수-적분4-1(186~240)eps교

Transcription

1 .. stat-istics 0 86

2 ;I+!( -ß) - 87

3 ~4 P(=x) ~ 4 ~47 E()V()() 3 ~6 48 ~5 B(p) 7 ~0 53 ~56 ~3 57 ~65 N(m )N(0) ~69 70 ~74 ç 4 ~9 ~5 75 ~78 6 ~7 88

4 79 ~8 pˆ 8 ~ ~ ~3 07 /3 38~4 P(=x) 89

5 08 statistic statistics statistics status state statistics AchewallG7977 statistik

6 GautJ60674 PettyW63687 HallyeyE65674 NeumaK QuételetLAJ GaltoF89 PeasoK GossetWS FisheRA

7 x ; 0º0;_x= x= ;0{0;= _;0!0;+50000_;0@0;+0000_;0%0; = ;4!; ;5@; (a+b) a b 45 9

8 P(=x) MP3 CD M C M MP3 C CD S={(M M)(M C)(C M)(C C)} S S M M (M, M) 0 (M, C) (C, M) (C, C) MP3 S M 0 {(M C) (C M)} S {=} {=} P(=) P(=)=;!; {(M M)} {(C, C)} P(=)=;4!;P(=0)=;4!; MP3 MP3 M S M MM 0 MC CM CC MP3 ;4@;=;!; P(=)=«C p«q«=03y P(=)=pq =03y P(=)= P(=)= ÂC ÂC«NC e -l l! =03y =03y P(=)=;!; =03y P(=)= C p q =kk+k+y 93

9 x x p P(=x )=p (i=3y) x x x yx«p p p yp«p(=x ) S R :S3 R x (s)=x S R S s {=x} P(=x) s x x x x yx«x P(=x )=p (i=3y) p (0 p ) p +p +y+p«= x p (i=3y) P(=x ) P(=x )=p (i= 3y) x <x <y<x«x x x y x«p(=x) p p p y p«j x xδ {x xδ} P(x xδ) P(x xδ)=; p k=i P( ) =P(=)+P(=) P(=x) 0 ;4!; ;!; ;4!; x x x y x«p(=x) p p p y p«x x x y x«p(=x ) (i=3y ) P(=x ) 0 P(=x ) (i=3y) yy [ ;I+!P(=x )= yy P(=x ) b= ;!;+;3!;+a= a=;6!; =;!;+;4!;=;4#; 3 ab P(=x) ;!; ;3!; a b 95 Gudbegiffe de Wahschei lichkeitsechug

10 P(=x) ;6!; ;6!; ;6!; ;6!; ;6!; ;6!; P( 5) =P(=)+P(=3)+P(=4)+P(=5) =;6!;+;6!;+;6!;+;6!; P(=k) C C = C 5 3 P(æ) 0 Cº_ C P(=0)= =; 8; yy C C _ C P(=)= =;!8%; yy C C _ Cº P(=)= =;!8);=; 4; yy C 0 P(=x) ; 8; ;!8%; ; 4; P(æ)=P(=)+P(=)=;!8%;+; 4;=;@8%; yy` yy` =;6$;=;3@; P( 5) 3 P( ) 3 0 Cº_ C P(=0)= =; 0; C C _ C P(=)= =;5#; C C _ Cº P(=)= =; 0; C P( )=P(=)+P(=) P( )=;5#;+; 0;=; 0; 0 P(=x) ; 0; ;5#; ; 0; ab P(=k)=a {;!;} a P(=x) 0 k= P(=k)= 0 k= 0 a a a=;6!;b= (k=3y0) ;A;{-{;!;} a {;!;} } = = -;!; a=;!0)@3$; 3a b 95

11 a+b ax+b E() V() () P(=)=;3 0; P(=3)=;3 0; P(=5)=;3 0; P(=)=;3 0; P(=4)=;3!0); 3 ;3 0; 6 ;3 0; 3 7 ;3 0; _3+_6+3_7+4_0+5_4 = 30 =_;3 0;+_;3 0;+3_;3 0;+4_;3!0);+5_;3 0; =_P(=)+_P(=)+3_P(=3) =_P(=)+4_P(=4)+5_P(=5) ;3!0); ;3 0; m _3+_6+3_7+4_0+5_4 m= =;; 5 ;; 30 =;3 0;[{-;; 5 ;;} _3+{-;; 5 ;;} _6+{3-;; 5 ;;} _7 +{4-;; 5 ;;} _0+{5-;; 5 ;;} _4] =;; 7º5 ;; 07 ' 07 =æ = = ' '3 5 N x x x yx«m m= ;K+! x N N x f f +f +f +y+f«=n m m= ;K+! x f k=3y N x x x y x«f f f y f«x x x y x«f f f y f«96

12 (x ) x x x y x«(f ) m f f m= ;I+!x f N f x N f =p N P(=x) x p x p f x p y y y f«x«p«n E() E Expectatio m mea x x x y x«p(=x) p p p y p«x p +x p +y+x«p«=;i+! x p E() m E()=x p +x p +y+x«p«=;i+!x p E() E()=x p +x p +y+x«p«=;i+! x p m f m= ;I+! x f =;I+! x N N 0 m= ;I+! x p P(=x) ;4!; ;!; ;4!; E() E()=0_;4!;+_;!;+_;4!;= 908Studet (PeasoK xy t E() P(=x) ;6!; ;6!; ;6!; ;6!; ;6!; ;6!; E()=_;6!;+_;6!;+3_;6!;+4_;6!; +5_;6!;+6_;6!; E()=;&; 97

13 3 E()=m x x x y x«p(=x) p p p y p«p(=x) 0 ;4!; ;!; ;4!; E()=0 ;4!;+ ;!;+ ;4!;= -m (-m) (x -m) p +(x -m) p +y+(x«-m) p«=;i+! (x -m) p V()=E( )-{E()} =0 ;4!;+ ;!;+ ;4!;- =;!; ()=" V()=Æ;!;= ' V() V Vaiace V() V()=E((-m) ) V() E()=3E( )=9 V()=;I+! (x -m) p () (sigma) Stadad deviatio s V()=;I+! (x -mx +m )p V()=;I+!x p -m ;I+! x p +m ;I+!p V()=;I+!x p -m m+m V()=E( )-m V() () ()=" V() 00 3 E()=;I+!x p P(=x) ;8!; ;8#; ;8#; ;8!; E()=0_;8!;+00_;8#;+00_;8#;+300_;8!;=50 V()=E( )-{E()} V()=0 _;8!;+00 _;8#;+00 _;8#;+300 _;8!;-50 E()=m V() () V()=;I+!x p -m =E( )-{E()} ()=" V() 3 V()=7500 yy` ()=" V()='ƒ7500=50'3 yy` 4 3 m = ;I+!(x -m) f N f x N f =p N = ;I+!(x -m) f = ;I+!(x -m) N =;I+! (x -m) p 3 (x ) (f ) P(=x) x f x p x f x p x f x p y y y y x«f«x«p«f N N V() () V()=E( )-{E()} =9-9=0 ()=" V(çΩ)='åß0 3 0 P(=x) ;7!; ;7$; ;7@; E()=0_;7!;+_;7$;+_;7@;=;7*; E( )=0 _;7!;+ _;7$;+ _;7@;=;; 7 ;; V()=E( )-{E()} =;; 7 ;;-;4^9$;=;4@9); ()=Æ;4@ 9);= '5 7 98

14 4 Y=a+bab Y E()=m E(Y)=E(ax+b) =ae()+b V(Y)=E({Y-E(Y)} ) =E({ax+b-(am+b)} ) =E((ax-am) ) =E(a x -a mx+a m ) =a E( )-a me()+a m =a E( )-a {E()} =a [E()-{E()} ] =a V() 4 Y=a+bab Y x x x y x«p(=x) p p p y p«y y =ax +b(i=3y) P(Y=y )=P(Y=ax +b)=p(=x )=p Y Y y y y y y«p(y=y) p p p y p«y E(Y)=;I+!y p =;I+! (ax +b)p =a;i+!x p +b;i+!p =ae()+b V(Y)=;I+!{y -E(Y)} p =;I+! [(ax +b)-{ae()+b}] p V(Y)=;I+![a{x -E()}] p =a ;I+!{x -E()} p =a V() (Y)=" V(Y)=" a V()= a () ab E(a+b)=aE()+b V(a+b)=a V(), (a+b)= a () E()=4V()= -+ E(-+)=-E()+=(-)_4+=-7 V(-+)=(-) V()=4_=8 (-+)= - ()=_'=' m =;I+!(x -m) p f(x) x m=x f(x)=;i+!(x -x) p f(x)=;i+!(x-x ) p x f'(x)=;i+!(x-x )p ={x;i+! p -;I+! x p } =(x-m) f(x) x=m 99

15 3 a+b E()= V()= 3+ E(3+)=3E()+=3_+=8 V(3+)=3 V()=9_=9 4 E()=V()= 3+ E(+3) V(+3) P(=x) ;6!; ;6!; ;6!; ;6!; ;6!; ;6!; 43 a+b E()=_;6!;+_;3!;+3_;3!;+4_;6!;=;%; V()= _;6!;+ _;3!;+3 _;3!;+4 _;6!;-{;%;} E()=_;6!;+_;6!;+3_;6!;+4_;6!;+5_;6!;+6_;6!;=;&; V()= _;6!;+ _;6!;+3 _;6!;+4 _;6!;+5 _;6!;+6 _;6!;-{;&;} =;#%; E(+3)=E()+3=_;&;+3=0 yy` V(+3)= V()=4_;#%;=: 3 : yy` P(=x) ;6!; ;3!; ;3!; ;6!; E(6+5) V(6+5) V()=;!!; E(6+5)=6E()+5=6_;%;+5=0 V(6+5)=6 V()=36_;!!;= a+b E()=0_;3 5;+_;3!5@;+_;3!5*;+3_;3 5;=;; 7 ;; E( )=0 _;3 5;+ _;3!5@;+ _;3!5*;+3 _;3 5; E( )=;; 7 ;; 0 3 P(=x) ;3 5; ;3!5@; ;3!5*; ;3 5; V()=E( )-{E()} =;; 7 ;;-;; 4 9 ;;=;4@9$; E(7-3)=7E()-3=7_;; 7 ;;-3=9 V(7-3)=7 V()=49_;4@9$;=4 E() V() E()=V()=;!; E()=;8%;V(8+7) a+b+;!;= a+b=;!; a b P(=x) b a ;!; E()=ab+ba+;!;=;8%; ab=; 6; E( )=a b+b a+ _;!;=ab(a+b)+;!;=;3!&; V()=E( )-{E()} =;3!&;-;6@4%;=;6ª4; V(8+7)=8 V()=64_;;6ª4;;=9 00

16 3 B(p) 3 P(=x) 0 3 C {;4!;} {;4#;} =;6@4&; A p A 0y P(=)=«C p q - (q=-p=0y) P(=) 0 y y «Cºp q««c p q««c p q«y «C p q«y «C«p«q (p+q)«(p+q)«=«cºp q«+«c p q - +y+«c p q - +y+«c«p«q 3 ;4!; ;4#; ab S={ab} (a) (b) 0 (a)= (b)=0 p P(=x) q p P(=x) ;6@4&; ;6@4&; ;6ª4; ;6 4; E() E()=0_;6@4&;+_;6@4&;+_;6ª4;+3_;6 4;=;4#; (q=-p) E()=0 q+ p=p E( )=0 q+ p=p V()=E( )-{E()} =p-p =p(-p)=pq 0

17 p Y P(Y=0)=-p=qP(Y=)=p Y E(Y)=pV(Y)=pq p B(p) Y E()=E{ Y }= E(Y ) E()= p=p V()=V{ Y }= V(Y ) V()= i= i= pq=pq Y YΔ i= i= i= i= V(Y +YΔ)=V(Y )+V(YΔ)) i= B( p) B Biomial distibutio «C! =!(-)! (-)! = (-)!(-)! = «C (-) «C =(-)«C =(-)«C =(-)«C ( æ) p B( p) B( p) 5 B {5 ;!;} B( p) 0 5 B( p) P(=)=«C p q - (q=-p =0 y ) E()=;R+) P(=)=;R+! «C p q - E()=;R+! «C p q - =p;r+!«c p - q (-)-(-) E()=p(p+q) - =p V()=E( )-m =;R+) «C p q - -m V()=;R+! (-)«C p q - +;R+! «C p q - -m V()=;R+! (-)«C p q - +p-(p) V()=;R+@ (-)«C p q - +p-(p) V()=(-)p ;R+@ «C p - q (-)-(-) +p-(p) V()=(-)p (p+q) - +p-(p) V()=(-)p +p-(p) =-p +p V()=p(-p)=pq K M M B{ } N ÂC P(=x)= x -ÂC«-x (, x=0y) NC«M E()=_ N M M N- V()= {- }_{ } N N N- 4 ;!; 0 B{0;!;} ;6!; 5 B{5;6!;} 0

18 p.7 B(p) E()V() (p+q=) p.7 B( p) E()=p V()=pq()='ƒpq (q=-p) E() (x+q)«(x+q)«= «C x q - yy x (x+q) - = «C x q - yy x E()= x(x+q) - = «C x q - yy «C p q - x=p p(p+q) - = «C p q«e()=p - V() V()=E( )-{E()} x (x+q) - +(-)x(x+q) - = «C x - q - yy x x(x+q) - +(-)x (x+q) - = «C x q - yy E( )= «C p q - x=p p(p+q) - +(-)p (p+q) - = «C p q - p+ p -p = «C p q«v() =0 =0 =0 =0 =0 E( )=p+ p -p =0 =0 =0 =0 - V()=E( )-{E()} V()=p+ p -p -(p) V()=p(-p) V()=pq =0 =900p=;3!; B {0 ;5!;} E()=0_;5!;= V()=0_;5!;_;5$;=;5*; ()=Æ 0_;5!;_;5$;= ' 0 5 B {00 ;5!;} B {300 ;4!;} 900 B{900 ;3!;} E()=900_;3!;=300 V()=900_;3!;_;3@;=00 ()="çv()=' 00=0' yy` yy` yy` 4 5 B{00;5!;} =00p=;5!; E()=00_;5!;=0 V()=00_;5!;_;5$;=6 ()='å6=4 B{300;4!;} =300p=;4!; E()=300_;4!;=75 V()=300_;4!;_;4#;= ()=æ 5 4 =;; ;;

19 5 B(000.7) =00p=0.7 E()=00_0.7=40 V()=00_0.7_0.3=4 ()=" V(ç)='4ß 5 p lim lim =p yy =p e N >N -p <e -p <h -p <h -p <h h lim P{ -p <h}= yy` % 00 ;6!; P(=) P(=)=«C {;6!;} {;6%;} =3050 P { -;6!; <0.}=0.784P { -;6!; <0.}= P(=0) P(=) P(=) P(=3) P(=4) P(=5) P(=6) P(=7) P(=) P(=8) P(=9) ;6!; P(=0) P(=) 0. P(=) P{ -;6!; <0.} P(=3) P(=4) P(=5) P(=6) =0 -;6!; <0. 0 P(=7) P(=8) 0.66y<<.66y P(=9) P { -;6!; <0.}=P(=)+P(=)= =0 -;6!; < y<<3.66 y =3 P{ -;6!; <0.6} 0 =P(=)+P(=)+P(=3) = =0.769 =30 -;6!; < y<<9.799 y =3y

20 P{ -;6!; <0.6} 30 =P(=)+P(=)+P(=3) +P(=4)+P(=5)+P(=6) +P(=7)+P(=8)+P(=9) = =0.976 =50 -;6!; < y<<6.333 y =3y6 P{ -;6!; <0.6} 50 =P(=)+P(=)+P(=3)+P(=4) +P(=5)+P(=6)+P(=7) +P(=8)+P(=9)+P(=0) +P(=)+P(=)+P(=3) +P(=4)+P(=5)+P(=6) = =0.998 P{ - <0.6} 6 4 P { -;6!; <0.} ;6!; 6 ;6!; A p A p A p A h lim P { -p <h}= P{ -;6!; <0.6} = % 00 0 % 00 E()=0V()=9()=3 B(000.8) =00p=0.8 E()=00_0.8=80 V()=00_0.8_0.=6 ()=" V()='å6=4 0 % 8 «C (0.) (0.8) - B(0.) =0 ()=" V()='ƒ_0ƒ._ 0.ß8=8 =400 «C (0.) (0.8) - =E() =0 E()=400_0.=80 05

21 4 5 3 x x x P(x x )=;5!;(x -x ) 3 P(3 5)=;5!;(5-3) =;5@; 0 5 [x x ](0 x x 5) [x x ] (x -x ) P(x x )=k(x -x ) k [0 5] y =P(0 x 5)=k(5-0) 5 k=;5!; P(x x )=;5!;(x -x ) (0 x x 5) 3 P(3 5)=;5!;(5-3)=;5@; f(x)= (a x b) b-a f(x)= e - x h (xæ0) h f(x)= e - (x-m) (- <x< ) ' p f(x)= x k- e - x h (x>0) h C(k) f(x)= (x>0) x h {+ } k+ h O x x 5 x 06

22 6 6 [a b] [a b] f(x) f(x) [] y=f(x) (a x b) x f(x)æ0 [] y=f(x) x x=ax=b :Ab f(x)dx= x [3] P(x x )=: f(x)dx (a x x b) x x P( x) x (cumulative distibutio fuctio c.d.f) F(x) F(x)=P( x) P(a b)=:ab f(x)dx f(x) (pobability distibutio fuctio p.d.f) f(x) F(x)=P(a b) =:Ab f(x)dx y y=f(x) P(x x ) O a x x b x P(=k)=P(k k)=:kk f(x)dx=0 P(a b)=p(a <b)+p(=b)=p(a <b) P(a b)=p(a< b)=p(a<<b) f(x)=ax (0 x 4) a P(0 ) :)4 f(x)dx=:)4 axdx=[;a;x ]4)=8a= a=;8!; P(0 )=:) ;8!;xdx=[; 6;x ])=;4!; f(x)=ax (0 x 3) a P( ) 8 9 f(x)=ax (0 x 3) a a :)3 f(x)dx=:)3 ax dx=[ x ]3) 3 =9a= a=;9!; f(x)=;9!;x (0 x 3) P( )=:! ;9!;x dx P( )=[; 7;x ]! P( )=; 7; 847 (849) p(x) 07

23 9 30 f(x)=ke kx (0 x ) k f(x)=;[k; ( x e) k P( 'e ) :!e f(x)dx= : ;[!;dx =l x +C P( 'e ):!e ;[K; dx=[klx]e!=k-0= :) f(x)dx=:) ke kx dx k= P( 'e )=:! 'e ;[!; dx=[lx]! 'e =;!;-0=;!; yy` yy` =[e kx ]) =e k -= k=l f(x)=ke kx (0 x ) k P{0 ;!;} P{0 ;!;}=:) ;!; l e x l dx =:) ;!; l x dx =[ x ] ;!; ) ='- 7 [a b] f(x) E() V() () f(x) (a x b) E()=:Ab xf(x)dx V()=E((-m) )=:Ab (x-m) f(x)dx 7 y=f(x)(a x b) E() V()=:Ab x f(x)dx-m (m=e()) ()=" V() E()=:Ab xf(x)dx=m V() V() =E((-m) ) =:Ab (x-m) f(x)dx =:Ab (x -mx+m )f(x)dx =:Ab x f(x)dx-:ab mxf(x)dx+:ab m f(x)dx =:Ab x f(x)dx-m:ab xf(x)dx+m :Ab f(x)dx =:Ab x f(x)dx-m m+m =:Ab x f(x)dx-m () ()=" V(çΩ) ()=æ:a b x f (x) dx -m f(x) a Dp=P(a a+dx)=:a a+dx f(x)dx=f(x )Dx x (a<x <a+dx) = f(x ) Dx 0 x a = f(a) f(a) =a f(a) P(=a) =a P(=a) P(=a)= lim P(a a+dx) Dx 0 P(=a)= lim :A a+dx f(x)dx=:aa f(x)dx=0 Dx 0 Dp Dx lim Dx 0 Dp Dx 08

24 8 30 E()V()() E()=:) xf(x)dx=:) x 3x dx E()=:) 3x dx=[;4#;x ])=;4#; f(x)=x (0 x ) E()=:) xf(x)dx E()=:) x dx=[;3@;x ])=;3@; yy` V()=:) x f(x)dx-m V()=:) x dx-{;3@;} V()=:) x f(x)dx-m V()=:) x 3x dx-{;4#;} V()=:) 3x dx-;ª6; 3 E()=[;!;x ])-;9$;=; 8; ()=" V()=Æ ; 8;= ' 6 yy` yy` f(x)=3x (0 x ) V()=[;5#;xfi ])-;ª6; V()=;8 0; ()=" V(çΩ)=Æ ;8 0;= 'ß5 0 k :_! f(x)dx=:_! kx dx :_! f(x)dx=[ k=;#; x ]_!=;3@;k= E()V() k 3 E()=:_! xf(x)dx=:_! x ;#;x dx E()=:_! ;#;x dx=[;8#;x ]_!=0 V()=:_! x f(x)dx-m V()=:_! x ;#;x dx-0 V()=:_! ;#;x dx V()=[; 0;xfi ]_!=;5#; f(x)=kx (- x ) k f(x)= ( x k) x k f(x)=e (0 x l) E()=:) l xf(x)dx=:) l x e dx E()=[xe ] l ) -:) l e dx E()= l-[e ] l ) E()= l - k=e 09

25 380 ~ ~ (g) 370 ~ ~ ~ ~ ~40 5 (g) N(m ) N(0) 370 ~ N(m ) N Nomal distibutio 430 ~ f(x)= e - (x-m) (- <x< ) 'ƒp m (>0) E()=m V()= m (>0) N(m ) (g) 400 ~ ~ ~ ~440 3 N(74 ) N(03 ) E()=0()=3 0

26 8 A p m f(x)= e - (x-m) (- <x< ) 'ßp y=f(x) x=m x=m x 'ßp x=m x : - f(x)dx= mx 5 N(5 ) 7 4 N(0 9) f(x)= e - (x-m) (- <x< ) 'ƒp [] x=m f(x) p [] x=m x ' p [3] x :_'" f(x)dx= m y=f(x) f(x) =0.8 = = O 3 m x f(x) O x m m x O m m=0 m= m=4 m x f()= e - (x-m) (- <x< ) 'ßp E() V() x-m=y E()= (y+m)e - y : dy 'ßp - E()= ye - y dy+ e - y : m : dy 'ßp - 'ßp - A= e - y : y dy =t 'ßp - ' A= : e -t dt 'ßp - A A = : e -x dx : e -y dy 'ßp - 'ßp - A = : : e -(x +y ) dxdy p - - x=cos hy=si h A = :) p dh:) e - d p A = p[-;!;e - ] ) p A = A= ( A>0) E()= [-e - y ] +m =m 'ßp - V()= (x-m) e - (x-m) : dx 'ßp - V()= y e - y : dy 'ßp - V()= (- y) {- e - y : y } dy 'ßp - V()= [[- ye - y ] 'ßp V()= 'ßp= 'ßp - : - + e - dy] y

27 9 N(m ) Y=a+b N(am+b (a) ) -m Z= E(Z)=0V(Z)= Z N(0) f(z)= e - z (- <z< ) 'ßp f(x)= e - (x-m) (- <x< ) 'ßp m=0= 9 N(m ) 0 N(0 ) Z N(0 ) Z f(z)= e - 'ƒp (- <z< ) z Z 0 z f(z) P(0 Z z)=:)z f(z)dz O z z Z N(0 ) f(z) -z O z z P(-z Z 0)=P(0 Z z) z y y 0.08 y P(Zæ0)=0.5 P(0 Z.5)= P(-.34 Z 0)=P(0 Z.34) = P(-3. Z 3.) =P(-3. Z 0)+P(0 Z 3.) =P(0 Z 3.) =_0.499=0.998 f(x)= e - x a (x>0) a (Expoetial distibutio) P(0 Z.65)= P(0 Z.96)= P(-.58 Z.58)=P(0 Z.58)=_0.495=0.990 Z P(Zæ0) P(0 Z.5) P(-.34 Z 0) P(-3. Z 3.) E() E()= xf(x)dx= xe - x :) :) a dx a E()= [-ax e - x ] - {-ae - x a :) a }dx a 0 a E()=- [a e - x a ] =a a 0 V() E(x )= x f(x)dx= x e - x :) :) a dx a E(x )= [-ax e - x ] - {-axe - x a :) a }dx a 0 a E(x )=:) xe - a dx E(x )=[-axe - a ] -:) {-ae - a }dx E(x )=[-a e - a ] =a x x x 0 0 V()=E( )-{E()} =a -a =a x

28 Z N(0 ) P(Z.58) P( Z ) P(Z.58)=0.5+P(0 Z.58) f(z) P(Z.58)= P(Z.58)=0.995 yy` N(m ) Z= N(0 ) -m P( Z )=P(0 Z )-P(0 Z ) P( Z )= P( Z )=0.359 yy` f(z) O.58 z N(3 ) m=3= -m Z= = -3 O z 4 N(5 3 ) 3 Z N(0 ) P(Z -) P(-.96 Z.58) N(7 3 ) P(4 0) E(a+b) =ae()+b (a+b)= a () N(m ) -m Z= -m m E(Z)=E { }= E()- =0 -m (Z)={ }= ()= Z 0 N(0 ) N(m ) N(0 ) Z= -m 5-7 Z= 3 P(4 0)=P{ } P(4 0)=P(- Z ) P(4 0)=P(0 Z ) P(4 0)=_0.343 P(4 0)=0.686 yy` f(z) - O N(0 5 ) P( 5) P(5 0) z 3 f (z) P(Z -) =0.5-P(- Z 0) =0.5-P(0 Z ) = =0.587 P(-.96 Z.58) =P(0 Z.96) =+P(0 Z.58) = = O z f (z) -.96 O.58 z 3 3 N(53 ) m=5=3 -m Z= = Z= P( 5)=P{ } 5 5 P( 5)=P(Z 3) P( 5)=0.5+P(0 Z 3) P( 5)= = P(5 0)=P{ } P(5 0)=P(- Z ) P(5 0)=P(0 Z )+P(0 Z ) P(5 0)= =

29 N(m ) -m Z= P( -m <) -m =P { <} =P( Z <) =P(0 Z<) =_0.343=0.686 ( 68.3 %) P( -m <)= ( 95.4 %) P( -m <3)= ( 99.7 %) 68.3 % 95.4 % 99.7 % P(æ80)=P { -7 æ 80-7 } 5 5 P( 80)=P(Zæ.6) P( 80)=0.5-P(0 Z.6) P( 80)= P( 80)= % yy` P(67 87)=P { } P(67 87)=P(- Z 3) P(67 87)=P(0 Z )+P(0 Z 3) P(67 87)= P(67 87)= _0.84=40 yy` m- m m+ x m- m m+ x m-3 m m+3 x 7 40 g 0 g 400 g (P(0 Z )=0.343) 6 N(m ) P( -m <.96) P( -m <.58) cm 5cm 75 cm % (P(0 Z.94)=0.4738) Z= -m % N(7 5 ) Z= -7 5 B { ;6!;} P(=)=«C {;6!;} {;6%;} - (=0 3 y ) =03050 P(=) P(=) Z= P( -m <.96) =P{M =P( Z <.96) M<.96} =P(-.96<Z<.96) =P(0 Z<.96) = P( -m <.58) =P{M =P( Z <.58) M<.58} =P(-.58<Z<.58) =P(0 Z<.58) = m -m -m f (z) f (z) -.96 O.96 z -.58 O.58 z 34 N(400 ) -40 Z= 0 P(æ400)=P(Zæ-)=P(0 Z )+0.5 = =0.843 N(65.35 ) Z= 5 P(æ75)=P(Zæ.94)=0.5-P(0 Z.94) = = cm.6% 4

30 0 (de Moive, A. ; 667~754) 733 (a+b)«(laplace, P. S. ; 749~87) - (x-p) pq lim «C p q«= e 'åp' pq B(p) -p Z= N(0) ' pq P(x x ) x -p -p x -p =P{ } ' pq ' pq ' pq x -p x -p =P{ Z } ' pq ' pq P(x x )=P(x <<x ) 0 9 P(x=) 0.3 =0 0. =30 =50 0. O B( p) B( p) p 'ƒpq N(p pq) (q=-p) B( p) N(p pq) (q=-p) pæ5 (-p)æ5 B{64 ;!;}m m=64_;!;=3 =Æ 64_;!;_;!;=4 N(3 4 ) B {00 ;4!;} N(a b) ab 3 4 B{00;4!;} m m=00 ;4!;=5 =00 ;4!; ;4#;=;; 4 ;; N{5;; 4 ;;} a=5b=;; 4 ;; B{30;!;} P(=7)= B(p) P(x x ) P(x x ) =P(x -0.5 x +0.5) x -0.5-p -p x -0.5-p =P{ } ' pq ' pq ' pq x -0.5-p x -0.5-p =P{ Z } ' pq ' pq Z pæ5 5

31 B(4000.8) m m=400_0.8=30='4ƒ00_0.8_0.=8 N(308 ) -30 Z= P(æ30)=P{ æ } 8 8 =P(Zæ-.5) =0.5+P(0 Z.5) = = m=p=' pq ( P(0 Z )=0.477) 00 B{00 ;!;} m m=00_;!;=50=æ 00_;!;_;!;=5 N(50 5 ) Z= P(æ40)=P { -50 æ } 5 5 P(æ40)=P(Zæ-) P(æ40)=0.5+P(0 Z ) P(æ40)= P(æ40)= yy` 0 80 % (P(0 Z.5)=0.3944) N(0 ) -0 Z= P(7 )=P{ } P(7 )=P(-.5 Z ) P(7 )=P(0 Z.5)+P(0 Z ) P(7 )= = B{400;5!;} m=400 ;5!;=80=Æ40 0 ;5!; ;5$;=8 N(808 ) -80 Z= P(7 96)=P{ } P(7 96)=P(- Z ) P(7 96)=P(0 Z )+P(0 Z ) P(7 96)= =0.885 N(0 ) P(7 ) B {400 ;5!;} P(7 96) N(405 ) P(43 44) B(p) ººC {;3!;} {;3@;} 800- (P(0 Z.5)=0.4938) z P(0 Z z) B{800;3!;} m=600=0 N(6000 ) Z= 640 = = ººC {;3!;} {;3@;} 800- =P( ) =P(- Z ) =

32 a +a+;a;= a +3a-=0(a+)(a-)=0 a=;!; ( 0 a ) E()=(-)_;4!;+0_;!;+_;4!;=0 E( )=(-) _;4!;+0 _;!;+ _;4!;=;!; V()=E( )-{E()} =;!;-0=;!; 50 A B{50 ;5!;} E()=50_;5!;=0V()=50_;5!;_;5$;=8 3 E()=:) x(4x-3x )dx P(=x) a a ;A; a A 0 % 50 A 3 f(x)=4x-3x (0 x ) 4 45 kg kg 40 kg 50 kg (P(0 Z.5)=0.4938) 5 60 % (P(0 Z )=0.477) E()=:) (4x -3x )dx E()=[;3$;x -;4#;x ])=; ; V()=:) x (4x-3x )dx-{; ;} V()=:) (4x -3x )dx-; 4ª4; V()=[x -;5#;xfi ])-; 4ª4;=;7 0; 4 N(45 ) -45 Z= P(40 50)=P { } P(40 50)=_P(0 Z.5) =_0.4938= B{60000 ;5#;} m=60000_;5#;=36000 =Æ 60000_;5#;_;5@;=0 N( ) Z= 0 P(æ3640)=P(Zæ) =0.5-P(0 Z ) = =0.08 7

33 ;8@0$; % 3 4 =300_;8@0$;= N(5 ) P( 3)+P(æ7) P(0 Z )=0.343) B(400p) B{00;!;} P( 60) P(0 Z )=0.477)

34 40 A

35 3 3 ºP =0 = ºP =0_9_8= P =5 =5 P =5_4= A B 7 7 P =7 =49 P =7_6=4 7_6 C = =! 0

36 3 3 3 (, ) (, ) (, ) (3, 3) ;#; 3 m 3 P(=x) m 3 ;3!; ;3!; ;3!; m=_;3!;+_;3!;+3_;3!;= = _;3!;+ _;3!;+3 _;3!;- =;3@; 3 m ++3 m= = = - =;3@; ;#; ;#; ;%; ;%; 3 h^ h (ubiased estimato) h ^ h ^ h h ^ h ^ (efficiet estimato) (sufficiet estimato) (cosistet estimato)

37 =;!;( + ) E( ) V( ) E( )=_;9!;+;#;_;9@;+_;9#;+;%;_;9@;+3_;9!; E( )= V( )= _;9!;+{;#;} _;9@;+ _;9#; E( )=;3!; +{;%;} _;9@;+3 _;9!;- E( )V( ) m= =;3@; E( )=mv( )= N y m N m= N i= N = ( -m) N i= =" y«s S = i= S = ( - ) - i= S=" S - E(S )= y p m m=e() =V()=() =;!;( + ) 9 E( ) V( ) E( )=, V( )=;3!; m= =;3@; E( )=m, V( )= y«=;!;( + +y+«)=;!;;i+! y «S = {( - ) +( - ) +y+(«- ) } - S = ;I+! ( - ) (S>0) - S y «P( =xæ) ;#; ;%; 3 ;9!; ;9@; ;9#; ;9@; ;9!; E( )=E( )=y=e(«)=m V( )=V( )=y=v(«)= 3 3 ;#; ;#; ;%; ;%; 3 (statistic) (estimato) m (estimate) xæ

38 3 y«e( )=E{ i= } = E( ) i= = (_m) ( E( )=m) =m E( )=m m y«3 V( + ) =V( )+V( ) E( )= {E( )+E( )+y+e(«)}= _m=m V( )={ } {V( )+V( )+y+v(«)}={ } _ = m E( ) V( ) E( )=mv( )= E( )=7, V( )=; 6;=;4!; m=3 = =5 E( )V( ) ( ) E( )=m=3 V( )= =; 5; ( )=øπv( ) ( )=Æ ; 5;=;5!; (Cetal Limit Theoem) «5 3fi =43 P O y«m S«= + +y+«s«-m Z= ' N(0) {;!;S«-m} S«-m Z= = = ' ' Z= -m ' -m ' N(0) N{m } 3

39 40 m=0 =6 =8 E( ) ( ) E( )=0 6 ( )= =;3@; '8å N{0{;3@;} } m N{m, } N{m, } æ E( )=5( )= =;#; ' 36 N {5 {;#;} } a=0b=;9$; N(a b) ab P( æ90) ( P(0 Z.5)=0.4938) -ay Z= -m ' m=300 =40 =00 E( ) ( ) E( )=m=300( )= 40 =4 ' 00 N(300 4 ) N() - Z= 0 3 P(0 3)=P(- Z ) P(0 3)=P(- Z 0)+P(0 Z ) P(0 3)=P(0 Z )+P(0 Z ) P(0 3)= P(0 3)= % 8.85% Z= P( æ90)=p(zæ-.5)=0.5+p(0 Z.5) P( æ90)= = yy` 4

40 4 4 m=45 =0 =00 E( ) ( ) E( )=m=45 0 ( )= = 'ß0å0 N(45 ) -45 Z= P( æ43)=p{ æ } =P(Zæ-) =0.5+P(0 Z ) = =0.843 m=50 =5 =36 E( ) ( ) E( )=m=50 5 ( )= =;6%; '3å6 m 8 = (P(0 Z 0.5)=0.95) Æ æ (P(0 Z )=0.343) m=70 =4 E( ) ( ) E( )=m=70 4 ( )= 'ß 4 N{70{ } } 'ß -70 Z= 4 'å P( æ7)=p ª Zæ 'å º = =0.5-P(0 Z 0.5)=P(Zæ0.5) 7-70 =0.5 4 'ß =36 5

41 3 PET PET /kg PET 007 PET N(m ) N{m, } Z= -m ' N(0) 4 PET PET 007 PET m m= 4 = PET 353 (estimatio) (hypothesis testig) 6

42 4 P(-.96 Z.96)=0.95 -m P ª º ' =P{-.96 -m.96 } ' ' =P { -.96 m +.96 } ' ' -.96 O.96 z =0.95 m [ -.96, +.96 ] yy ' ' % m P(-.58 Z.58)= % m [ -.58, +.58 ] ' ' 95 % 95 % m f(z)... m N(m ) m 95 % [ -.96, +.96 ] ' ' 99 % [ -.58, +.58 ] ' ' xæxæ xæ [xæ-.96 xæ+.96 ] 'ß 'ß N(m 8 ) m 95 % m '6å4 '6å m 4.96 [ ] N(m 4 ) 8 50 m 99 % (P( Z.58)=0.99) s m 95 % ( P( Z.96)=0.95) =00xÆ=500s=50 m 95 % s m ' 00 ' m m [ ] yy` 4 k z k Z P(-z k Z z k )=k -z f (z) m y N{m{ } } 'ß -m Z= Z N(0) 'å P{ -z k m +z k }=k 'ß 'ß m O k z z x [x -z k x +z k ] 'ß 'ß m s m 00k(%) s s [x -z k x +z k ] 'ß 'ß 43 =8 x =50 m 99 % _ m _ '8å '8å m [ ] 7

43 44 =64x =0.76s= _ m _ '6å4 '6å m [ ] m 95 % (P( Z.96)=0.95) 40 g m 95 % 0 g ( P( Z.96)=0.95) 45 m 99 % _.58_ 'ß 5 _.58_ 'ß æ(.58_5) =66.4 æ m 95 % 40 _.96.96_ 0 'ß 'ß _.96_40 æ{ } = yy` 3 5g m 99 % g (P( Z.58)=0.99) N(m 0 ) m 95 % ( P( Z.96)=0.95) =400x =70=0 m 95 % _ m _ '4ß0å0 '4ß0å m [ ] g g m 95 % P( Z.96)=0.95 [ ] kg m 99 % m 0.5 kg P( Z.58)=0.99 P{ -.58_ m +.58_ }=0.99 'ß 'ß P{-.58_ m-.58_ }=0.99 'ß 'ß P{ m-.58_ }=0.99 'ß m 0.5 kg.58_ 0.5 'ß æ

44 4 A B A 70 % 60 % 35 % B 30% 40% 65% 5 A AB A 50_0.7+75_0.6+00_0.35=5 A 5 B 0 A B 3 B 3 B 5 a p a Y Y B(p) Y Y Y yy = Y i i= B(p) ^p= ^p popotio ^p N{p a p a ^p ^p= ^p p a a p B(p) E()=pV()=p(-p) ^p E(^p)=E{ }= E()= p=p V(^p)=V{ }= V()= p(-p)= p(-p) p(-p) ^p-p Z= p(-p) æ } ^p Z N(0) Z p p ^p ^p-p Z= æ ^p(-^p) Z N(0) p 00k (%) p( Z z k )=k [p^-z k æ ^p(-^p) ^p+z k æ ^p(-^p) ] ^pæ5(-^p)æ5 9

45 ^p p(-p) N{p } ^p N(0) p ^p Z= N(0) P(-.96 Z.96)=0.95 ^p-p Z= p(-p) æ ^p-p ^p(-^p) æ f(z) ^p-p P ª ^p(-^p) º æ -.96 O.96 ^p(-^p) ^p(-^p) =P -.96æ ^p-p.96æ ^p(-^p) ^p(-^p) =P ^p-.96æ p ^p+.96æ =0.95 z p p 95 % ^p(-^p) ^p(-^p) [^p-.96æ ^p+.96æ ] 99 % ^p(-^p) ^p(-^p) [^p-.58æ ^p+.58æ ] 00 MP3 80 MP3 p 95 % ( P( Z.96)=0.95) =00 ^p=0.8 p 95 % æ 0.8(-0.8) 00 p æ 0.8(-0.8) p [0.76, ] yy` ^p(-^p) [^p-.96æ ^p+.96æ ^p(-^p) ] 95 % p P(-.58 Z.58)= % p p 99 % (P( Z.58)=0.99) ^p(-^p) [^p-.58æ ^p+.58æ ^p(-^p) ] p 95 % ( P( Z.96)=0.95) =400^p=0. 99 % p 0.(-0.) 0.(-0.) æ p æ p 0.56 [ ] =00^p= % p æ 0.5(-0.5) 00 p æ 0.40 p [ ] 0.5(-0.5) p 99 % P( Z.58)=0.99 p 95 % [ ] p ^p= P( Z.96)= (-0.5) P p æ =0.95 p ^p æ 0.05 æ

46 + = ;#; ;%; ;#; ;%; 3 3 ;%; 3 ;&; P(0 Z 0.5)= m 95 % (P( Z.96)=0.95) 4 ;%; 3 ;&; 4 ;#; ;%; 3 ;&; 4 4 A 300 A 5 A p 99 % (P( Z.58)=0.99) P( =x ) ; 6; ;8!; ; 6; ;4!; ; 6; ;8!; ; 6; m=70 =4 =36 E( )=70 4 ( )= =4 ' 36 N(70 4 ) Z= P( æ7)=p(zæ0.5) P( æ7)=0.5-p(0 Z 0.5) P( æ7)= P( æ7)= m 95 % m ' 400 ' m ^p=;3@0@0%;=;4#; p 99 % ;4#;{-;4#;} 300 ;4#;-.58 æ p ;4#;+.58 æ p ;4#;{-;4#;} 300 3

47 P(=0)= Cº {;!;} {;!;} =;8!; 3 P(=)= C {;!;} {;!;} =;8#; P(=)= C {;!;} {;!;} =;8#; P(=3)= C {;!;} {;!;} =;8!; 80 % P(=x) ;8!; ;8#; ;8#; ;8!; 3 f(x)=a(-x )(0 x ) a P{0 ;!;} B(00 0.8) E()=00_0.8=80 V()=00_0.8_0.= (P(0 Z )=0.343) 3 f(x) :) a(-x )dx=a [x-;3!;x ]) :) a(-x )dx=a {;3@;-0}= a=;#; P{0 ;!;}=:) ;!; ;#;(-x )dx P{0 ;!;}=;#; [x-;3!;x ]) ;!; P{0 ;!;}=;!6!; E()=:) x ;#;(-x )dx E()=;#;:) (x-x )dx E()=;#; [;!;x -;4!;x ])=;8#; V()=:) x ;#;(-x )dx-{;8#;} V()=;#; : 0 (x -x )dx-;6ª4; V()=;#; [;3!;x -;5!;xfi ])-;6ª4; V()=;5!;-;6ª4; V()=;3 ª0; 4 N(5 3 ) Z= -5 3 P(æ8)=P(Zæ) =0.5-P(0 Z ) = =

48 5 P(-k Z k)= a 00 k 5 ' 400 k ' 400 k.96 a 95 a 95 5 N(m 5 ) 400 m a % 0.98 a (P( Z.96)=0.95) km 4000 km km (P(0 Z.5)=0.4938) E( )= ( )= =00 ' 400 N( ) Z= P( æ9500)=p(zæ-.5) =0.5+P(0 Z.5) = = P( <9500) p 95 % (P( Z.96)=0.95) 7 p 95 % p 95 % ^p=;!4$0$0);=;5#; p 95 % ;5#;{-;5#;} 400 ;5#;-.96 æ p ;5#;+.96 æ p ;5#;{-;5#;}

49 % % % ^p=0.55 p 95% =00^p= æ 0.55(-0.55) 00 p æ p [ ] 0.55(-0.55) 00 CONFIDENCE ^p=;5#0);=0.6 p 95% =50^p= æ 0.6(-0.6) 50 p æ p [ ] 0.6(-0.6) 50 Alpha 0.05Stadad-dev Size.96 ' ^p=0.06=600.96æ p^(-p^) =0.06 p95 % [ ] 34

50 P(=k)=ak(k+) (k=3y0) a 4 k f(x)= k x ( x e) E()V() a E(+)=4a+E( )=E()+3a V() A B 3 B A P(0 Z )= Y B(5p) B{ -p} 4 E(Y)=E()V(Y)=V(4) p 6 50 g 0 g 4700 g P(0 Z 3)=

51 0 P(=k)= ak(k+) k= 0 k= 0 =a (k +k)= k= a=;44!0; E(+)=4a+ E()=a V()=E( )-{E()} =E()+3a -(a) =a-a =-(a-) + V() a= 5 A B N(3) yy -3 Z= yy A B 4 P(æ4)=P(Zæ)=0.5-P(0 Z ) P(æ4)= =0.587 yy 0 % 0 % 60 % 3 Y B(5p)B{ 4 -p} E()=5pE(Y)=;4!; (-p) V()=5p(-p)V(Y)=;4!; (-p)p E(Y)=E()V(Y)=V(4) E(Y)=E()V(Y)=6V() ;4!; (-p)=0p ;4!; (-p)p=80p(-p) =30p=;9*; yy yy 4 e k dx=[kl x] e : =k= f(x)= x x e e E()= : x dx= : dx=[x] e =e- x e V()= : x x dx-(e-) V()= : e x dx-(e -e+) V()=[;!; x ] e -(e -e+) V()=-;!; e +e-;#; 6 0 N{50{ } } 'å -50 Z= yy 0 'å 4700 g 4700 æ4700 æ P{ æ }=PªZæ º 0 'å P{ æ }= P(Zæ-3)= 'å =00 =-3 yy yy 0 % 50 % 30 % 36

52 a 3 P(=x) a-a a 3a -a ;3!; ;4!; ;5!; ;6!; 5% N(m ) P(-5 m)=0.343 P(-5 5)=0.885 m P(0 Z )=0.343P(0 Z )= O 0% 00 O 5 P(0 Z.5)= P(0 Z )=

53 (a-a )+a+(3a -a)= a +a-=0(a+)(a-)=0 a=;!; ( aæ0) 900 B( ) m m=900_0.05=95 ='ƒ900_0.05_0.95= O B(000.) m m=00_0.=0 =00_0._0.8=6 yy N(04 ) -0 Z= 4 yy P( 5)=P{ 4 4 } =P(Z.5) =0.5+P(0 Z.5) = = yy 30 % 30 % 40 % 4 0 ab P(=x) a ;!; b E()=;!;+bE( )=;!;+4b V()=E( )-{E()} =;!;+4b-{;!;+b} V()=-4{b-;4!;} +;!;{0 b ;!;} b=;4!; 00_;4!;= m P(-5 x m)=p{ Z 0}=0.343 P(-5 x m)=p(- Z 0) -5-m =- yy P(m 5)=P(-5 5)-P(-5 m) P(m 5)= = m P(m x 5)=P{0 Z } 5-m = yy m=-;3%;=;; 3º;; E( )=66V( )= = yy ' 36 N(66 ) -66 Z= yy P( 70)=P{ } P( 70)=P(Z )=0.5+P(0 Z ) P( 70)=0.977 yy 0 % 0 % 60 % 38