cd the ket luan gi ve bao cao cua phan xifdng A ? 156 157 TAI LIEU THAM KHAO [1] Dang Hung Thang, Md ddu ve ly thuyet xdc sudt ud cdc ling dung, NXBGD 1997 [2] Jay L Devore, Probability and statistics for engineering and the sciences, Wadsworth Publishing Company, 1995 [3] William Feller, An introduction to probability theory and its applications, Volume I, II, John Wiley & Sons, Inc 1959, 1966 [4] Alan H Kwanli, C Stephen Guynes and Robert J Pavur, Introduction to business statistics, a computer integrated approach West Publishing Company, 1996 h 158 159 MUC LUC DAI CUONG VE GIAI TICK TO HOP Tap hap Cac phep toan tap htfp Quy tSc dem giai tich to hop 11 Bai tap 13 Choong : DAI CUONG VE XAC SUAT 15 Hien tOOng ngau nhien 15 Xac suat 17 Xac suat co dieu kien 22 Looc Bernoulli 33 Dai cOOng ve bien so ng^u nhien va phifOng phap lay mau 35 Bai tap 38 Choong : BIEN SO NGAU NHIEN 43 Xac dinh bien so ng^u nhien 45 A Bien so ngau nhien red rac 48 B Bien so ng^u nhien lien tuc 51 Tham so dac trifng cua bien ngau nhien 54 Vecto ng^u nhien 63 A VectO ng&u nhien red rac 63 B Vecto ng£u nhien lien tuc 67 Tham sb dac trOng vecto ngbu nhien 69 160 Bai tap 73 Chifong : PHAN PHOI XAC SUAT 76 Phan phdi nhi thdc B(n;p) 77 Phan phdi sieu boi H(N, K,n) 80 Phan phdi Poisson 83 Phan pho'i chuan 86 Phan phoi Gamma va Chi-binh phiTOng, % 91 Nguyen ly xac suat Idn va nho 94 Bai tap 96 ChiWng : MAU THONG KE & UdC LUONG THAM SO 99 Ly thuyet lay mau 100 Ly thuyet iTdc liftfng 108 Bai tap 126 ChifPng : KIEM DINH GIA THUYET 130 Cao loai loi lam 130 So sanh cac bang so lieu 132 So sanh tham so' dac trtfng cac bang so lieu 140 Bai tap 151 TAI LIEU THAM KHAO 156 MUG LUC 157