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2 0 THEME!!!_!_!_!_!=_6=8 pp. ~8!!!_!=70 0, P =_=, 0, _=9, _=9,, +9+9=0 6 6!=70, f, l, w, r P _!= = = THEME (-)!=!!!_!=6 (-)!=!!!_!= 6 (-)!=! 6_!=6_= 6 (6-)!=!=0 0_=60, 6! 6 = =60

3 _ e, t l, r 6! =80!! THEME a= P = = b= P = =6 a+b=+6=88 7 P = =8,, 6, 8, 0 P = =,,,,,,, P = =6 THEME A Q 6! =0!! Q B! =6!! 0_6=0 a, b a+b=9, a, b a=9, b=0, a=7, b=, 7, 8! =8 7!! a=, b=,, 7! =!! a=, b=,, 6! =0!! fi a=, b=,,! =!! = D B C A B A,,! =0!!! A C D B A C B, A D B! =, A C D B! 0-(+-)=

4 0, 0, 0, P _ P =8,,, 0, 0 _= _=, 8+=0 p. 9 P B P, Q, R A Q B A P B, A Q B, A R A R B A P B _= A Q B 7! 6! _ =!6!!! A R B 8! _ =8!7!,, ++8= (-)!=! _!=0 n P«=«n P + P + P +y+ P« y+«æ0 («-) æ0, «æ76 - næ7 0 7

5 0 8 0 THEME pp. ~ f()= < f X Y f( )<f( ) X Y X, 6 7, Y f()<f()<f() Y,, 6, 7 f(), f(), f() f C _ C =_=8 8 9 ªC =8 C =0 C = 8-(0+)=70 C =0, 7 C = 0+= 7 C = C =6, a a=-6+=6 7 C = C = b b=-= b-a=-6= THEME,,,,, = 6 H = C =ªC =6 9, 9 Hª= ª Cª= Cª= C = (+y+z) π yœ z p+q+r=7, p, q, r, y, z 7 H = C =ªC =ªC =6,,,, 6, 7, 8, 9 f() f() f() f() f() 6, 7, 8, 9,,,, H = C = C = C =6

6 THEME 6 S(6, ) C _ C _ C _ =! C _ C _ C =60 C _ C _ C _ =! S(6, )=+60+=90 X S(, ) C _ C = C _ C =0 S(, )=+0= Y! S(, )_=_=0 0 S(, ) S(, ) C _ C + C _ C =+0= C _ C _ C _ + C _ C _ C _!! =0+= a=, b= a+b=+=0 X S(, ) C _ C _ C _ =0! C _ C _ C _ =! S(, )=0+= Y P S(, )_ P =_=600 THEME 6=6 =+=+=+ =++=++=++ =+++=+++ =++++ =+++++ a=p(6, )=, b=p(6, )= c 6 c=p(6, )+P(6, )+P(6, )+y+p(6, 6) =+++++ = a+b+c=++ a+b+c=6 P(, ) ++, ++ P(, )= 6

7 7=7 =6+=+=+ =++=++=++=++ =+++=+++=+++ =++++=++++ =+++++ = , ++++, +++++, P(0, )+P(0, )+P(0, ) 0=0 =9+=8+=7+=6+=+ =8++=7++=6++=6++ =++=++=++ =++ P(0, )+P(0, )+P(0, )=++8= a= 0 P(0, ) 0=6++++=++++ =++++ =++++ =++++ =++++ =++++ b=p(0, )=7 a+b=+7= p.,, 6, 8 C,,,, 7, 9 C C _ C _!=960, y, z =a+, y=b+, z=c+, a, b, c +y+z= a++b++c+=, a+b+c=8 a+b+c=9, y, z a+b+c=9 a, b, c Hª= ª Cª= Cª= C = 8 C _ C _ C _ =0! C _ C _ C _ =80! 0+80=90 8 8=8 =7+=6+=+=+ =6++=++=++ =++=++ =+++=+++=+++ =+++=+++=+++ P(8, )+P(8, )+P(8, )+P(8, ) =+++= 7

8 0 pp. 7~0 THEME (+) C (+)fi Cß ß fi ß (+) (+)fi C Cß fi ß ±ß r+s= r, s (r, s) (, 0) (0, ) C Cºfi + Cº C =96+80=76 THEME {a + }6 C (a ) { }6 - r = C a fl 7r-8= r= C a =60a =0 a = a= 8 { + } ««( +)««C ( )«=«C =fl n-r=6 r=n- fl «C«=«C = n(n-) =, n -n-=0 (n+6)(n-7)=0 n=7 n C ( ) { } - r = C fi r-=0 r= C =00 (-y+z)fl 6! 6! π (-y)œ (z) = (-)œ π yœ z p!q!r! p!q!r! (, p+q+r=6, pæ0, qæ0, ræ0) y z p=, q=, r= y z 6! (-) =60_9_=080!!! {+;[!;} C {;[!;} = C yy` ( +){+;[!;}, ;[!; r-7=- r=;%; r 0 r 7 ;[!; r-7=-, r= C =, = {- } C fi {- }r = C (-) fi yy` (a -){- } a - -r=- r= C (-) =- 0 8

9 THEME -r= r= r 0 r 0 a {- }=-0a 0-0a=0 a=- (+) (+ ) (+) (+ ) (+) C () = C (+ ) Cß ß ( )ß = Cß ß ß ß = Cß ß ß (+) (+ ) C Cß ß ß = C Cß ± ß ± ß r+s= r, s (r, s) (0, ), (, 0) Cº C + C Cº =+07 Cº C + C Cº =8 «C =«C +«C Cº= Cº C = Cº+ C + C +y+ C k= = Cº+ C + C +y+ C = C + C + C +y+ C = C + C +y+ C = C + C = C B C = C = C + C + C + C + C + C = C + C + C + C + C = C + C + C + C = C + C + C = C + C = C THEME º C + º ºC + ººªC +y+ ºº C + ºº C = º C + º ºC + ººªC +y+ ºº C + ºº C = º C + º ºC + ººªC +y+ ºº C + ºº C = º C + º C = º C 9 =(0-) 9 = Cº(-) + C 0 (-) + C 0 (-) 9 =+y+ C Cº(-) + C 0 (-) 00 Cº(-) + C 0 (-) =-+ 0=09 9 «Cº+«C +«C +y+«c«=««c +«C +y+«c«=«-«cº=«- 00<«-<00 n=8 =0, aº=0 +0, (+)+(+) +y+(+) = (+){(+) -} (+)- (+) -(+) = yy` 9

10 a fi fi (+) fl (+) C r=6 C =6 a fi (+)fi, (+)fl, (+), (+), (+), (+) fi, a = C + C + C + C +ªC + ºC a = a =6 C C C C C N= C + C + C +y+ C + C = fi = log N=log = (+) C (+) = Cº+ C + C +y+ C = = Cº+ C + C +y+ C y` =- 0= Cº- C + C -y- C y` - =( C + C +y+ C ) p. C + C +y+ C = Cº+ C + C +y+ C = C r=0 = C r=0 =(+) = = =log =log = (+i) =( ºCº- ºC + ºC - ºC +y- ºC + ºC º) +( ºC - ºC + ºC - ºC +y+ ºC - ºC ª)i = ºCº- ºC + ºC - ºC +y- ºC + ºC º ºC - ºC + ºC - ºC +y+ ºC - ºC ª=0 (+i) =i (+i) =(i) =- =-0 (a+b)«=«cºa«+«c a«b+y+«c«b «p«a=, b=-;!; p«={-;!;} «= «m p«=;!;+ +y+ n= μ {-} μ a«= - a«=- μ {- }- = <0 μ μ μ >000 m 0 0

11 0 pp. ~6 THEME A 0 a= = B 0 b= = a 00 = = b 00 THEME a=; 0 0 0;=0.7 C A B A B, A C C = C 0 7 7! 6! 6!_! 6!_! =;7@; 7! 6!,!,!_!!_! =;@;! 9 ªC, C _ C C _ C = ªC b=; 0ª0 0;=0.9 a+b= a+b=0.68 p_ =9p OFA ' OFA= _ = ABCDEF 9' _6= 7' 9' 7' 7' ' = = 9p 8p p ABCD a_a=6a OP æa 6a -pa =(-p)a (-p)a -p = 6a B A a B A C 60 O O a F D a E D C

12 THEME -7a+a =0 (-a)(-a)=0 =;A; =;A; a a A, B P(A)=; 0 0;, P(B)=; 0 0;, P(A;B)=;0*0; P(A'B)=P(A)+P(B)-P(A;B) P(A'B)=; 0 0;+; 0 0;-;0*0; P(A'B)=; 0º0;=;!; 7 A, B P(A)=; 0 0;, P(B)=; 0 0;, P(A;B)=;0^0; P(A'B)=P(A)+P(B)-P(A;B) +a+b=0 Dæ0 D=a -bæ0 b ;!;a b ;!;a (a, b) b b= ;; a 6 O 6 a a= 0 a= (, ) a= (, ), (, ) a= (, ), (, ), (, ), (, ) a= (, ), (, ), (, ), (, ), (, ), (, 6) 6 a=6 (6, ), (6, ), (6, ), (6, ), (6, ), (6, 6) 6 P(A'B)=; 0 0;+; 0 0;-;0^0; ;!6(; P(A'B)=;!0!0); P(A'B)=;!0!;,, A, B, C C C P(A)= =;%6;, ªC C C P(B)= =; º6;, ªC C Cº P(C)= =;!6; ªC A, B, C P(A'B'C)=P(A)+P(B)+P(C) P(A'B'C)=;%6;+; º6;+;!6; P(A'B'C)=; 6; THEME 6 P _6! A P _6! P(A)= =; ; 8! -P(A)=-; ;=;ª; A A Ç ºC P(A Ç ºC )= =; ; C P(A'B'C)=;!!; P(A)=-P(A Ç )=-; ;=; ;

13 A A Ç 8 C =6 (,, ), (,, ), (,, ), (,, 6),, (, 6, 7), (6, 7, 8), (7, 8, 9), (8, 9, 0) 6 8 _6= P(A Ç 8 )=-P(A)=- ºC P(A Ç )=-;*0;=;!$; ;@6$;=;7#; C =9 a=, b= {,,, } a+b=9, {6, 7, 8, y, } C «C C C - =, = «C 7 «C 7 6 = (n+)(n+) 7 (n+)(n+)= n= C _ C =8 C _ C =; 9 ;=; 6 ; C -a a = + -6 b b=6a b=a b=a a, b (a, b) (, ), (, 6) 6 =6 p. 7 7,,,,,, 7! =0!_!_!,,,,, 6! =90!_!_! 90 = 0 7 ; 6;=; 8; -; 8;=;!8&;

14 0 THEME P(A)=; 0 0;, P(B)=; 0 0;, P(M;A)=; 0º0;, P(F;B)=; 0º0; pp. 9~ A B P(A;B)=0._0.6=0.8 P(AÇ ;B)=0.7_0.= P(A;B) P(A B)= P(B) P(A B)= = 0. 6 P(M;A) P(M A)= =;9%; a= P(A) P(F;B) P(F B)= =;!); b= P(B) a+b=+=8 P(A'B)=P(A)+P(B)-P(A;B) P(A;B)= =0. P(A;B) P(A B)= P(B) 0. P(A B)= =;@; 0. P(A Ç ;B Ç )=-P(A'B)=;!; P(A'B)=;#; P(A B)= P(A;B) P(B) =;6!; THEME E, A, B, C A, B, C P(A;E)=P(A)P(E A) C P(A;E)=;!;_ =; 0; C P(B;E)=P(B)P(E B) C P(A;E)=;!;_ =; 8; C P(C;E)=P(C)P(E C) C P(A;E)=;!;_ =;!; C P(E)=P(A;E)+P(B;E)+P(C;E) P(E)=; 0;+; 8;+;!;=;!^; P(B;E) P(B E)= =; ; P(E) P(A;B)=;6!;P(B) P(A'B)=P(A)+P(B)-P(A;B) ;#;=;!;+P(B)-;6!;P(B) A A, B B E ;6%; P(B)=; ; P(B)=;!; P(A;E)=P(A)P(E A) C _ C P(A;E)=;!;_ ºC P(A;E)=;9@0$;

15 P(B;E)=P(B)P(E B) C _ C P(A;E)=;!;_ =;9@0%; ºC P(E)=;9@0$;+;9@0%;=;9$0(; P(A;E) P(A E)= P(E) ;9@0$; P(A E)= =;@9$; ;9$0(; A A B B E P(A;E)=P(A)P(E A) P(A;E)=; 0 0;_;0%0;=;0(0; P(B;E)=P(B)P(E B) P(A;E)=; 0 0;_;0#0;=;0#0#0; P(E)=;0(0;+;0#0#0;=;0&0*0; P(B;E) P(B E)= P(E) ;0#0#0; P(B E)= =;!6!; ;0&0*0; A, B P(A;B)=;8%;_;7$;=;@6); THEME A B n P(A)=, P(B A)= n+ n+ n P(A;B)=P(A)P(B A)= n+ n+ n P(A;B)= (n+)(n+) n =;!; (n+)(n+) (n+)(n+)=0n, n -n+=0 (n-)(n-)=0 n= n= n A, B P(A)=; ;, P(B A)=; ; P(A;B)=P(A)P(B A) P(A;B)=; ;_; ;=; ; A, B A B A Ç B Ç P(A Ç ;B Ç )=P(A Ç )P(B Ç ) ={-P(A)}{-P(B)} P(A Ç ;B Ç )=0.6_0.=0.8 -P(A Ç ;B Ç )=-0.8=0.8 P(A Ç ;B)=;8#;_;7%;=;!6%; P(B)=P(A;B)+P(A Ç ;B) P(B)=;@6);+;!6%;=;#6%;=;8%; P(A;B) P(A B)= =;7$; P(B) A, E, P(E)=P(A;E)+P(A Ç ;E) 9 (, 6), (, ), (, ), (6, ) 0 (, 6), (, ), (6, ) (, 6), (6, )

16 ,, ;ª6;=;!;,,, y 9 6 L C {;!;} {;!;} ;!;=; ; ;!;+{;#;} _;!;+{;#;} _;!;+y N ;!; = =;7$; -{;#;} C {;!;} {;!;} ;!;=; ;, ; ;+; ;=; 6; THEME C {;@;} {;!;} =; º; C {;@;} {;!;} =; º; C {;@;} fi {;!;} =; ; ; º;+; º;+; ;=;8^$; ;!; C {;!;} {;#;} =; 6;=; 8; ;!;, ;@; A P, C {;!;} {;@;} =; º; m=, n=80 m-n=-80=6 p. A, B P(A B)=0.8, P(A B Ç )=0., P(B)=0.7 P(A)=P(A;B)+P(A;B Ç ) P(A)=P(B)P(A B)+P(B Ç )P(A B Ç ) P(A)=0.7_0.8+0._0. P(A)= P(A)=0.669 P(A)=a, P(B)=b (a>b) P(A'B)=P(A)+P(B)-P(A;B) ;@;=a+b-;6!; a+b=;6%; yy A, B P(A;B)=P(A)P(B) ab=;6!; yy, a=;!;, b=;!; ( a>b) P(B)=;!; 6

17 P(r)= ºC {;!;} {;!;} = ºC {;!;} ºC {;!;} P() = P(6) ºC {;!;} = 70!!! 70! 6!! =;#^; 06 pp. ~8 P A P D, 9,, y 9 a, b =9 a+b= yy` [ a+b=9 yy` a=, b= THEME X 0, C P(X=0)= =;7!; C C C P(X=)= =;7$; C =;7!;, =;7$; a+a +;A;= C {;!;} {;!;} =; 6; = a+b= [ a+b= a=0, b= Cº{;!;} {;!;} fi =; ;, ; 6;+; ;=;!!; yy` yy` a +a-=0 (a-)(a+)=0 a=;!; ( a>0) P(0)=k, P()=k, P()=k, P()=k P(0)+P()+P()+P()= k+k+k+k=k= k=; ; P(Xæ)=P(X=)+P(X=) P(Xæ)=k+k P(Xæ)=9k P(Xæ)=9 ; ; P(Xæ)=;ª; 7

18 X X P(X=) ; 0; ; 0; ; 0; ; 0; X -0X (X-0)(X-00) 0 0 X 00 P(X -0X+000 0) =P(0 X 00) P(0 X 00)=P(X=0)+P(X=00) P(0 X 00)=; 0;+; 0; P(0 X 00)=; 0; P(0 X 00)=;!; THEME 0,,, 0,,,, 6, X X 0 P(X=) ;6!; ;6@; ;6@; ;6!; E(X)=0_;6!;+_;6@;+_;6@;+_;6!; E(X)=;#; P(X=)+P(X=)+P(X=)+P(X=) = ;k!;+;k@;+;k#;+;k$;=, : kº:= k=0 X X P(X=) E(X)= ; 0;+ ; 0;+ ; 0;+ ; 0; E(X)=;#0);= E(X )= ; 0;+ ; 0;+ ; 0;+ ; 0; E(X )=: º0º:=0 ; 0; ; 0; ; 0; ; 0; V(X)=E(X )-{E(X)} V(X)=0- = r(x)=" V(X)= X 0,, Cº C P(X=0)= =; ; C C C P(X=)= =; ; C C Cº P(X=)= =; ; C X E(X)=0_a+_;!;+_b= b+;@;= b=;$; b=;9$; a+;!;+b= a=-;!;-;9$;=;9@; a+b=;*;+;$; a+b=: ;;= X 0 P(X=) ; ; ; ; ; ; X E(X)=0 ; ;+ ; ;+ ; ;=;@; E(X )=0 ; ;+ ; ;+ ; ;=;$; V(X)=E(X )-{E(X)} =;$;-{;@;} =;!^; 6 " r(x)=" V(X)=æ = 8

19 THEME THEME E(Y)=E(X-)=E(X)-= E(X)= E(X)= V(Y)=E(Y )-{E(Y)} =7- =0 V(Y)=V(X-)= V(X)=0 V(X)= E(X)-V(X)=-= X B{6, ;!;} E(X)=6_;!;=, V(X)=6_;!;_;@;=6, r(x)=' 6=6 E(X)+r(X)=+6=60 E(X)=7, r(x)= E(-X+)=-E(X)+ =-+ ;#; ;!;, X X B{, ;#;} =- r(-x+)= - r(x) r(-x+)=_=8 P(Xæ)=P(X=)+P(X=) P(Xæ)= C {;#;} ;!;+ C {;#;} E(-X+)+r(-X+) P(Xæ)=;!*6(; =-+8 =- X B{0, ;!;} E(X)=(-)_; 0;+0_; 0;+_; 0;+_; 0; E(X)=; 0; X+ E(Y)=E{ }=E{;#;X+} E(X)=0_;!;=0, V(X)=0_;!;_;!;=0 E(X )=V(X)+{E(X)} =0+00=0 E(Y)=;#;E(X)+ E(Y)=;ª0;+=;$0(; E((X-) )=E(X )+E(-X)+9 E((X-) )=E(X )-E(X)+9 E(X)=, V(X)=6 E(Y)=E(aX+b) =ae(x)+b =a+b= V(Y)=V(aX+b)=a V(X) V(Y)=6a =6 a = a= ( a>0) a= b=- a-b=+= yy` E((X-) )=_0-_0+9 E((X-) )=09 P(X=r)= C {;!;} {;#;}, r=0,,, y, 6 E(Y)=E( ) 6 E(Y)= C {;!;} {;#;} r=0 6 E(Y)= C {;%;} {;#;} r=0 E(Y)={;%;+;#;} E(Y)=fl =6 9

20 X X 0 P(X=) ;6!; ;6$; ;6!; p ;!; ;@; ;!; 0_;!;+(00-9)_;!;+(00-0)_;!; =0+;%;+:ª : E(X)=0_;6!;+_;6$;+_;6!;=, =;:!):%; V(X)=0_;6!;+ _;6$;+ _;6!;- =;!; E(X)-V(X)=-;!;=;@; k=,,, X P(X=k) 0 ;6!; ;6@; ;6#; E(X)=_0+_;6!;+_;6@;+_;6#;=: º: E(X )= _0+ _;6!;+ _;6@;+ _;6#; =: : f(a)=e((x-a) ) f(a)=e(9x -6aX+a ) f(a)=9e(x )-6aE(X)+a f(a)=a -0a+0 f(a)=(a-0) + a=0, f(0)= = X X B(67, 0.9) X P(X=)= C (0.9) (0.0)fl (=0,,, y, 67), Xæ66 P(Xæ66)=P(X=66)+P(X=67) P(Xæ66)= C (0.9)fl fl (0.0)+ C (0.9)fl P(Xæ66)=67_0.0_0.0+_0.0 P(Xæ66)= ;!; 60 ;!; ;!;_;!;=;!; 0

21 07 pp. ~ P(X k) y k ;; =;!;_k_{;!;- +;!;} y=f() k ;;-;; =;#; O k 8 k -6k+8=0, (k-)(k-)=0 k= ( 0<k<8) THEME f()=k(-) =, y=f() ;!;_(+)_k= = ;!;_(k+9k)_= k= k=; ; ;!;_6_6k= 8k= k=; 8; P( X 6) =;!; {;9!;+;!;} k=;7!; 0 6 f()=; ; P( X )=-P(0 X ) P( X )=-;!; ; ; P( X )=;@8%; y=f() =;9*; y 9k k O y=f() =6 y ;; ;9; y=f() O 6 THEME P(60 X 0)=P{ Z } 0 0 P(60 X 0)=P(- Z ) P(60 X 0)=P(- Z 0)+P(0 Z ) P(60 X 0)=P(0 Z )+P(0 Z ) P(60 X 0)= P(60 X 0)=0.979 P( X-m r)=p(-r X-m r) X-m P( X-m r)=p{- } r P( X-m r)=p(- Z ) P( X-m r)=p(0 Z ) P( X-m r)=_0.987=0.997 X- Z= Z N(0, ) k- P(Xæk)=P{Zæ }=P(Yæk) k- =k, k-=k k=-;#; f(a)=p(a X a+) =a =a+ =m, =0 a+(a+) =0 a=- a O a+ f(-)=p(- X )=P(0 X ) f(-)=_0.77=0.9

22 THEME X X N(, ) Z= N(0, ) 8- P(Xæ8)=P{Zæ } P(Xæ8)=P(Zæ-) P(Xæ8)=0.+P(0 Z ) P(Xæ8)= P(Xæ8)=0.98 X X N(9, ) Z= N(0, ) P( X 6) =P{ Z } =P(- Z ) =_P(0 Z ) =_0. =0.68 Z cm 6 cm 68 % X- 7 6 X X N(0, 6 ) Z= Z N(0, ) 6-0 P(X>6)=P{Z> } 6 P(X>6)=P(Z>) X-9 P(X>6)=0.-P(0 Z ) P(X>6)=0.-0. Z 00_0.98=96 X-0 6 P(X>6)=0.87 THEME X X N(, ) ;9(^0;=; 0;=0.0 a P(Xæa)=0.0 X- Z= Z N(0, ) a- P{Zæ } a- =0.-P{0 Z }=0.0 a- P{0 Z }=0. P(0 Z.6)=0. a- =.6, a-=7.8 a=86.8 X X B{0, ;@;} E(X)=0_;@;=60, V(X)=0_;@;_;#;=6 7 X N(60, 6 ) X-60 Z= Z N(0, ) P(Xæ7)=P{Zæ }=P(Zæ) 6 P(Xæ7)=0.-P(0 Z ) P(Xæ7)= =0.08 X B(00, ; 0;) E(X)=00 ; 0;=0, V(X)=00 ; 0; ;ª0;=9

23 00 X N(0, ) X-0 Z= Z N(0, ) P(7 X 9)=P{ Z } P(7 X 9)=P(- Z ) P(7 X 9)=P(0 Z )+P(0 Z ) P(7 X 9)= P(7 X 9)=0.8 X X B{6, ;!;} E(X)=6 ;!;=, X B{00, ;!;} E(X)=00 ;!;=80, V(X)=00 ;!; ;$;=6 00 X N(80, 8 ) X-80 Z= Z N(0, ) 8 P(66 X 98) =P{ Z } 8 8 =P(-.7 Z.) =P(0 Z.7)+P(0 Z.) = =0.97 V(X)=6 ;!; ;!;=6 6 X N(, ) X- Z= Z N(0, ) 0- P(Xæ0)=P{Zæ }=P(Zæ) 6 X X B{0, ;@;} p. C C D E(X)=0 ;@;=00, V(X)=0 ;@; ;!;=00 0 X N(00, 0 ) X-00 Z= Z 0 N(0, ) k-00 P(Xæk)=P{Zæ } 0, k-00 = 0 k=0 X, Y N(0, ), X-m N(, ) Z= r a=p(-<x<)=p(-<z<), b=p(<y<) - - b=p{ <Z< } b=p(0<z<) c=p(-<y<-) c=p{ <Z< } c=p(-<z<-)

24 c<b<a O -K+=0 D 0 D=K -æ0 K æ K N(0, ) P( K æ)=-p( K ) P( K æ)=-p(0 K ) P( K æ)=-0.68 P( K æ)=0. z 08 THEME pp. 7~0 X X X P(X=) ;8$; ;8!; ;8@; ;8!; C X X B{9, ;!;} E(X)=_;8$;+_;8!;+_;8@;+_;8!; E(X)= E(X)=9_;!;=8, V(X)= _;8$;+ _;8!;+ _;8@;+ _;8!;- V(X)=9_;!;_;#;=6 9 X N(8, 6 ) -8 P(Xæ)=P{Zæ } 6 P(Xæ)=P(Zæ-) P(Xæ)=0.+P(0 Z ) P(Xæ)=0.+0. P(Xæ)=0.8 V(X)=;%; ;%; E(X )=, V(X )= =; ; E(X )+V(X )=+; ;=;@(; E(X)=90, r(x)= 6 E(X )=E(X)=90, r(x )=;6$;=;@; E(X )_r(x )=90_;@; E(X )_r(x )=60 E(X)=_0.+_0.+_0.+_0. E(X)=. V(X)= _0.+ _0.+ _0.+ _0. V(X)=-. V(X)=0.89

25 n V(X )=; 0ª0; 0.89 V(X )= =; 0ª0; n= n X ;!; E(X)= ;!;+7 ;!;+9 ;!;+ ;!;+ ;!; E(X)=: :=9 V(X)= ;!;+7 ;!;+9 ;!;+ ;!;+ ;!; V(X)=-9 V(X)=89-8=8 E(X )=9, V(X )=;*;= N(80, 0 ) 0 n X N{80, } n X -80 Z= Z 0 'ßn N(0, ) P(80 X 8) =P Z ª 0 0 º 'ßn 'ßn 'ßn =P{0 Z } =0.9 P(0 Z.)=0.9 'ßn =., 'ßn=7 n=9 THEME N(0, 0 ) 9 X N{0, N(0, 0 ) Z= Z N(0, ) N(0, ) 6 X N{0, }, 6 N(0, ) Z= Z N(0, ) P(8 X ) X =P{ Z } =P(- Z )=P(0 Z ) }, 97-0 P{X <: : 9 : :}=P(X <97)=P{Z< } 0 P{X <;:#9%:&:#;}=P(Z<-.)=0.-P(0 Z.) P{X <;:#9%:&:#;}=0.-0.9=0.0 X -0 =_0.= X N{000, } n X -000 Z= Z 00 'ßn N(0, ) 77 P{X æ900+ } 'ßn 'ßn =P Zæ 00 'ßn 'ßn =P{0.9- Z 0}+P(Zæ0) 'ßn =P{0.9- Z 0}+0.æ0.8 'ßn P{0.9- Z 0}æ0. P(0 Z 0.8)=0. 'ßn 'ßn , æ. 'ßnæ.9 næ8.0 n 9

26 THEME n 6.8 kg, 0. kg 9 % 0. kg 0. 0., 'ßnæ_= 'ßn næ6 6, = 'ßn 'ßn=9 n=8 THEME n p^=; 0º0;=0.8 9 % kg,. kg m 99 % 0..-_ m.+_ ' 00. m.6 N(0, ) Z a P( Z k)= a % 00 k_ = k=0' ' 0 n, k_ = _0'_ = 'ßn 'ßn 'ßn=0' n= ' 00 a % n 'ßn, p=;!;, n p^ N{p, } ;!;_;!; = n= n 0 0._0.8 0._ æ p 0.+æ p^-p æ 0.-_0.0 p 0.+_ p 0. 9 % p-p^ æ 0.8_0. æ 0.0, 'ßnæ0 n næ00 p^(-p^) n 00 p^(-p^) n pq n p^=; 0º0;=0., n=00 ;!; 6 n=0_6=800 0._0.9 p-0. æ = p 0.6 6, 9 % 00 = ' 6 00 n, 99 % 00 = 600 'ßn 'ßn 6 % : 6 : p. 6

27 X X X P(X=) ;6#; ;6@; ;6!; E(X)= ;6#;+ ;6@;+ ;6!;=: 6º:=;%; V(X)= ;6#;+ ;6@;+ ;6!;-{;%;} V(X)=: 6º:-: 9 :=;9%; 600 p=0.0 p^ 0.0_0.96 E(p^)=0.0, r(p^)=æ = p^-0.0 Z= Z N(0, ) P{p^æ;6 0 0;}=P{Zæ } P{p^æ; 0 0;}=P(Zæ.) P{p^æ; 0 0;}=0.-P(0 Z.) P{p^æ; 0 0;}= E(X )=E(X)=;%; P{p^æ; 0 0;}=0.006 ;9%; V(X )= =; 8; E(X )+V(X -)=;%;+ V(X ) E(X )+V(X -)=;%;+9 ; 8; E(X )+V(X -)=;%;+;%; E(X )+V(X -)=: 6 : : 6 : X N(m, ) X N{m, ;!;} X X a a`% P( Z k)= 00 00, 0, a % m 0-k_ m 0+k_ ' 00 ' m k_ =9.7 ' 00 k=.06 P(-.06 Z.06)=0.8_= % a=96 7

28 09 pp. ~ , (-)!=!= _ = (-)!=!=6 P = 6_= (-)!=!=6 _6=0,,,, P = = P = =6 P = = P = =7,, -(6+6-7)= X Y Y,,, = P = =6 X Y Y,,, y= P = +y=6+=88 b a, a, b, c, c! =0 0!_! C =9 C = C = C = C = 9-(+)=, y, z +y+z=0 (æ, yæ, zæ) -=X, y-=y, z-=z X+Y+Z=7 (Xæ0, Yæ0, Zæ0) X+Y+Z=7 X, Y, Z (X, Y, Z) H =ªC =ªC =6 8

29 9 0 C _ C _ C _ =0! C _ C _ C _ =! S(, )=0+= S(, )= C _ C _ C _ C _ =0! S(, )= +0+=6 X S(6, ) X C _ C _ C _ C _ =0! C _ C _ C _ C =!! X 0+=6 Y P 6_ P =6_0= =7+=6+=+=+ 8=6++=++=++ 8=6++=++=++ +, ++, ++, ++ P =!=0 a+b+c=6 H = C = C =8,, P = =8 P = =6 6_6=6 {a + } C (a ) ± = C a k-7=- k= C a fi = a =96 '7 a =;7!; a= ( a>0) 7 {+ } C { } = C yy` (++ ){+ } 9

30 8-r= r= r 0 r 8 8-r=0 r= C 9 = 8-r=- r= r 0 r 8 _ C = C (+)fi C (+) Cß ß ß (+)fi (+) C Cß ß ß = C Cß ß ±ß r+s= r, s (r, s) (0, ), (, ), (, ), (, 0) Cº C + C C + C C + C Cº = =7 6 a«= C + C + C +y+«c a«= Cº+ C + C +y+«c«a«= Cº+ C + C +y+«c«a«= C + C +y+«c«a«= C +y+«c«a«=«c«a«=«c (n+)n(n-)(n-) a«=! a«(n+)n(n-)(n-) lim = lim n n n!n = =! 8 9 (+)«=«Cº+«C +«C +y+«c««y` = «=«Cº+«C +«C +y+«c«yy` =- 0=«Cº-«C +«C -«C +y+(-)««c«yy` + «Cº+«C +«C +y=«, (+ )+ (+ ) +y+ (+ ) y` (+ ), +, 0 (+ ){(+ ) -} =(+ ) -(+ ) (+ )- (+ ) (+ ) C ( ) = C = r=8 r= C (+ ), (+ ) (+ ) C (+ ) C (+ ) ºC C + C + C +y+ ºC = C P(S)= 0 P(A) 0 P(S)-P(A) 0 P(A), 0 P(B) 0 P(A)+P(B) P(A)+P(B)< P(A'B)=P(S)=P(A)+P(B)-P(A;B) P(A)+P(B)-P(A;B)= P(A)+P(B)=+P(A;B) P(A;B)+0 0

31 0 X={,, }, Y={,,,, } f:x Y P =, <X, <X + f( )+f( ) P = =60 60 =;!@; p=, q= p+q=+=7 7 n «C C 0 AÇ ;BÇ, (A'B)Ç P((A'B)Ç )=-P(A'B) P((A'B)Ç )=- P((A'B)Ç )= a, e a, e, b, c, d, a, e «C n(n-) =, = C 0 n(n-)==7_6 n=7 a, b, c, d, e a e! =60! a e a, e A, B A={(, ), (, ), (, ), (, )} B={(, ), (, ), (, 6), (, ), (, ),! a, e 60-! =; 0;=; 0;! (6, )} 6 P(A)= =, P(B)= =, P(A;B)=; 6;=; 8; P(A'B)=P(A)+P(B)-P(A;B) P(A'B)= + -; 8;=;9@; A, B 0 P(A)= =, P(B)= =, 0 0 P(A;B)= = 0 0 P(A'B)=P(A)+P(B)-P(A;B) P(A'B)= + - = 0

32 0 ;!; 6 7; ; : º: pp. 6~ A B P(B)=0.6, P(A;B)=0.8 P(A;B) P(A B)= P(B) P(A B)= = 0.6 6, P(B)=P(A;B)+P(AÇ ;B) P(B)= + = ;!; A, B P(A;B)=P(A)P(B) P(A Ç ;B)=P(B)-P(A;B) =P(B)-P(A)P(B) P(A Ç ;B)=(-P(A))P(B) P(A Ç ;B)=P(A Ç )P(B) A Ç, B A, B P(A)>0, P(B)>0 P(A;B)=P(A)P(B)>0 A, B P(A;B)=0 A, B B, C A, C A, B, C P(A)=;!;, P(B)=;!; A, B P(A)= =, P(B A)= = 9 8 P(A;B)=P(A)P(B A) P(A;B)= = 6 P(AÇ )= =, P(B AÇ )= 9 8 P(AÇ ;B)=P(AÇ )P(B AÇ ) P(AÇ ;B)= = 8 P(A;B)=;6!;=P(A)P(B) P(B)=;!;, P(C)=;!; P(B;C)=;6!;=P(B)P(C) A, B B, C P(A;C)=0+P(A)P(C) A, C ;!;, y : ;!; : ;@;_;@;_;!;

33 : ;!;+;!; +;!; ;!; +y= -;9$; 8 X -6X+8<0 (X-)(X-)<0 <X< P(X -6X+8<0)=P(<X<) P(X -6X+8<0)=P(X=) ;!;+;!; {;@;} +;!; {;@;} +y=;#; 6 7 A, B, C A, B, C A, B, C P(AÇ ;BÇ ;CÇ ) =P(AÇ )P(BÇ )P(CÇ ) =(-0.6)_(-0.6)_(-0.8) =0._0._0. =0.09 = 6 k= y +y=, +y=8 =, y= 8 C { } { } 9 9 _ C { } { } = 6 _ C { } { } = P(X=)= = = C 0 a+b+c= yy` E(X)= 0 a+ b+ c= b+c= yy` V(X)=E(X )-{E(X)} V(X)= 0 a+ b+ c-= b+c= yy` a=, b=, c= 6 6 abc= X 0,, Cº C P(X=0)= =, C C C P(X=)= =, C 8 C Cº P(X=)= = C 8 X, 7 + = X P(X=) 0 ; ; ;!8%; ; 8;

34 X E(X)= E(X)= = 8 E(X )= E(X )= 8 7 V(X)=E(X )-{E(X)} = -{ } 8 V(X)=E(X )-{E(X) }= 'ß r(x)=" V(X)=æ = 8 X,, C C P(X=)= =, C C C P(X=)= =, C C Cº P(X=)= = C X X P(X=) ;!; ;#; ;!; X E(X)= + + = E(Y)=E(X-) E(Y)=E(X)- E(Y)= -= X B{90, } E(X)=90_ =0, V(X)=90 =0 E(aX+b)=aE(X)+b E(aX+b)=0a+b= yy` V(aX+b)=a V(X)=0a =80 0a =80 a = a=( a>0) y` b=- ab=_(-)=-0, ;!; X B{00, ;!;} E(X)=00_;!;=0, V(X)=00_;!;_;!;=, r(x)= E(Y)=E{;!;X-0}=;!;E(X)-0=0, V(Y)=V{;!;X-0}=; ;V(X)=;@%;=, r(y)= Y 0, y=f() k= k= f() ( ;@; (0 ) f()= { -; 0;+;!; ( ) 9 f()= y 0 ;; P(Xæ) ;0; y=f() O 9 P(Xæ)= (-) = 0 0

35 6 7 X-0 Z= Z N(0, ) 0 8 P(6 X )= P{ Z }= P(-0. Z 0.)=0.08 P(0 Z 0.)=0.08 P(0 Z 0.)=0. -0 P(Xæ)=P{Zæ }=P(Zæ0.) 0 P(X>)=0.-P(0 Z 0.) P(X>)=0.-0. P(X>)= X- Z= Z N(0, ) P(0 X a)= a- P{ Z } a- =P{- Z } a- =P(- Z 0)+P{0 Z } a- =P(0 Z )+P{0 Z } a- =0.+P{0 Z }=0.88 a- P{0 Z }=0.77 P(0 Z )=0.77 a- = a=6 X X X-68 N(68,. ) Z=. Z N(0, ) P(Xæ7.)=P{Zæ }. P(Xæ7.)=P(Zæ) P(Xæ7.)=0.-P(0 Z ) P(Xæ7.)=0.-0.= _00=8 9 X B{80, } 6 E(X)=80_ =0, 6 V(X)=80 = X N(0, ) X-0 Z= Z N(0, ) P( X 0)=P{ Z } P( X 0)=P( Z ) P( X 0)=P(0 Z )-P(0 Z ) P( X 0)= P( X 0)=0. X X B{00, } E(X)=00_ =0, V(X)=00 =6 00 X N(0, ) X-0 Z= Z N(0, ) P(Xæa)=0.0 a-0 P{Zæ }=0.0 a-0 P(Zæ0)-P{0 Z }=0.0 a-0 0.-P{0 Z }=0.0 a-0 P{0 Z }=0.8 P(0 Z )=0.8 a-0 =, a-0=8 a=8

36 0 X X 00 0 cm X P(X=) ;9!; ;9!; ;9!; ;9!; ;9!; ;9!; ;9!; ;9!; ;9!; 7 cm m 9% m 7+ ' 00 ' 00 7 m 76 E(X)=;9!;(++y+9)=, V(X)=;9!;( + +y+9 )- V(X)=: º: X E(X )=, : º: V(X )= =;%; E(X )+V(X )=+;%;=: º: : º: N(0, ) Z a P( Z k)= a% 00 6 k_ =6 k=' 0 ' 0 n _' 0_, 'næ6' 0 'n næ080 N(0, 6 ) 6 n X N{0, } n X -0 Z= Z 6 'n N(0, ) P(X æ0)= P Zæ ª 6 º 'n 'n =P{Zæ- } 6 'n =P{- Z 0}+P(Zæ0) 6 'n =P{0 Z }+0. 6 =0.98 'n P{0 Z }=0.8 6 : p= n=6 p^ 0._0. E(p^)=, r(p^)=æ = 6 n p^ p^-;!; N{, } Z= Z ; ; N(0, ) ;@;-;!; P{p^æ }=PªZæ º ; ; P{p^æ }=P(Zæ) P{p^æ }=0.-P(0 Z ) P{p^æ }=0.08 P(0 Z )=0.8 'n =, 'n= 6 n= 6

37 pp. 60~ ;!%; (-)!=!=6!= 6_= P =_=0 P =_ = = a, b, c, c, c, d, d 7! =0!! a, b A 6 A, c, c, c, d, d 6! =60!! a b!= a b 60_=0 0-0=00 6 7,, H = C = C =, H = C = C = = ªC C = =7+=6+=+=+ 8=6++=++=++ 8=++=++=++ P(8, )+P(8, )+P(8, )=++=0 (a+) C a (+) Cßß (a+) (+) C a Cßß = C Cßa ±ß r+s= r, s (r, s) (0, ), (, 0) 0 Cº C a + C Cº a =0 7a +a =0, (a-)(7a +0a+0)=0 a= ( a ) 7

38 8 0 =(0+) = º C 0 k=0 = º Cº0 + º C 0 + º C 0 +y+ º C º 0 =+00+00( º C 0 +y+ º C º 0 ) 00( º C 0 +y+ º C º 0 ) (,, ), (,, ), y, (8, 9, 0) 8 (,, ), (,, 6), y, (6, 8, 0) 6 (,, 7), (,, 8), (, 6, 9), (, 7, 0) 9 (+)+(+) +(+) +y+(+) y` +, +, 0 (+){(+) -} (+) -(+) = (+)- yy` (+) (+) C r= C (,, 9), (, 6, 0) ; º0;=;6!; ºC =0 (,,, ), (,,, ), y, (7, 8, 9, 0) 7 (,,, 7), (,, 6, 8), (,, 7, 9), (, 6, 8, 0) (+)+(+) +(+) +y+(+) (,, 7, 0) (+) +(+) +(+)fi +y+(+) C + C + C +y+ ºC = Cº+ C + C +y+ ºC ; 0;=; ;,, 0 = C + C + C +y+ ºC = C + C + C +y+ ºC = ºC + ºC = C = C ºC =0 (,, ), (,, 8), (,, 9) A B 6 A A A, B C C P(A)= =; ;, ªC C C P(B)= =; ; ªC A, B P(A'B)=P(A)+P(B) ;#0;=; 0; P(A'B)=; ;+; ;=;!&; 8

39 0 n (0-n) ; 9; º «C (0-n)(9-n) = =; 9; ºC 0_9 (0-n)(9-n)=0=_ E(X)= E(X)= a+ ;!;+ b= a+b= yy`, a=b=;!; a +b =; 6;+; 6;=;8!; n= A 0 %, B 60 % 0._ _0.0 B 0.6_0.0 =;#6);=;!%; ;!%; 0._ _0.0 A, B P(A;B)=P(A)P(B) 7 9,, 9 X 0,,, Cº C P(X=0)= =; ;, ªC C C P(X=)= =;!8%;, ªC C C P(X=)= =; ;, ªC C Cº P(X=)= =;8 ; ªC X X 0 (-P(A))(-P(B)) =-P(A)-P(B)+P(A;B) =-{P(A)+P(B)-P(A;B)} =-P(A'B) P(X=) X ; ;!8%; ; ; ;8 ; E(X)=0 ; ;+ ;!8%;+ ; ;+ ;8 ;= P(A;B Ç )=P(A)-P(A;B) P(A;B Ç )=P(A)-P(A)P(B) P(A;B Ç )=P(A)(-P(B)) P(A;B Ç )=P(A)P(B Ç ) P(A Ç B Ç )=P(A Ç B)=P(A Ç )=-P(A), -P(A B)=-P(A) P(A Ç B Ç )=-P(A B) 8 E(8X-)=8E(X)- E(8X-)=8_-= ;!; X,, B{0, ;!;} E(X)=0_;!;=;%;, 6 fi ;!;! 0 {;!;} = =;8!);!! fi a+;!;+b= a+b=;!; yy` V(X)=0_;!;_;#;= V(X)=E(X )-{E(X)} E(X )=V(X)+{E(X)} E(X )= + =

40 9 f()=a - =0, =6 ;!; a y a ;6; ;9; O ;; y=f() 6 N(00, 0 ) 00 X 0 N{00, }, N(00, ) 00 X -00 Z= Z +;!; a= N(0, ) P(98 X 0) 9a= a=;9!; =P{ Z } P{;#; X } =P(- Z 0)+P(0 Z.) =P(0 Z )+P(0 Z.) =0.+0. P{;#; X }=;!;_;#;_;6!;+;!; ;9!; =0.77 P{;#; X }=;7!#; 0 P(Y )=P(X+ )=P{X : :} 6-0 P(Y )=P(X 6)=P{Z } P(Y )=P(Z )=0.+P(0 Z ) P(Y )= =0.977 X B{68, ;!;} E(X)=68_;!;=6, V(X)=68_;!;_;@;= 68 X N(6, ) X-6 Z= Z N(0, ) P(Xæa)= a-6 P{Zæ }= a-6 P(Zæ0)-P{0 Z }= a-6 P{0 Z }=0. P(0 Z.)=0. a-6 =. a= p p=0.7, n=00 p^ 0.7_0. E(p^)=0.7, r(p^)=æ =; 0; 00 n p^ p^-0.7 N{0.7, }, Z= Z 0 0 N(0, ) P(0.7 p^ 0.8) =P Z ª º 0 0 =P(- Z ) =P(0 Z ) =0.9 n 0 p^= = _0.8 æ 0.0 n , 'n 'n 0.0 'næ næ

확통Ⅰ-0(001~007)OK

확통Ⅰ-0(001~007)OK CONTENTS I 5 5 7 9 6 40 45 47 48 49 5 5 II 57 6 68 70 7 7 77 8 8 84 85 88 89 6 III 9 97 04 4 09 5 7 9 6 4 45 46 47 50 5 5 70 7 7 8 7 I / / 여러 개의 물건을 여러 개의 상자에 나누어 담는 경우의 수는 그 순서와 방법에 따라 달라진다. 9 4.. A B