- 1 -Tm tt cng thc - 1 -XSTK Tm tt cng thc Xc Sut - Thng K I.Phn Xc Sut 1.Xc sut c in -Cng thc cng xc sut: P(A+B)=P(A)+P(B)-P(AB). -A1, A2,, An xung khc tng i P(A1+A2++An)=P(A1)+P(A2)++P(An). -Ta c oA, B xung khc P(A+B)=P(A)+P(B). oA, B, C xung khc tng i P(A+B+C)=P(A)+P(B)+P(C). o( ) 1 ( ) PA PA = . -Cng thc xc sut c iu kin: ( )( / )( )PABPA BPB= , ( )( / )( )PABPB APA= . -Cng thc nhn xc sut: P(AB)=P(A).P(B/A)=P(B).P(A/B). -A1, A2,, An c lp vi nhau P(A1.A2..An)=P(A1).P(A2)..P( An). -Ta c oA, B c lp P(AB)=P(A).P(B). oA, B, C c lp vi nhau P(A.B.C)=P(A).P(B).P(C). -Cng thc Bernoulli:( ; ; )k k n knB knp Cpq= , vi p=P(A): xc sut bin c A xy ra mi php th v q=1-p.-Cng thc xc sut y - Cng thc Bayes oH bin c gm n phn t A1, A2,, Anc gi l mt php phn hoch caO 1 2. ; , 1,...i jnAA i j i j nA A A= u, = e + + + = O oCng thc xc sut y : 1 1 2 21( ) ( ). ( / ) ( ). ( / ) ( ). ( / ) ... ( ). ( / )ni i n niPB PA PB A PA PB A PA PB A PA PB A== = + + + o Cng thc Bayes: ( ). ( / )( / )( )i iiPA PB APA BPB=vi 1 1 2 2( ) ( ). ( / ) ( ). ( / ) ... ( ). ( / )n nPB PA PB A PA PB A PA PB A = + + +2.Bin ngu nhin a.Bin ngu nhin ri rc -Lut phn phi xc sut vi( ), 1, .i ip PX x i n = = =Ta c: 11niip== v f({a f(X) b}=iia x bP ps )ss s Xx1 x2xn Pp1p2pn - 2 -Tm tt cng thc - 2 -XSTK -Hm phn phi xc sut ( ) ( )iX ix xF x PX x p s s -K vng 1 1 2 21( . ) . . ... .ni i n niEX x p x p x p x p== = + + + 1 1 2 21( ( )) ( ( ). ) ( ). ( ). ... ( ).ni i n niE X x p x p x p x p == = + + + -Phng sai 2 2( ) ( ) VarX EX EX = vi 2 2 2 2 21 1 2 21( ) ( . ) . . ... .ni i n niEX x p x p x p x p== = + + + b.Bin ngu nhin lin tc. -f(x) l hm mt xc sut ca X ( ) 1+ =}f x dx , {a X b} ( ).baP f x dx s s =} -Hm phn phi xc sut ( ) ( ) ( )xXF x PX x f t dt= < = } -Mode 0ModX x = Hm mt xc sut f(x) ca X t cc i ti x0. -Median 1 1( ) ( )2 2exe X eMedX x F x f x dx= = =}. -K vng EX . ( ) x f x dx+= }. ( ( )) ( ). ( ) E X x f x dx += } - 3 -Tm tt cng thc - 3 -XSTK -Phng sai 2 2( ) ( ) VarX EX EX = vi 2 2EX . ( ) x f x dx+= }. c.Tnh cht - ( ) , ( ) 0 E C C Var C = = , C l mt hng s. -2( ) , ( ) EkX kEXVarkX k VarX = =- ( ) EaX bY aEX bEY + = +- Nu X, Y c lp th 2 2( ) . , ( ) EXY EX EYVaraX bY a VarX b VarY = + = +- ( ) X VarX = : lch chun ca X, c cng th nguyn vi X v EX. 3.Lut phn phi xc sut a.Phn phi Chun 2( ~ ( ; )) X N o-( ) XO= , EX=ModX=MedX= , 2VarX =-Hm mxs 22( )21( , , )2xf x e = Vi0, 1: = =221( )2xf x e= (Hm Gauss) -(a X b) ( ) ( )b aP s s = o o vi 2201( )2txx e dt =} (Hm Laplace) -Cch s dng my tnh b ti tnh gi tr hm Laplace, hm phn phi xc sut ca phn phi chun chun tc Tc vMy CASIO 570MSMy CASIO 570ES Khi ng gi Thng kMode(tm)SDMode(tm)STAT 1-Var Tnh2201( )2txx e dt =} 221( )2= }txFx e dt Shift 3 2 x ) = Shift 3 1 x ) = Shift 1 7 2 x ) = Shift 1 7 1 x ) = Thot khi gi Thng kMode 1Mode 1 Lu :( ) 0, 5 ( ) = + Fx x b.Phn phi Poisson( ~ ( )) X P -( ) XO= ,EX . odX=k -1 k VarX M = = s s-(X=k)=e ,!kP kk e- 4 -Tm tt cng thc - 4 -XSTK c.Phn phi Nh thc( ~ ( ; )) X B np-( ) {0..n} XO= , EX=np, VarX=npq, ModX=k ( 1) 1 ( 1) n p k n p + s s +-(X=k)=C . . , q p 0 ,k k n knP p q k n k=1 ,s se -Nu( 30; 0,1 0, 9; 5, 5) > < < > > n p np nqth 2~ ( ; ) ( ; ) ~ o X B np Nvi . , n p npq =o = 1(X=k) ( ), 0 ,kP f k n k~s seo o (a X s 0,1, < n p npth~ ( ; ) ( ) ~ X B np Pvinp =(X=k) e ,!kP kk ~e -Nu( 30, 0, 9, 5) > > < n p nq (X=k) e ,( )!n kP kn k~ e vinq = d.Phn phi Siu bi( ~ ( ; ; ))AX HNN n-( ) {max{0; ( )}..min{n;N }}A AX n N N O= -EX=np, VarX=npq1N nN vi ANpN= , q=1-p. - ( 1)( 1) 2 ( 1)( 1) 212 2A AN n N nModX k kN N+ + + + + += s s+ +. -(X=k)= , ( )A Ak n kN N NnNC CP k XC e O-Nu20Nn>th ~ ( ; ; ) ( ; )AX HNN n B np ~vi ANpN= . (X=k) C . . , ( ), 1k k n knP p q k X q p~e O= . - 5 -Tm tt cng thc - 5 -XSTK XY= S tm tt cc dng phn phi xc sut thng dng: n30, np20np=ANN, q=1-p n30, np 5 > , nq 5 >0,1 : Chp nhn Ho. - 1: , :o o oH H = > ( ) 0, 5 , .oxz z z no= o =o - Nuz zo> : Bc b Ho, chp nhn H1. - Nuz zos : Chp nhn Ho. Trng hp 2. ( o cha bit,30 n > ) - 1: , :o o oH H = = 2 21( ) , .2oxz z z nso o= =- Nu 2z zo> : Bc b Ho, chp nhn H1. - Nu 2z zos : Chp nhn Ho. - 1: , :o o oH H = < ( ) 0, 5 , .oxz z z nso= o =- 10 -Tm tt cng thc - 10 -XSTK - Nuz zo< : Bc b Ho, chp nhn H1. - Nuz zo> : Chp nhn Ho. - 1: , :o o oH H = > ( ) 0, 5 , .oxz z z nso= o =- Nuz zo> : Bc b Ho, chp nhn H1. - Nuz zos : Chp nhn Ho. Trng hp 3. ( o cha bit, n : Bc b Ho, chp nhn H1. - Nu ( 1; )2nt tos : Chp nhn Ho. - 1: , :o o oH H = < ( 1; ), .onxt t ns oo =- Nu ( 1; ) nt t o< : Bc b Ho, chp nhn H1. - Nu ( 1; ) nt t o> : Chp nhn Ho. - 1: , :o o oH H = > ( 1; ), .onxt t ns oo =- Nu ( 1; ) nt t o> : Bc b Ho, chp nhn H1. - Nu ( 1; ) nt t os : Chp nhn Ho. b)Kim nh t l. - 1: , :o o oH p p H p p = =2 21( ) , , .2 (1 )oo of p kz z f z nn p po o= = = - Nu 2z zo> : Bc b Ho, chp nhn H1. - Nu 2z zos : Chp nhn Ho. - 1: , :o o oH p p H p p = : Chp nhn Ho. - 1: , :o o oH p p H p p = >( ) 0, 5 , , .(1 )oo of p kz z f z nn p po= o = = - Nuz zo> : Bc b Ho, chp nhn H1. - Nuz zos : Chp nhn Ho. c)Kim nh phng sai. Trng hp 1. ( cha bit) - Nu cha cho s m cho mu c th th phi s dng my tnh xc nh s. - 2 2 2 21: , :o o oH H o = oo = o2 21( 1;1 )212 no oo _ = _ , 2 22( 1; )22 nooo _ = _, 222( 1)on s=o - Nu 2 222 21

_ > _

_ < _: Bc b H0, chp nhn H1. - Nu 2 2 21 2_ s _ s _ : Chp nhn Ho. - 2 2 2 21: , :o o oH H o = oo < o2 21 ( 1;1 )1n oo o _ = _ , 222( 1)on s _ =o - Nu 2 21_ < _ : Bc b H0, chp nhn H1. - Nu 2 21_ > _ : Chp nhn Ho. - 2 2 2 21: , :o o oH H o = oo > o2 22 ( 1; ) n oo _ = _ , 222( 1)on s _ =o - Nu 2 22_ > _ : Bc b H0, chp nhn H1. - Nu 2 22_ s _ : Chp nhn Ho. 4.Kim nh so snh tham s. a)Kim nh so snh gi tr trung bnh. Trng hp 1. (1 2, o o bit) - 1 2 1 1 2: , :oH H = = 1 22 22 2 1 21 21( ) ,2x xz z zn no o= =o o+ - 12 -Tm tt cng thc - 12 -XSTK - Nu 2z zo> : Bc b Ho, chp nhn H1. - Nu 2z zos : Chp nhn Ho. - 1 2 1 1 2: , :oH H = < 1 22 21 21 2( ) 0, 5 ,x xz z zn no= o =o o+ - Nuz zo< : Bc b Ho, chp nhn H1. - Nuz zo> : Chp nhn Ho. - 1 2 1 1 2: , :oH H = > 1 22 21 21 2( ) 0, 5 ,x xz z zn no= o =o o+ - Nuz zo> : Bc b Ho, chp nhn H1. - Nuz zos : Chp nhn Ho. Trng hp 2. (1 2, o ocha bit, 1 230 n n ,> ) - 1 2 1 1 2: , :oH H = = 1 22 22 2 1 21 21( ) ,2x xz z zs sn no o= =+ - Nu 2z zo> : Bc b Ho, chp nhn H1. - Nu 2z zos : Chp nhn Ho. - 1 2 1 1 2: , :oH H = < 1 22 21 21 2( ) 0, 5 ,x xz z zs sn no= o =+ - Nuz zo< : Bc b Ho, chp nhn H1. - Nuz zo> : Chp nhn Ho. - 1 2 1 1 2: , :oH H = > 1 22 21 21 2( ) 0, 5 ,x xz z zs sn no= o =+ - 13 -Tm tt cng thc - 13 -XSTK - Nuz zo> : Bc b Ho, chp nhn H1. - Nuz zos : Chp nhn Ho. Trng hp 3. (1 2o= ocha bit, 1 2, 30 n n < ) - 1 2 1 1 2: , :oH H = = 1 21 2( 2; )221 2,2 1 1( )n nx xt tsn no+ oo =+, vi 2 22 1 1 2 21 2( 1). ( 1).2n s n ssn n + =+ - Nu 1 2( 2; )2n nt to+ > : Bc b Ho, chp nhn H1. - Nu 1 2( 2; )2n nt to+ s : Chp nhn Ho. - 1 2 1 1 2: , :oH H = < 1 21 2( 2; )21 2,1 1( )n nx xt tsn n+ oo =+, vi 2 22 1 1 2 21 2( 1). ( 1).2n s n ssn n + =+ - Nu 1 2( 2; )2n nt to+ < : Bc b Ho, chp nhn H1. - Nu 1 2( 2; )2n nt to+ > : Chp nhn Ho. - 1 2 1 1 2: , :oH H = > 1 21 2( 2; )21 2,1 1( )n nx xt tsn n+ oo =+, vi 2 22 1 1 2 21 2( 1). ( 1).2n s n ssn n + =+ - Nu 1 2( 2; )2n nt to+ > : Bc b Ho, chp nhn H1. - Nu 1 2( 2; )2n nt to+ s : Chp nhn Ho. b)Kim nh so snh t l. 1 2 1 21 21 2 1 2, ,k k k kf f fn n n n+= = =+ - 1 2 1 1 2: , :oH p p H p p = =1 22 21 21( ) ,2 1 1(1 ).( )f fz z zf fn no o= = + - Nu 2z zo> : Bc b Ho, chp nhn H1. - Nu 2z zos : Chp nhn Ho. - 14 -Tm tt cng thc - 14 -XSTK - 1 2 1 1 2: , :oH p p H p p = : Chp nhn Ho. - 1 2 1 1 2: , :oH p p H p p = >1 21 2( ) 0, 5 ,1 1(1 ).( )f fz z zf fn no= o = + - Nuz zo> : Bc b Ho, chp nhn H1. - Nuz zos : Chp nhn Ho. c.Kim nh so snh phng sai. - 1 2, cha bit nn tnh s1 v s2 t mu (s dng my tnh) nu bi cha cho. - 2 2 2 21 2 1 1 2: , :oH H o = oo = o-211 1 2 2 1 222, ( 1; 1;1 ) , ( 1; 1; )2 2sf f f n n f f n nso o= = = -Nu 12f ff f<

>: Bc b Ho, chp nhn H1. -Nu 1 2f f f s s : Chp nhn Ho. - 2 2 2 21 2 1 1 2: , :oH H o = oo < o- 211 1 222, ( 1; 1;1 )sf f f n ns= = o-Nu 1f f < : Bc b Ho, chp nhn H1. -Nu 1f f s : Chp nhn Ho. - 2 2 2 21 2 1 1 2: , :oH H o = oo > o- 212 1 222, ( 1; 1; )sf f f n ns = = -Nu 2f f > : Bc b Ho, chp nhn H1. -Nu 2f f s : Chp nhn Ho. 5.H s tng quan mu v phng trnh hi quy tuyn tnh mu. - 15 -Tm tt cng thc - 15 -XSTK a. H s tng quan mu: 1 1 12 2 2 21 1 1 1( ) ( )n n ni i i ii i in n n ni i i ii i i in x y x yrn x x n y y= = == = = == Phng trnh hi quy tuyn tnh mu:xxy A B = +vi 1 1 12 21 1( )n n ni i i ii i in ni ii in x y x yBn x x= = == == v 1 1.n ni ii iy B xAn= == . b. Trong trng hp s dng bng tn s: Ta tnh theo cng thc thu gn nh sau: H s tng quan mu: 1 1 12 2 2 21 1 1 1( ) ( )k k ki i i i i i ii i ik k k ki i i i i i i ii i i in n x y n x n yrn n x n x n n y n y= = == = = == Phng trnh hi quy tuyn tnh mu:xxy A B = +vi 1 1 12 21 1( )k k ki i i i i i ii i ik ki i i ii in n x y n x n yBn n x n x= = == == v 1 1.k ki i i ii in y B n xAn= == . ix1x2x kxiy1y2y kyin1n2n kn- 16 -Tm tt cng thc - 16 -XSTK c. S dng my tnh b ti tnh h s tng quan mu v phng trnh hi quy tuyn tnh mu: Tc vDng CASIO MSDng CASIO ES Bt ch nhp tn sKhng cn Shift Mode+4 1 Khi ng gi Hi quy tuyn tnh Mode(tm)REG Lin Mode(tm)STAT A+BX Nhp s liu 1x, 1yShift , 1nM+ kx, kyShift , knM+ 1in=th ch cn nhn ix, iyM+ XYFREQ 1x= kx= 1y= ky= 1n= kn= Xa mn hnh hin thACAC Xc nh: -H s tng quan mu (r) -H s hng: A -H s n (x): B Shift 2 3 = Shift 2 1 = Shift 2 2 = Shift 1 7 3 = Shift 1 7 1 = Shift 1 7 2 = Thot khi gi Hi quyMode 1Mode 1 Lu : My ES nu kch hot ch nhp tn s phn L thuyt mu ri thkhng cn kch hot na. .