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 6 4 A B 0 I.

(Galilei, G. ; 564~64) 9 0 6 0 7 9 5 +=5

A, B A B m n A B m+n 0 5 7 5 7 5 (, 4), (, ), (, ), (4, ) 7 (, 6), (, 5), (, 4), (4, ), (5, ), (6, ), 4+6=0 0 0 0 4 9 0 500 00 6 50 5 650 I.

(Einstein, A. ; 879~955) Imagination is more important than knowledge. _=6 A m B n A, B m_n 0 A, B, C A B C A B C

0 08 08 08= _ 08,,,,, 4 08 _4= 04 400 540 05 (a+b)(x+y+z) (p+q+r)(a+b)(x+y) 5 4 4 5 4? 4 I.

988 6 (end) 4 4 A, B, C, D 4 B A A A A C B C C B D B D D D C 4_= n r (r n) «P P Permutation n r «P 5

«P n r n (n-) y r {n-(r-)} y r n (n-) (n-) y (n-r+) «P «P =n(n-)(n-)y (n-r+) «P =n(n-)y(n-r+) r ( M { M 9 n r «P =n(n-)(n-)y (n-r+) 0<r n n r n r «P P =5_4=0 P =6_5_4=0 0 P P 0 0 8 5 6 I.

n n «P r=n «P«=n(n-)(n-)y n n n! n n (factorial) n! n!=n(n-)(n-)y r<n «P «P =n(n-)(n-)y (n-r+) = n(n-)(n-)y (n-r+)(n-r)(n-r-)y (n-r)(n-r-) y = r=0 r=n n! (n-r)! «Pº=, 0!= n r «P = n! (n-r)! ( 0 r n) «P«=n!, «Pº=, 0!= 0 5!! P 7

0 r n «P =n «P (n-)! n! n «P =n = =«P {(n-)-(r-)}! (n-r)! «P =n «P «P n, (n-) (r-) n n, (n-) (r-) «P «P =n «P 04 r n- «P =«P +r «P 0 6 5 5!! 5!_!=40 40 05 5 a, b, c, d, e a, c, d d, e 8 I.

6 4 A, B, C, D 4 ABCD DABC 4 ABCD DABC 4 ABCD DABC 4 ABCD DABC, CDAB, BCDA A D C B B D A C D B C A C B A D 4 4! 4 4 4! 4 =!=6 n (n-)! n n! =(n-)! n 9

0 6 6 6 6 6! =5!=0 6 5 (5-)!! (5-)!_!=48 4 4 4! _4!=4 0 48 4 06 9 6 0 I.

0 9 0 0_0_0_0=0 n r «P P Product P (pi) n r «P «P n r n n y, r n «P «P =n_n_y_n=n ( M { M 9 r n r «P =n n r r n «P

a, b 5 P =fi = 4 P =4 =64 07 6 4 04 A A 4 P _ P = _4 =576 576 08 5 0,,,, 4 09 I.

5 a, a, b, b, b aabbb aabbb a a, a aabbb aabbb b b, b, b aabbb aabbb a a, a! b b, b, b! 5 a, a, b, b, b aabbb, ababb, abbab, abbba, baabb babab, babba, bbaab, bbaba, bbbaa 0 aabbb a a, a b b, b, b a a b b b a a b b b a a b b b a a b b b a a b b b a a b b b a a b b b a a b b b a a b b b a a b b b a a b b b a a b b b!_! 5 a, a, b, b, b x a a, a b b, b, b x_(!_!) 5 5! x_(!_!)=5! 5! x= =0!_!

n p q y, r n n! p!q!yr! p+q+y+r=n 0 05 P Q Q P a, b P Q a 4 b 7! =5!_4! P a b b a b b a Q 5 P R Q R Q P 4 I.

I- A B m, n A, B A B A, B n r n! =;:::::::; (n-r)! ( 0 r n) «P«=, «Pº=, 0!= n! n r «P = n p, q y, r n ;:::::::: ( p+q+y+r=n) p!q!yr! x, y 4 (x, y) n r 7! n!=70 =0 (7-r)! 6 A, B 5

4 5 6 5,,, 4, 5 000 7 6 C, O, F, F, E, E C 8 6 6 9 4, y 6 I.

a, a, b, c A, A, B, C 4 a, a, b, c a, a, b, c a, A 7

6 6 64... A={a, b, c, d} B,A n(b)= B C,A a<c C (a+b) (a+b+c) 8 I.

00 (Lotto) 45 6 6 5 5 6 D 4 6 n r (r n) «C C Combination n r «C 9

«C n r «C r r! «C _r! n r «P «C _r!=«p «C = «P r! r n «C «P n(n-)(n-)y(n-r+) n! «C = = = r! r! r!(n-r)! 0!= «Cº= n r «P n! «C = = ( 0 r n) r! r!(n-r)! «Cº= n r n r «C 5_4 6_5_4 C = =0 C = =0 0 C C 0 I.

0 0 8 5 0 0 r n r «C =n «C n «C =n = =r n! (r-)!(n-r)! =r «C (n-)! (r-)!{(n-)-(r-)}! n! r!(n-r)! r «C =n «C n r n r «C r r r «C n n (n-) (r-) «C n «C r «C =n «C 04 «C =«C«(0 r n) «C =«C +«C ( r n-)

0 6 4 4 4 4 6 4 C 4 C 6! 4! C _ C = _ =5_6=90 4!_!!_! 6 6! 90_6!=90_70=64800 90 64800 05 7 4 4 06 I.

0 8 4 4 8 4 C 4 C 4! C _ C _ =5! 5 07 0 4, (,, ) (,, ) (,, ), (,, ), (,, ), (,, ) 4 n r «H H Homogeneous product «H

«H, (,, ), (,, ), (,, ), (,, ) H =4 0,, (,, ), (,, 4), (,, 4), (,, 4) 4,,, 4 H = C H = C = C r n r>n n r «H =«C n r n r «H H = C = C = H = C =0 08 H H 09 6 5 4 I.

04 x+y+z=5 x, y, z 5 (x, y, z) (,, ) xyyyz, (0,, ) yyyzz, (0, 0, 5) zzzzz, y x, y, z 5 H = C =,,,, +, + +, +, x+y+z=5 (,, ) 5 + 7! 5!_! = C = 0 x+y+z=9 (a+b+c) (a+b)fl (x+y+z) X={,, } Y={4, 5, 6, 7} f f()æf()æf() f f()>f()>f() f()=f()>f() f()>f()=f() f()=f()=f() 5

6 4 8 6 A={a, b, c, d} A a b c d A {a}'{b, c, d}, {b}'{a, c, d}, {c}'{a, b, d}, {d}'{a, b, c} 4 A {a, b}'{c, d}, {a, c}'{b, d}, {a, d}'{b, c} A={a, b, c, d} 4+=7 6 I.

S(n, k) S Stirling numbers n k S(n, k) S(4, )=7 k n n k S(n, k) 0 S(4, ) S(5, ) 0 5 5 ( )( ) 5 S(5, ) ( ) C _ C _ C _ =0! ( ) C _ C _ C _ =5! S(5, )=0+5=5 5 0 4 6 7

6 5 6=++++ 6 6 6 6=5+=4+=+ 6 6=4++=++=++ n k P(n, k) P Partition P(n, k) P(6, )=, P(6, )= k n n k n k P(n, k) 0 P(6, 4) P(8, 5) 8 I.

0 4 9 4 9 9 4 P(9, 4) 9 4 9=6+++=5+++=4+++ =4+++=+++=+++ P(9, 4)=6 6 04 0 7 7=+a+b (a, b ) 6=a+b 6 7=a+b+c (a, b, c ) a=a'+, b=b'+, c=c'+ 4=a'+b'+c' 4 P(7, )=P(6, )+P(4, ) P(8, 4)=P(7, )+P(4, 4) 9

7 (Pascal, B. ; 6~66) (a+b)«( ; 8~98) 4 (+x) (+x) x C (+x) x (+x) =+x+x +x x C (a+b)«(a+b) (a+b) =(a+b)(a+b)(a+b)(a+b) =(a +ab+b ) =a +4a b +b +(a b+ab +a b ) =a +4a b+6a b +4ab +b a b 4 (a+b) b a a b C 40 I.

a, a b, a b, ab, b Cº, C, C, C, C (a+b) (a+b) = Cºa + C a b + C a b + C ab + C b (a+b)(a+b)(a+b)(a+b) a a a a Cºa a a a b a a b a a b a a b a a a ) M } C a b M 0 b b b b C b (a+b)«a«b n (a+b) r b (n-r) a a«b «C (a+b)«n (a+b)«=«cºa«+«c a«b+y+«c a«b +y+«c«b«n = ;R+)«C a«b «C =«C«a«b a b«(a+b)««cº, «C, y, «C, y, «C««C a«b (a+b)«=«cºa«+«c a«b +y+«c a«b +y+«c«b«ø 0 (a+b)fl 4

0 (a+b)fi (a+b)«a«b (a+b)fi = Cºafi + C a (b)+ C a (b) + C a (b) + C a(b) + C (b)fi = afi +5a (b)+0a (4b )+0a (8b )+5a(6b )+ (bfi ) =afi +0a b+40a b +80a b +80ab +bfi 0 (-a)fi (x+y)fl 0 {x- x } fl x (a+b)«a«b «C {x- } fl x C (x)fl {- } = C fl (-) xfl x = C fl (-) xfl xfl =x 6-r=4 r= x x C _fi _(-)=-576-576 0 {x - } fl {x + } x x 4 I.

(a+b)fi 4 6 4 (a+b)«n=,,, y (a+b) Cº C (a+b) Cº C C (a+b) Cº C C C HjK (a+b) Cº C C C C 4 6 4 (a+b)fi Cº C C C C C 5 0 0 5 (x+)fl (x+)fl =xfl +6xfi +5x +0x +5x +6x+ 4 6 4 5 0 0 5 6 5 0 5 6 04 (a+b) (x-) 4

0 «Cº+«C +«C +y+«c«=«(+x)«x «C (+x)«=«cº+«c x+«c x +y+«c«x«x= «=«Cº+«C +«C +y+«c«05 «Cº-«C +«C -y+(-)««c«=0 «Cº+«C +«C +y=«c +«C +«C +y=«06 ( ºCº) +( ºC ) +( ºC ) +y+( ºC º) = ºC º (x+) (x+) =(x+) 44 I.

I- n n r «P =;:::; ( 0 r n) r! n r «H = n k S(n, k) n k P(, ) n (a+b) «= r=0 : (a+b) «n r «C =56 C = C 9 (x+)fi {x+ } x fi 45

4 0 5 5 0 6 0 7 7 8 {x- } xfi x x 9 5 7 6 fi f(x)= CÆ{ } 0 { } (x=0,,, y, 5) f()+f()+f(5)+y+f(5) 46 I.

5 5 5 5 6 4 5 5 5 0 6 47

85 (Guthrie, F. ; 8~899) 4 4 00. 890 (Heawood, P. J.) 5. 90 (Franklin, D.) 5 4. 968 (Ore, O.) (Stemple, J.) 40 4 976 (Appel, K.) (Haken, W.) 4 48 I.

I. 0 x, y - x, 0 y 4 (x, y) 05 n «C =«C «P =60 «C «C =«C +«C 0 5 5 0 5 0 40 06 X={,,, 4} Y={,,, 4, 5, 6} f f() f() f() f(4) 0, 7777 f()<f()<f()<f(4) 04 A B B 07 5 A 966 04 06 050 76 I. 49

08 6 4 [~7] 7 a, b, c, d, e, i, u e, i, u 7 09 4 8 0,,,,,,, 4 A _0 A 6 0 5 4 4 n n n 6 n 7 X={,,, y, n} A, B (A, B) A,B n 50 I.

5 4 0 7 (+x) (+x )«x 5 (+x) (+x )«6 0 6 n 4 4 5 6 I. 5

50 0 00 5 I.

0 0,,,, 4, 5, 6, 7, 8, 9 0 =00 00,,,, 44, 55, 66, 77, 88, 99 0 0fl =000000 555 0 =00 x+y=9 (x, y<{0,,,, 4, 5, 6, 7, 8, 9}) 0 =0000 x+y+z+w=8 (x, y, z, w<{0,,,, 4, 5, 6, 7, 8, 9}) I. 5

II / /

% 987 5~0 mm 70 % 0 7 70 %.. 0 0 4 6 SECRET 6 E 56 II.

A, B A B (Fermat, P. ; 60~665) S S={,,, 4, 5, 6} {, 4, 6} {}, {}, {}, {4}, {5}, {6} 0 57

S A, B A B A'B A, B A;B S S A B A B A'B A;B A, B A, B A;B= A, B A A A AÇ S S A B A AÇ 0 0 A, 5 B A={, 6, 9}, B={5, 0} A;B= A, B A AÇ={,, 4, 5, 7, 8, 0} 0 4,,, 4 5 A B A B A'B 58 II.

P(A) P Probability A P(A) S A A n(a) P(A)= = S n(s) A (Laplace, P. S. ; 749 ~87) S A P(A)= n(a) n(s) 59

0 R, R, R, B, B S S={R R, R R, R R, R B, R B, R B, R B, R B, R B, B B } n(s)=0 A A={R B, R B, R B, R B, R B, R B } n(a)=6 6 = 0 5 5 C C _ C C 6 = = 0 5 C _ C 5 0 04 5,,,, 4 60 II.

= 0 50 00 00 00 500 9 9 4 78 00 0.45 0.8 0.4 0.9 0.4 0.4 0.5 0.4 0. 0 00 50 00 00 00 500 0.4 n A r«n A r«n p p 6

n n 500 00 =0.4 A p A 00 500 r«n r«n p 05 500 000 70 56 68 975 4 8 6.5..5.5.5 0.7 0.8 6 II.

0 A P(A)= A P(A)=0 S A S 0 n(a) n(s) n(s) P(A)= 0 P(A) S n(s) n( ) P(S)= =, P( )= =0 n(s) n(s) n(a) n(s) 6

A 0 P(A) S P(S)= P( )=0 0 0 0 4 5 A, B n(a), n(b) n(a'b)=n(a)+n(b)-n(a;b) A B n(a'b) n(a) n(b) P(A'B)= = + - n(s) n(s) n(s) =P(A)+P(B)-P(A;B) n(a;b) n(s) 64 II.

A, B P(A;B)=0 P(A'B)=P(A)+P(B) A, B P(A'B)=P(A)+P(B)-P(A;B) A, B P(A'B)=P(A)+P(B) 0 0 0 6 9 6 A, 9 B A;B 8 P(A'B)=P(A)+P(B)-P(A;B) 5 7 = + - = 0 0 0 0 C, D C, D P(C'D)=P(C)+P(D) C = + ºC C _ C ºC 7 5 = + = 9 9 9 7 0 9 65

0 5 8 6 0 9 8 0 = 0 0 0 5 0 - = 5 5 A, AÇ P(A'AÇ )=P(A)+P(AÇ ) P(A'AÇ )=P(S)= A AÇ P(AÇ )=-P(A) 66 II.

A AÇ P(AÇ )=-P(A) 0 4 5 A AÇ C 4 P(AÇ )= = = ªC 84 0 P(A)=-P(AÇ )=- = 0 04 7 05 (de Mere, C. ; 607~684) 5 { } =0.509 6 4 67

A S S A, B A AÇ P(A)=;::::::; n(s) n 0 P(A) P(S)= P( )= P(A'B) =P(A)+P(B)- A, B P(A'B)=P(A)+P(B) P(A Ç )=- A r«r«;::; n 0 0 A 0 B A B A, B A, B P(A)=, P(B)=, P(A;B)= 4 6 P(A'B) 68 II.

II- 4 5,,, 4, 5 5 4 700 5 5 6 5 5 A, B, C, D, E A B 7 6 8 A B A B 9 a, b, c ax -bx+cæ0 x 69

(Pacioli, L. ; 445~57) 5 4 70 II.

(Zipf, G. K. ; 90~950) the of the and the r..9 5.4 r.8 8 8 0 8 55..9.8 5.4 7

5~4 8.6 %.9 % 6.4 % 5~9 0. % 0~4 8.5 %.. 4 a, b, c, d a 7 II.

50 7 9 8 0 0 50 S A n(a;b) n(a) A A A;B B n(a;b) n(a) 7 = 9 n(a;b) n(a;b) n(s) = = P(A;B) n(a) n(a) P(A) n(s) 7

0 A B A B P(B A) P(B A) P(B A)= P(A;B) P(A) A B P(B A)= P(A;B) P(A) P(A)>0 A B P(A)=0.9, P(A;B)=0.8 P(A;B) 0.8 P(B A)= = = 8 P(A) 0.9 9 0 5 7 0 4 5 7 8 74 II.

A, B A, B P(A;B)=P(A)P(B A) P(A)>0 P(A;B)=P(B)P(A B) P(B)>0 A, B(P(A)>0, P(B)>0) P(A;B)=P(A)P(B A)=P(B)P(A B) 0 5 0 A, B 5 P(A)= 0 4 P(B A)= 9 5 4 P(A;B)=P(A)P(B A)= _ = 0 9 9 9 0 4 75

0 60 % 40 % 0.7 0. A B D P(A)=0.6, P(B)=0.4, P(D A)=0.7, P(D B)=0. P(D)=P(D;A)+P(D;B)=P(A)P(D A)+P(B)P(D B) =0.6_0.7+0.4_0.=0.46 P(D;A) P(A)P(D A) 0.6_0.7 P(A D)= = = = P(D) P(D) 0.46 04 A, B 70 % 0 % A, B %, % A 7 0 76 II.

6 a b a b 6 q a u q a u 4 A, B P(B A), P(B) P(B A), P(B) P(B A) P(B) P(A;B) ;7#;_;7$; 4 P(B A)= = = =P(B) P(A) 7 ;7#; A, B P(B A)=P(B) A, B P(B A)+P(B) A, B 77

A, B P(A)>0, P(B)>0 A B P(B A)=P(B) P(A;B)=P(A)P(B A)=P(A)P(B) P(A;B)=P(A)P(B) A, B P(A)>0, P(B)>0 A, B A B P(A;B)=P(A)P(B) P(A)>0, P(B)>0 0 A, B, C A B, A C P(A)=, P(B)=, P(C)= P(A;B)=, P(A;C)= 6 6 6 6 6 P(A;B)+P(A)P(B) A B P(A;C)=P(A)P(C) A C 0 4 6 8 4 A, B A B 0 S A, B P(B A)=P(B AÇ ) A B P(A)>0, P(B)>0 78 II.

5 Z Y ZY 5 C 5 { } 5 { } 6 6 C { } 5 { } 6 6 4 5 Z Z Y Y Y Z Y Z Y Y Z Y Y Z Y 4 Z Y Y Y Z 5 Y Z Z Y Y 6 Y Z Y Z Y 7 Y Z Y Y Z 8 Y Y Z Z Y 9 Y Y Z Y Z 0 Y Y Y Z Z A p n A r «C p (-p)«(, r=0,,, y, n) 79

0 5 C { } { } 80 = 4 0 Cº{ } { } fi, C { }{ } -[ Cº{ } { } fi + C { }{ } ]= 4 80 4 4 0 0 4 0 987654 0 0 (Galton board) (Galton, S. F. ; 8~9) 80 II.

II- A, B A P(A;B)=P(A) B =P(B) P(A;B) P(B A)=;::::::::; P(A)>0, P(B)>0 P(A)>0 A B P(A;B)=P(A) P(A)>0, P(B)>0 A p n A r «C p ( ) «r=0,, y, n) 0 0 4 A B P(A)=, P(A;B)= 4 P(B) 8

4 0.7 0.4 5 5 4 6 A, B A 4 5 A 7 8 P x y P y 4 P O 4 x 8 II.

4 8

8 (Bayes, T. ; 70~76) 76 0 0.95 0.0 0.0 0.49 (context) 84 II.

II. 0 k, m, o,,, 7 k, m, o 05 0 50 55 06 9 0 A, B P(A)=P(B), P(A'B)= P(A), P(B) 4 07 40 40 5 04 8 -, -, 0,,,,, a, b ab=0 08 60 % 7 64 7 4 64 5 64 0.0 0.005 II. 85

09 0 4 A, B, C C 5 5 [~6] 4, 5, 6 4 5 P(A)>0, P(B)>0 A, B A, B P(AÇ BÇ )=-P(A) A, B P(AÇ B)=-P(A B) A, B AÇ, BÇ A, B A, B A, B A, B 4 4 4 4 5 A, B A 6,,,,, B 6,,,,, A 6 86 II.

5 6 8 A, B, C 5 A B C A, B, C 9 8 7 4 A A A 4 4 5 6 II. 87

(Sports Analyst) 00 4 (Sabermetrician) 88 II.

(Bertrand, J. L. F. ; 8~900) ' O ' O x x x (x, y) II. 89

III / /

생태학자들은 임의의 야생동물에 GPS 추적 장치를 부착하여 그들의 습성을 파악한다.

65 90 85 80 75 70 65 60 55 50 45 40 5 0 5 0 5 0 5-500,000-400,000-00,000-00,000-00,000 0 00,000 00,000 00,000 400,000 500,000... 4. 50 cm 55 ~ 60 0.04 60 ~ 65 7 0.4 65 ~ 70 70 ~ 75 75 ~ 80 80 ~ 85 0 5 5 50 0. 0. 0. 0. 9 III.

0 0.0 0.07 0. 0.08 0.06 0.04 0.0 0 4 5 6 7 8 9 0 H, T S X X X 0,, S HH, HT TH, TT 0 S HH HT TH TT X R 0 9

X, Y, Z x, y, z 0 0 AB C AC X S H X R H T S S={H, TH, TTH, y} X TH TTH y y 0 AB C AC X A C 0 B S S={x 0 x 0} X 0 0 0 9 0 OX 94 III.

X X X x P(X=x) X 0,, Cº _ C P(X=0)= = C 0 C _ C P(X=)= = C 5 C _ Cº P(X=)= = C 0 X X x, x, x, y, x«p(x=x )=p (i=,,, y, n) x, x, x, y, x«p, p, p, y, p«x P(X=x ) X X P(X=x) p X P(X=x ) x x x y x«p p p y p«p«p p O x x x y x«x 95

X x, x, x, y, x«p(x=x ) (i=,,, y, n) 0 P(X=x ) P(X=x )= n ;I+! 0 4 X X X X 0,,, Cº_ C C _ C P(X=0)= =, P(X=)= = C 5 C 5 C _ C 8 C _ Cº 4 P(X=)= =, P(X=)= = C 5 C 5 CÆ_ C Æ P(X=x)= (x=0,,, ) C Xæ 8 4 P(Xæ)=P(X=)+P(X=)= + = 5 5 5 CÆ_ C Æ P(X=x)= (x=0,,, ) C 5 0,,, 4, 5 X X P(X ) 96 III.

500 0 ;50!0; 5 0 5 5 479 500 = = 0_+5_5+_5 500 5 5 =0_ +5_ +_ =0. 500 500 500 { _ } E(X) E Expected value m mean X X x x y x«p(x=x ) p p y p«n ;I+! x p X E(X) m 97

X P(X=x )=p (i=,,, y, n) X E(X) E(X)=m= ;I+! x p n 0 5 X X X X,,, 4, 5 4 P(X=)=, P(X=)= _ = 5 5 4 5 4 4 P(X=)= =, P(X=4)= _ = 5 4 5 5 4 5 4 P(X=5)= = 5 4 5 X X P(X=x) 4 5 5 5 5 5 5 E(X)=_ +_ +_ +4_ +5_ = 5 5 5 5 5 0 500 00 98 III.

x, f, n m x x f f { _ } k m= = ;I+! x f n x f n f p n k m= ;I+! x p = { _ } = {(x -m) f +(x -m) f +y+(x -m) f } = ;I+!(x -m) f n n k f n =p k ;I+!(x -m) p X (X-m) X P(X=x ) x x y x«p p y p«n E((X-m) )=;I+!(x -m) p m=e(x) V(X) V Variance X V(X) 99

r(x) r standard deviation s (sigma) V(X) " V(X) X r(x) X X P(X=x )=p (i=,,, y, n) X V(X) r(x) n V(X)=E((X-m) )=;I+!(x -m) p m=e(x) r(x)=" V(X) V(X) (x -m) X V(X) V(X)=E((X-m) ) n = ;I+!(x -m) p n = ;I+!(x p -mx p +m p ) ;I+! x p =m, ;I+! p = n = ;I+! x p -m +m n = ;I+! x p -m n n =E(X )-{E(X)} V(X)=E(X )-{E(X)} E(X ) X {E(X)} X 00 III.

0 X X X X P(X=x) 4 5 0. 0.7 0.5 0. 0.05 X E(X)=_0.+_0.7+_0.5+4_0.+5_0.05=.8 X V(X)=E(X )-{E(X)} = _0.+ _0.7+ _0.5+4 _0.+5 _0.05-.8 =. r(x)='ƒ. V(X)=., r(x)='ƒ. 0 X X P(X=x) ;:; P(X=x) 0.4 0. P(X=x) 0.4 0. O x O x O 4 x 0

ax+b X P(X=x )=p (i=,,, y, n) Y=aX+b(a, b, a+0) E(Y)=E(aX+b) n = ;I+!(ax +b)p n =a;i+! x p +b;i+! p =ae(x)+b n Y=aX+b(a, b, a+0) V(Y)=V(aX+b) n = ;I+![(ax +b)-{ae(x)+b}] p E(X)=m n = ;I+! a (x -m) p n =a ;I+!(x -m) p =a V(X) r(y)=" V(Y)=" a V (X)= a r(x) a, b(a+0) E(aX+b)=aE(X)+b V(aX+b)=a V(X) r(ax+b)= a r(x) 0 X 7 6 X X- 0 III.

ax+b 0 X m, r Z= X-m r E(X)=m, V(X)=r X-m m m m E(Z)=E{ }= E(X)- = - =0 r r r r r X-m V(Z)=V{ }= V(X)= r = r r r E(Z)=0, V(Z)= 04 X m, r X-m T=50+0{ } r T 60 0 75 )= )+ _ (I.Q.) 00 5-0

(Maxwell, 500 000 m/s J. C. ; 8~879) 7 æ (m/s) 00 ;::::::::::::; 5 ~ 0 0 ~ 5 5 ~ 0 0 ~ 5 5 ~ 0 0 ~ 5 6 6 8 6 00 0.6 0. 0.6 0.8 0. 0.06 0.0 0.044 0.06 0.04 0.0 0 5 0 5 0 5 0 5 0 5 X X 5 5 X X 0 5 P(0 X<5)=0. 04 III.

= _ = ;::::::::::::; 0.06 0.04 0.0 P(0 X<5) 0 5 0 5 0 5 0 5 ;::::::::::::; 0.06 ;::::::::::::; 0.06 P(0 X<5) 0.04 0.0 0.04 0.0 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 x x X a b P(a X b) x x=a, x=b f(x) X 05

f(x) X f(x)(a x b) f(x)æ0 f(x) x f(x) P(a X b) f(x) a, b(a a b b) P(a X b) f(x) x O a a b b x=a, x=b x f(x) f(x) X x P(X=x)=0 P(a X b)=p(a X<b)=P(a<X b)=p(a<x<b) 0 X f(x)=a(-x)(0 x ) a P( X ) f(x)æ0 aæ0 f(x) x, y a f(x) x x=0 a= a= f(x) x x=, x= 9 4 P( X )= _{ + }_= 9 9 f(x) a O f(x) ;:; O f(x)=a(-x) f(x)=;:;(-x) 9 x x 9 0 X f(x)=a x- (0 x ) P{ <X< } a 06 III.

I X(a x b) P(a X b)(a a b b) f(x) x x=a, x=b f(x) O a P(a X b) f(x) a b b x P(a X b)= :Ab f(x)dx f(x) f(x)æ0 :Ú f(x)dx= f(x) f(x) 0 X f(x)=ax (- x ) a P{0 X } f(x)æ0 aæ0 x :_! f(x)dx= :_! ax dx=a[ ]_!= a a + = a= x P{0 X }= x dx=[ ]) ;!; :) ;!; = 6 6 0 X f(x)=a(4x-x )(0 x ) a P( X ) 07

X V(X) = :Ú (x-m) f(x)dx =E(X )-m X f(x)(a x b) X E(X) V(X) r(x) E(X)= :Ú xf(x)dx V(X)=E((X-m) )=:Ú r(x)=" V(X) (x-m) f(x)dx m=e(x) X ax+b a b a+0 E(aX+b)=aE(X)+b V(aX+b)=a V(X) r(x)= a r(x) 0 X f(x)= x(0 x ) X x x 4 x x E(X)= :) dx=[ ])=, E(X )=:) dx=[ ])= 6 8 6 V(X)=E(X )-{E(X)} =- = 9 9 4 E(X)=, V(X)= 9 0 X f(x)= 4 (-x )(- x ) X X+ 08 III.

4 00 7 5 8 8 0.9 5 X X X P(X=x)= CÆ0.9 0.fi (x=0,,,, 4, 5) A p n A X X 0,,, y, n X P(X=x)=«CÆp q«(q=-p, x=0,,, y, n) 09

B(n, p) B Binomial distribution B(n, p) X X A 0 A X A X B{0, } 6 6 0 0.4 0 X X 8 X X 0 B(6, 0.) X= Xæ B(6, 0.5) P(X=x) 0.6 0.4 0. 0 n=6 p=0. 0 4 5 6 x 0 III.

0 0 0 40 0 5 X X B{0, } 40 X<4 P(X<4)=P(X=0)+P(X=)+P(X=)+P(X=) = ºCº{ } 4 { } 4 + ºC { }{ } + ºC { } 4 { } + ºC { } 4 { } 5 5 5 5 5 5 5 5 54 4 = =0.879 5 5 0.879 0 0. 0 0.8 =0.009, 0.8 =0.007 0 X X B{0, } 0 0_ =5 E(X)=5 X B(n, p) X E(X)=np V(X)=npq ( q=-p)

X B(n, p) E(X)=np V(X)=npq q=-p r(x)=' npq 04 0.8 50 X X n X X X n X n n X 6 B{n, } 6 6 n X n 6 6 III.

B{0, } 6 X P(X=x) 0.0 0.907 B{0, } 6 X P(X=x) 0.68 4 0.847 5 0.9 6 0.60 7 0.098 B{50, } 6 50 n=0 X X P{ - <0.}=P{ -0.< < +0.} n 6 6 0 6 =P(0.666y<X<.666y) = ;?+! P(X=x) = ºCÆ{ } 5 ;?+! { } 0-x =0.67 6 6 n=0 X P{ - <0.}=P(<X<8) n 6 7 = ºCÆ { } 5 ;?+# { } 0-x =0.785 6 6 n=50 X P{ - <0.}=P(.y<X<.y) n 6 = ºCÆ { } 5 ;?+$ { } 50-x =0.9455 6 6 X n P{ - <0.} n 6 0. 0.0, 0.00, y X n p A p n A X p X n n n

(Gauss, K. F. ; 777~855) 8 X X X f(x)=.7 - x r f(x) 'ß pr x, r f(x)=;:;::;.7 -;::; r ' pr x f(x) x=0 f(x) O x m x X f(x) N(m, r ) N Normal distribution f(x)= e - ' pr x, m, r, e.788y ) X N(m, r ) (x-m) r x=m r m f(x) ::::: ' pr m=0 m= m=4 O 4 x y 4 III.

m r f(x) ;::::: ' p ;:::: ' p ;::::: ' p O m r=;:; r= r= x N(m, r ) f(x)= e - (x-m) r ' pr x=m x x=m r m ' pr m r f(x) x f(x) ::::: ' pr O f(x)= ::::: ' pr m -;:::::::; e (x-m) r x X N(m, r ) X E(X) V(X) r(x) E(X)=m, V(X)=r, r(x)=r 05 4 x 5

X m, r Z= X-m r 0, X Z Z X X N(m, r ) X Z 0 N(0, ) X N(m, r ) X Z= X-m r N(0, ) f(z) f(x) m=0, r= Z N(0, ) f(z) f(z)= e - z z ' p z P(0 Z z) 70 f(z) P(0 Z z) f(z)=;:;::;e -;:; ' p O z z z 6 III.

P(0 Z.96)=0.4750 P(0 Z )=0.4987 z 0.00 y 0.06 y.9 0.4750.0 0.4987 0 Z N(0, ) P(Z.96) P(-.5 Z 0.) z=0 P(Z.96) =P(Z 0)+P(0 Z.96) =0.5+0.4750=0.9750 P(-.5 Z 0.) =P(-.5 Z 0)+P(0 Z 0.) =P(0 Z.5)+P(0 Z 0.) =0.4+0.79=0.55 f(z) O.96 z f(z) -.5 O 0. z 0.9750 0.55 06 Z N(0, ) P(Z -.65) P(Zæ) P(-.58 Z.58) P(-0.5 Z ) 07 Z N(0, ) c P( Z c)=0.80 P(Zæc)=0.958 7

P(a X b) a-m b-m =P{ Z } r r X N(m, r ) P(a X b) X X-m Z= r 0 X N(, 4) P( X 6) X- X, '4= Z= N(0, ) - X- 6- P( X 6)=P{ } f(z) =P(- Z.5) =P(- Z 0)+P(0 Z.5) =P(0 Z )+P(0 Z.5) =0.4+0.4=0.7745 - O.5 z 0.7745 08 X N(0, 5) P(X ) P(7 X 7) (de Moivre, A. ; 667 754) 7 «CÆp q«n 8 III.

04 0 00 78 0.5 68 88 % P(0 Z.04)=0.5 X 68-78 88-78 P(68 X 88)=P{ Z } 0 0 =P(- Z ) =_P(0 Z ) =_0.4=0.686 68.6 % 0 45 00 45 c P(Xæc)= =0.5 00 c-78 c-78 P{Zæ }=0.5 P{0 Z }=0.5-0.5=0.5 0 0 c-78 =.04 c=88.4 0 88.4 68.6 % 88.4 09 (I.Q.) 00 6 48 % % P(0 Z.88)=0.47 9

5,,, 4, 5 n 5 X X B{n, } n=0, 0, 50 n 5 P(X=x) P(X=x) P(X=x) 0.6 0.6 0.6 0.4 n=0 p=0. 0.4 n=0 p=0. 0.4 n=50 p=0. 0. 0. 0. 0 0 456789 x 0 0 456789 x 0 0 4 6 80468 x B(n, p) n B{n, } n=50 X 0 40 5 40 P(0 X 40)= ºCÆ{ } 4 { } fi x=0 5 5 B(n, p) 0 III.

X B(n, p) n X N(np, npq) q=-p n npæ5, nqæ5 05 40 % 600 5 55 600 X X n=600, p=0.4 B(600, 0.4) npæ5, nqæ5 n np=40, npq=44 X N(40, 44) 5-40 55-40 P(5 X 55)=P{ Z } =P(-.5 Z.5) =_P(0 Z.5) =_0.944=0.7888 0.7888 0 40 % 50 75 45 69

P(X=x)X X x f(x) X P(a X b) f(x) P(X=x )=p (i=,,, y, n) x x=a, x=b = x p n i= V(X)= -{E(X)} r(x)= V(X) X B(n, p) E(X)= V(X)=npq r(x)= npq q=-p X N(m, r ) X X-m Z=;:::::: r N(0, ) X x+ P(X=x)= (x=0,,,, 4) 5 X X f(x)(- x 0) P(- X 0) f(x) f(x) - O x III.

III- X B{0, } X+5 4 4 X N(0, 6) P(6 X ) P(X 4) 5 6 6 X X P(X 4) 6 X X X X P(X=x) 0 4 ;8!; a ;8!; b ;4!; ax+b 7 X x+ P(X=x)= (x=-, 0,, ) 0 Y=aX+b(a>0) 0 a, b

8 a(+x) (- x<0) X f(x)=[ a(-x) (0 x ) a P(- X ) 9 a 8 5 a 0 00 g, 5 g 8 400 g 700 g 0.0 0 0 0.99 =0.94, 0.99 =0.904 78 78 0 % 400 4 III.

0 5 5 X X 5

... 80 X X X N(5, 9) P( X-5 6) 6 III.

960 5 (Brix) (Brix, A. F. ;798~870) 7

0 0 500 0 8 III.

00 65 00 00 6 65 00 6 00 4,,, 4 4 X X (X, X ) (X, X ) (, ) (, ) y (, ) (, ) y (, ) (, ) y (4, ) (4, 4) y.5 y y X X m, r, r n X, X, y, X«X, S, S 9

X, S, S X, S, S n n X = ;I+! X, S = ;I+!(X -X ), S="çS n n- S n ;I+! (X -X ) n- X P(X=x) X P(X=x) 4 4 4 4 4 ;:; 4 O 4 x m r 5 5 m=e(x)=, r =V(X)= yy` 4 (X, X ) X, X X = X +X (X, X ) X (X, X ) X (, ) (, ) (, ) (, 4) (, ) (, ) (, ) (, 4).5.5.5.5 (, ) (, ) (, ) (, 4) (4, ) (4, ) (4, ) (4, 4).5.5.5.5 4 X x X X,.5,, y, 4 P(X =x ) X X P(X =x ).5.5.5 4 6 6 6 4 6 6 6 6 ;:; P X= ;:; x ;::; 4 6 ;::; 6 O 4 ;:; x 0 III.

X E(X ) V(X ) 5 5 E(X )=, V(X )= yy` 8 X E(X ), V(X ) m, r E(X )=m, V(X )= r n X ;:; P X= ;:; x ;:; 4 ;::; 6 ;:; 8 ;::; 6 O 4 ;:; x X=;:;(X +X ) X=;:;(X +X +X ) N(m, r ) n X N{m, r n } m, r n X, X, y, X«X E(X )=m, V(X )= N(m, r ) X N{m, r n } r n n X N{m, r n } n næ0

0 60 g, g 5 0 8 6 9 0 X X X X N(8, 6) 9 X X N{8, X Z= X -8 6 0-8 P(X æ0)=p Zæ =P(Zæ) ª 6 º =0.5-0.4=0.587 6 9 } N(0, ) 0.587 0 A B 40 0 A B 6 45 III.

N n m x N p population ratio n n N p m x p^ x m m p^ p^ P^ p^ n X p^ p^= X n

00 00 p p= = 00 00 40, 4 p^ 4 p^= = = 40 0 0 00 50 A 6 A p^ 00 4 p^ p^= X n X n X 0,,, y, n p X B(n, p) p^ X E(p^)=E{ }= E(X)= np=p n n n X pq V(p^)=V{ }= V(X)= npq= q=-p n n n n Z= p^-p æ ;: nœ: n X-np Z= Z 'ƒnpq p^= X B(n, p) n X N(np, npq) p^ N{p, } p^ Z= p^-p N(0, ) æ ;: nœ: X n pq n 4 III.

p n p^ pq p^-p N{p, } Z= N(0, ) n æ ;: nœ: q=-p n npæ5, nqæ5 0 0.8 00 0.9 00 p^ p=0.8 npæ5, nqæ5 n p^ N{0.8, p^-0.8 Z= N(0, ) 0.8_0. æ 00 P(p^æ0.9)=P Zæ ª 0.9-0.8 0.8_0. æ 00 P(p^æ0.9)=P(Zæ.5)=0.5-0.498=0.006 º 0.8_0. 00 } 0.006 04 6 % 00 0 5

95 (Gallup, G. H. ; 90~984) (Boulding, K. E. ; 90~99) r g m g n X X X -m P -.96.96 ª r 'n =P{X - m X + }=0.95 º 6 III.

a, b(a<b) {x a<x<b} {x a x b} {x a<x b} {x a x<b} (a, b), [a, b] (a, b], [a, b). N(m, r ) n X N{m, Z= X -m r 'n N(0, ) Z -.96.96 P(-.96 Z.96)=0.95 X -m P -.96.96 ª r 'n º } X r r =P{X -.96 m X +.96 }=0.95 yy` 'n 'n r n -.96 f(z) O 0.95.96 z r r m [X -.96, X +.96 ] 'n 'n 0.95 X x r r [x -.96, x +.96 ] 'n 'n m 95 % X x m m 95 % n 95 % m ;:; x-.96;::; r 'n ;:; x m ;:; x ;:; x ;:; x y ;:; x«0.95 ;:; x ;:; x+.96;::; r 'n ;:; x 7

0 m 99 % r r [x -.58, x +.58 ] 'n 'n N(m, r ) n X x m r r 95 % [x -.96, x +.96 ] 'n 'n r r 99 % [x -.58, x +.58 ] 'n 'n r n (næ0), S s r n 99 % _.58 95 % _.96 r 'n r 'n m _.96;::; r 'n _.58;::; r 'n 0.95 0.99 ;:; x n [0, 00] 00 % n 8 III.

0 5 5 75 5 5 95 % n=5, x =75, r= 5 95 % [75-.96_, 75+.96_ ] 'ß5 'ß5 [70.96, 79.704] [70.96, 79.704] 0 9 44 64 95 % 0 64 G= 0.6, 0. G 99 % G 04 m g, g r m 95 % _.96 'n 0.5 n 9

TV (people meter) TV 40 648 p^ n p q=-p p^-p P -.96.96 =P{p^- p p^+ }=0.95 ª º æ ;: nœ: p^ 648 p^= = =0.5 40 p^ n N{p, p^ Z= p^-p N(0, ) æ ;: nœ: pq n } Z p pq p^-p æ p, q p^, q^(q^=-p^) Z= n æ p^q^ n N(0, ) p 40 III.

Z -.96.96 P(-.96 Z.96)=0.95 p^-p P -.96.96 ª æ p^q^ n p^q^ p^q^ =P{p^-.96æ p p^+.96æ }=0.95 n n º -.96 f(z) O 0.95.96 z p^ p^ p^q^ p^q^ [p^-.96æ, p^+.96æ ] n n p 95 % 05 p 99 % p^q^ p^q^ [p^-.58æ, p^+.58æ ] n n n p^ n p p^q^ p^q^ 95 % [p^-.96æ, p^+.96æ ] n n p^q^ p^q^ 99 % [p^-.58æ, p^+.58æ ] n n q^=-p^ n np^æ5 nq^æ5 n æ p^q^ æ n 4n 4

0 (Not in Education, Employment or Training ; NEET) 5~4 00 7 95 % 00 p^ 7 p^= =0.6 00 np^æ5, nq^æ5 n 95 % 0.6_0.64 0.6_0.64 [0.6-.96æ, 0.6+.96æ ] 00 00 [0.9, 0.47] [0.9, 0.47] 06 (Blind test) A, B 00 45 A % A 99 % 07 95 % _.96æ 0. n 4n 4 III.

III- m, r n X N{m, } p n p ^ N{p, } q=-p N(m, r ) n x m 95 %: [x- ;::;, r x+ ;::;] r 'n 'n 99 %: [x-.58;::;, r x+.58;::;] r 'n 'n n p ^ n p 95 %: pq ^^ pq ^^ [p-.96 ^ ;::;, n p+.96 ^ ;::;] n 99 %: [p-.58 ^, p+.58 ^ ] q=-p ^ ^ N(0, 6) 00 84 0. 99 % 4

X 0 g, 5 g 6 X P(X 8) P(8 X ) 4 0 % 00 0 % 0 % 5 n 80 50 95 % [70., 89.8] n 6 6 8 95 % 7 9 p n p^ p^-p 0.5"çp^q^ 0.95 n q^=-p^ 44 III.

000 50 0 40 %, 00 50 %, 500 0 % 95 % 95 % A 0 0 A 0. A 0.4 A 95 % 45

40 7648 5 46 III.

III. 0 5,,, 4, 5 5 05 X f(x)=kx(0 x 4) k X 8 6 5 6 P(X ) 7 8 0 X X X 5 P(X=x) a ;9%; a 06 P(x, y),,, 4 x 5, 6 y 0 P x X X P y Y Y-5 0 X 8X+ 07 80 g, 0 g 00 5490 g 04 X E(X)=a, E(X )=6a V(X)=9 4X- 000 III. 47

08 00 g, 5 g 6 00 6 N(00, 5) 5 4 [~6] X P(X=0)=-P(X=) E(X)=V(X) P(X=0) P(X=0)+ 6 X f(x) (0 x a) f(x) ;:; 5 O ;:; f(x) ;:; 5 a x 09 00 P{a X 5 a} a 5.5 kg, 0.6 kg 95 % 9 000 00 % 0 0 6 65 0 6 95 % 48 III.

4 80 % 00 6 64 p 99 % 70 5 [ -a, +a] 5 5 n p 95 % [ -b(n), +b(n)] 4 4 a 5 0 6 6 b(n) a n 4 4 5 6 III. 49

B(n, p) N(m, r ) X B{50, } P(4 X ) 6 P(4 X ) 4, X N(5, 6) P(X ) P(X ) X B(50, 0.4) P(0 X 0) X N(0, 6) P(0 X 0) 50 III.

A, B A 000 B 000 500 B 000 500 _000- _500=50 50 B 000 A 000 500 A 50 A, A A III. 5

I 08 09 500 0 0 0 50 p.0 0 p.5 p. 0 0 m+n, m_n, «P, n!,,, n-, n, n! 0 0 04 05 0 0 0 04 05 06 07 5 5 4 6 p.5 4 840 70 670 0 400 (n-)! r(n-)! «P +r «P = + (n-r-)! (n-r)! (n-)! = {(n-r)+r} (n-r)! n! = (n-r)! =«P «P =«P +r «P 6 4560 440 5 96 4 5 6 7 5 6 64 P 4 4! P _4!=44 4 (4-)!! (4-)!_!=,, 5 5 _5 =75 4 5 4_5 =500 C 5!!_! 5! _ =90!_! 5

8 9 P 4! P =80 4! =4! 4! =4! 4! =6!_! 4! =! fi 4! =! ~fi 4+4+6++=8 4! a, a, b, c +4=6 _6=7 0 0 0 p.8 6 8 a +a b+ab +b a +b +c +ab+bc+ca p.9 5 5 0 56 0 a, a, b, c 4!! = p.7 0 B, C, A, A C, B, A, A A, B, C, A B, A, C, A A, C, A, B C, A, A, B 4 04 05 06 n! «C«= (n-r)!{n-(n-r)}! = =«C «C =«C««C +«C (n-)! (n-)! = + (r-)!(n-r)! r!(n-r-)! (n-)! = {r+(n-r)} r!(n-r)! n! = r!(n-r)! =«C «C =«C +«C 0 6 40 n! (n-r)!r! I. 5

07 600 06 08 09 0 6 0 5 5 55 8 5 70 «C n r+0, r+n «C = n! r!(n-r)! n p.6 0 6 5 0 0 65 04 8 «C, «C, n,k«c a«b p.45 p.40 0 Cº, C, C, C, C, C, C C afl b 0-5a+0a -0a +5a -afi 64xfl +9xfi y+40x y +60x y +60x y +xyfi +yfl 0-0 04 a +7afl b+afi b +5a b +5a b +a bfi +7abfl +b x -8x +8xfl -56xfi +70x -56x +8x -8x+ 05 (+x)«=«cº+«c x+«c x +y+«c«x«x=- 0=«Cº-«C +«C -y+(-)««c««cº+«c +«C +y+«c«=«yy` «Cº-«C +«C -y+(-)««c«=0 yy` + («Cº+«C +«C +y)=««cº+«c +«C +y=«- («C +«C +«C +y)=««c +«C +«C +y=«4 5 6 8 7 7 4xfi +405x +70x +90x +5x+ 0 5 xfi +5x +0x+ + + x x 0 ºC ºC - ºC - ºC =900 ºC ºC 0 ºC C ºC -5_ C =00 x, y, z x+y+z=0 x, y, z H =ªC =6 xfi 54

7 8 9 S(7, ) 7 5 4 4 5 C _ C _ C _ =! 4 C _ C _ C =05 C _ C _ C _ =70! C _ C _ C _ =05! {x- S(7, )=+05+70+05=0 x }7 C x {- } = C (-) x x x =xfi r= xfi x =x r= C _(-)=-4 x C _(-) =-80 5 { C _ C _ C _ }_!=0! 6 C =0 0 { + } fi 0_0=600 = Cº{ } fi + C { } { }+y+ C { } fi =f(0)+f()+y+f(5) yy` fi {- + } =- Cº{ } + C { } { }-y+ C { } =-f(0)+f()-y+f(5) yy` + { f()+f()+y+f(5)} ={ + } +{- + } =+{ } f()+f()+y+f(5)= [+{ } ] p.47 C C _ C C C _ C ( C _ C _ C )_( C _ C _ C )=40 C C _ C C _ C _ C =0 40+0=440 C C C C ªC ( C _ C )_( C _ C )_ªC =540 440+540=980 0 I. 55

p.49 (n+)n 0 x 6 y 5 6_5=0 (n-)(n-) = + (n-)(n-) n -9n+8=0 0 P =fi = n= n=8 0 0 9 næ4 n=8 0 06 07 P =96 H =ªC =6 P =4 P =60 C =5 H = C = 9_0_0 =900 08 6 4 C 04 P B 4 P, C Q, R Q A B A R (-)! A B C _ C _(-)!=0! 7!_6! =76 09 A P B A Q B 6! 7! _ =700!_! 4!_! { C _ C _ }_!=6!! 6_!_!=4 05 A R B 76--700-=04 «C =«C«=«C n-=5 n=8 n(n-)(n-)(n-) =60_ (n-)(n-)(n-)! 0 4 x, y, z, w x+y+z+w=5 x, y, z, w H = C =64 n(b)=k B A (A, B) «C «C _ k 0 n næ4 n=60_ n=0! ;Kn+) «C _ =(+)«=«56

a, b, c, d 5 P(0, 4) yy` 4!=4 yy` 0 4 e, i, u a, b, c, d 0=7+++=6+++ e, i, u =5+++=5+++ P =60 =4+4++=4+++ =4+++=+++ 4_60=440 yy` =+++ yy` a, b, c, d 4 0,,,,,, 4 P(0, 4)=9 P(n, k) 0 4 yy` 7! =840 yy`! 0,,,,,, 4 7! =60 yy`!_! _0 A 840+60=00 yy` 6 x, y x+y=0x, y (x, y) (0, 5), (, 4), (4, ), (6, ), (8, ), (0, 0) yy` A Cº+ C + C + C +ªC + ºC º=89 yy` 4 «C yy` n_«c «C -n_«c -n=6 n -9n +0n-96=0 n=8 n= '4å7i n yy` n n=8 yy` n 4 7 (+x) C x (+x )««Cß x ß C _«Cß_x ± ß yy` x r+s= r=0, s= Cº_«C =n r=, s=0 C _«Cº=6 x n+6 yy` n+6= n=6 yy` (+x) (+x )«x n I. 57

II p.68 n(a),,, 0, P(A;B), P(A) p.56 5 A={, 6, 9}, B={,, 5, 0} A, B 0 0 0 04 05 0 0 0 04 05 p.57 H T {HH, HT, TH, TT} {HH}, {HT}, {TH}, {TT} A, B {(, ), (, ), (, ), (, 4), (, 4), (4, ), (4, ), (4, 4)},,,,,, 4! 4! + =6+4=0!_!!!! + =6!! 6 = 0 5 0.56 p.6 0 7 4 56 0.49 9 4 4 5 6 7 4 5 6 P =0 700 P + P =4+4=48 48 = 0 5 A B A, B P(A'B)=P(A)+P(B) A A B B P(A'B)=P(A)+P(B)-P(A;B) 4!+4! 4!+4! = + - 5! 5! 7 = + - = 5 5 0 0 (5-)!_! (6-)! C C = + C C 0 = + = = 5 - = 5 5!_! 5! 58

8 9 =6 A B A B 6 =6 a, b, c b -4ac 0 9! 5!_4! 5! 4! _ =60!_!!_! 6!! _ =60!_!! 5!! =0!_!! 60 60 0 90 5 + - = = 6 6 6 6 7 b 4 ac >ac b a, c (a, c) b= (, ), (, ), (, ) b=4 (, ), (, ), (, ), (, ), (, ) 5 b=5 (, ), (, ), (, ), (, 4), (, 5), (, 6), (, ), (, ), (, ), (, ), (, ), (4, ), (5, ), (6, ) 4 b=6 60 6 60 6 0 6 b 4 (, ), (, ), (, ), (, 4), (, 5), (, 6), (, ), (, ), (, ), (, 4), (, ), (, ), (4, ), (4, ), (5, ), (6, ) 6 ~ +5+4+6 78 - = = 89 6 6 08 p.70 5 + _ = 4 5 _ = 4, 4 4 : 5 _ +{ } 5 +{ _ }= 6 5 +{ _ }= 6 5, 6 6 :5 II. 59

7 0 0 0 04 0 0 0 p.7 6 0 p.7 0 9 4 7 4 7 p.77 A B B=(A;B)'(AÇ ;B) (A;B);(AÇ ;B)= P(B)=P(A;B)+P(AÇ ;B) P(B)=P(A)P(B A)+P(AÇ `)P(B AÇ ) =P(A)P(B A)+P(AÇ `)P(B A) ={P(A)+P(AÇ `)}P(B A) =P(B A) A B 7 6 64 64 4 5 6 A B P(A)=0.7, P(AÇ )=0. P(B A)=0.7, P(B AÇ )=0.4 P(B)=P(B;A)+P(B;AÇ ) =P(A)P(B A)+P(AÇ )P(B AÇ ) =0.7_0.7+0._0.4=0.6 A B P(A)= 5, P(B)= P(A;BÇ )=P(A)-P(A;B) =P(A)-P(A)P(B) =P(A){-P(B)} =P(A)P(BÇ ) A, BÇ P(A;BÇ ) P(BÇ A)= = P(A) 8 A 0 Cº{ }0 { }4 = 5 5 65 6 A C { }{ } = 5 5 65 8 6 8 -{ + }= 65 65 65 4 =P(BÇ )=- = 4 4 P(A)P(BÇ ) P(A) 7 p.8 A, B, C R P(A), P(B A), P(A B), P(B), -p P(R)=P(R;A)+P(R;B)+P(R;C) =P(A)P(R A)+P(B)P(R B) +P(C)P(R C) = _+ _0+ _ = 60

8 P(R;A) ;!; P(A R)= = = P(R) ;!; P (, ) C { } { } 80 = 4 P (, ) C { } { } 60 = 79 P (, ), (, ) C { } { } 80 _ = 79 80 60 80 + - = 0 4 79 79 79 p.8 S G T P(S)=, P(G)= 8 0 P(T)=P(T;S)+P(T;G) =P(S)P(T S)+P(G)P(T G) = _0+ _= 6! =60! p.85 0 0 04 05 06 07 k, m, o 4! _!=7! 7 = 60 5 50 P(A'B)=P(A)+P(B) 4 =P(B)+P(B) P(B)=, P(A)=P(B)= a=0 A b=0 B P(A'B)=P(A)+P(B)-P(A;B) 5 = + - = 8 8 64 64 0 9 - = 49 49 ªC =6 6 7 6 5 0 = 7 7 7 49 6 6 6 4 6 -{ + }= 6 6 5 7 5 8 4 40 40 80 II. 6

A B ;8@0*; P(A;B) 8 P(B A)= = = P(A) 4 ;8$0#; 08 C { }{ } = 4 4 56 8 08 67 -{ + }= 56 56 56 08 A 4! 4, 4, 4, 5 =4! B P(A)=0.6, P(AÇ )=0.4 P(B A)=0.0, P(B AÇ )=0.005 P(B)=P(B;A)+P(B;AÇ ) =P(A)P(B A)+P(AÇ )P(B AÇ ) =0.6_0.0+0.4_0.005 4, 4, 5, 5 4! =6!_! 4, 5, 5, 5 4! =4! 4, 4, 5, 6 4! =! 4, 5, 5, 6 4! =! =0.04 P(B;AÇ ) 0.00 P(AÇ B)= = = P(B) 0.04 7 4! 4, 5, 6, 6 =! 4+6+4+++=50 yy` 09 A A B B C C E =8 yy` 50 8 yy` P(E)=P(E;A)+P(E;B)+P(E;C) 7 = + _ + = 4 4 4 4 4 4 64 4 5 4 5 ;6ª4; P(E;C) 9 P(C E)= = = P(E) 7 ;6#4&; 4 A, B A, B 0 5 4 6 _ +{ } _ = 8 C P(C)=P(C;A)+P(C;B) =P(A)P(C A)+P(B)P(C B) = _{ _ }+ _{ _ } = yy` 7 0 Cº { }0 { }4 = 8 4 4 56 P(C;A) ;8!; 9 P(A C)= = = yy` P(C) ;7!#; 6

5 6 A A B C B P(A)= 4 8 P(B)= + - = 9 9 P(A;B)=P(A)P(B) 8 6 = _ = yy` 9 7 6 - = yy` 7 7 A B C A A B C A yy` A B C A A A C _ C _ C = 8 C { }{ }+ C { }{ }+ C { }4{ }0 = yy` 8 4 4 45 5 + = = yy` 8 8 8 9 A III 0 0 0 0 0 04 0 p.9 0. 0. 0. 0 7.6 cm p.9 x- P(X=x)= (x=,,, 4, 5) 5 9 5 55 60 65 70 75 80 85cm p.97 50 4 E(X)=, r(x)= E(X-)=, r(x-)=8 E(T)=50, r(t)=0 65 p.04 5 7 III. 6

0 0 4 0 0 0 04 05 06 07 08 09 0 a=, P( X )= 8 E(X)=0, V(X)= E(X+)=, V(X+)= 9 5 p.09 B(0, 0.4) B{8, } 0 0.0466 E(X)=40, V(X)=8 0.0495 0.990 0.08 0.6687.5 -.7 0.0548 0.6449 0. % 5 0.08 0.006 0.97 E(X), E(X ), np, p. 4 5 6 7 0.58 0.84 X, 4, 5, 6 C C P(X=)= =, P(X=4)= = C 0 C 0 C 6 C 0 P(X=5)= =, P(X=6)= = C 0 C 0 Æ C P(X=x)= (x=, 4, 5, 6) C P(X 4)=P(X=)+P(X=4) +a+ +b+ = 8 8 4 a+b= E(X)= yy` 0_ +_a+_ +_b+4_ = 8 8 4 a+b= X = + = 0 0 5 = 4 yy` a=, b= 8 8 V(X)=0 _ + _ + _ 8 8 8 + _ +4 _ - 8 4 X X P(X=x) - 0 0 0 0 4 0 E(X)= 4 5 46, V(X)= 5 E(X+5)=0, V(X+5)= 5 4 E(X)=(-)_ +0_ +_ +_ = 0 0 0 0 V(X)=(-) _ +0 _ + _ 0 0 0 4 + _ - 0 = 64

8 E(Y)=aE(X)+b=a+b a+b=0 yy` V(Y)=a V(X)=a a = a= a>0 a= a= f(x)æ0 f(x) f(x) b=- aæ0 x - _4_a= a= f(x) P(- X )=-_{ }= 4 4 4 a O x C _0.0046_0.9954 =0.04 X X n=400, p=0. B(400, 0.) npæ5, nqæ5 n np=40, npq=6 X N(40, 6) X -40 P(X )=P{Z } 6 =P(Z -) =0.00 9 0 X X B{8, a a+ } a E(X)=5 8_ =5 a=5 a+ X X N(00, 5 ) 8 Y=8X E(Y)=8E(X)=400 r(y)=8r(x)=0 Y N(400, 0 ) P(400 Y 700) 400-400 700-400 =P{ Z } 0 0 =P(0 Z.5)=0.498 X X B(0, 0.0) P(Xæ)=-P(X<) =-P(X=0)-P(X=) =- ºCº_0.99 - ºC _0.0_0.99 =0.0046 p.5 5 5-X(X=0,,, y, 5) 6X, 4(5-X) 6X>0+4(5-X) X> X B{5, P(X>)=P(X=4)+P(X=5) } = C { } { }+ C { } fi { } = 4 p.6 E(X)=0V(X)=5 0.9544 III. 65

p.7 0 4 =0= = 5 5 0 [0.7, 0.49] 0 0 0 04 0 0 0 04 05 06 07 p.9 9 =60, = 5 0.08 8 5 0.8944 5 p.6 P(-.58 Z.58)=0.99 m 99 % r r [x -.58, x +.58 ] 'n 'n [6.5, 65.47] [0.555, 0.6645] 554 P(-.58 Z.58)=0.99 p 99 % p^q^ p^q^ [p^-.58æ, p^+.58æ ] n n [0., 0.578] 85 4 5 X N(0, 5) X N{0, 8-0 P(X 8)=P ª Z º ;4%; P(8 X ) 5 } 6 =P(Z -.6)=0.0548 8-0 -0 =P Z ª º ;4%; ;4%; =P(-.6 Z.6) =_P(0 Z.6)=0.8904 00 p^ p=0. npæ5, nqæ5 n p^ N{0., 0._0.8 00 P(0. p^ 0.) } 0.-0. =P Z ª 0._0.8 æ 00 =P(0 Z.5)=0.498 x =80 s=50 95 % 0.-0. æ 0._0.8 00 50 50 [80-.96_, 80+.96_ ] 'n 'n 50.96_ =9.8 n=00 'n º p.4 6 6 p^ p^=0.5 np^æ5, nq^æ5 n p^ r pq p^q^,,.96,.96,, p^q^ n n n n N{0.5, 0.5_0.5 } 6 66

95 % 0.5_0.5 0.5_0.5 [0.5-.96æ, 0.5+.96æ ] 6 6 [0.7, 0.66] 95 % [6578, 48] [0.9, 0.47] 7 p^-p n Z= æ p^q^ n N(0, ) P( p^-p 0.5øπp^q^) p^-p 0.5øπp^q^ =P ª º æ p^q^ æ p^q^ n n =P( Z 0.5'n) æ0.95 0.5'næ.96 næ6.4656, 6 p.45 50 0 0 00 5 500 5 X 50 X x = (0_0+00_5+500_5)=04 50 S s = {0_(0-04) +5_(00-04) 49 = 9600 49 9600 0'ƒ96 s=æ = 49 7 +5_(500-04) } 95 % 0'ƒ96 [ 7 7 04-.96_, 04+.96_ ] 'ß50 [65.78, 4.8] 0'ƒ96 'ß50 0 0 0 X 0,, Cº_ C P(X=0)= = C 0 C _ C 6 P(X=)= = C 0 C _ Cº P(X=)= = C 0 CÆ_ C Æ P(X=x)= (x=0,, ) C P(X )=P(X=0)+P(X=) 6 9 = + = 0 0 0 5 a + +a= 9a +9a-4=0 9 a= a=- aæ0 a= 5 E(X)=_ +_ +5_ = 9 9 X X P(X=x) 6 0 8 6 E(X)=0_ +_ +_ +_ 6 6 6 6 E(X)= 5 8 4 +4_ 4 6 p.47 0 4 5 6 6 0 6 8 6 6 6 4 6 6 +5_ 6 III. 67

6 0 8 6 V(X)=0 _ + _ + _ + _ 6 6 6 6 E(X)= 665 4 4 5 +4 _ +5 _ -{ } 6 6 8 665 V(8X+)=8 V(X)=8 _ =665 4 Y N(8000, 000 ) 5490-8000 P(Y 5490)=P{Z } 000 =P(Z -.5)=0.006 000 000_0.006= 04 V(X)=E(X )-{E(X)} 9=6a-a a= E(X)= E(4X-)=4E(X)-=9 08 6 N{00, 5 } 6 05 f(x)æ0 f(x) f(x) x x=4 kæ0 f(x) 4k f(x)=kx O 4 x 09 n=00 x =.5 s=0.6 95 % 0.6 0.6 [.5-.96_,.5+.96_ ] ' 00 ' 00 [.84,.676] 06 _4_4k= k= X B{0, 40 E(X)=0_ = 8 } 0 0 p^ 6 p^= =0. 0 95 % 0._0.7 0._0.7 [0.-.96æ, 0.+.96æ ] 0 0 [0.8, 0.6] r(x)=æ 0_ _ = Y B{0, E(Y)=0_ = 0 r(y)=æ 0_ _ = ' 0 ' 0 E(Y-5)=E(Y)-5=5 r(y-5)=r(y)='å0 } P(X=0)=a E(X)=0_a+_(-a) =-a P(X=)=-a E(X )=0 _a+ _(-a) =-a V(X)=E(X )-{E(X)} =(-a)-(-a) =-a +a yy` yy` 07 X X E(X)=V(X) -a=(-a +a) N(80, 0 ) 00 Y=00X E(Y)=00E(X)=8000 r(y)=00r(x)=000 a= a= a+ a= P(X=0)= yy` 68

E(X) a V(X) a P(X=0) 70-80 P(Xæ70)=P{Zæ } 4 =P(Zæ-.5)=0.998 yy` f(x) x _(+a)_ = a= yy` 5 5 5 P{a X a}=p{ X } = _ = yy` 5 5 5 r Z k r k = k yy` ' 0 kr=6'å0 X X a n r k =6 yy` 'n X X N(65, 00) 9 8 c 8 P(Xæc)= =0.9 yy` 000 ' 0 kr=6'å0 =6 n=40 'n 40 6 yy` P{Zæ c-65 0 }=0.9 c-65 P{0 Z }=0.5-0.9=0.8 0 yy` 6 99 % p^= 5 ;5!;_;5$; a=.58 =0.9 64 yy` 4 c-65 =.8 c=76.8 0 76.8 yy` X X 4 4 n=00, p= B{00, } 5 5 yy` 95 % p^= b(n)=.96 b(n) a ;4!;_;4#;.96 0.9 n næ4.84y ;4!;_;4#; n 4 yy` n 44 yy` npæ5, nqæ5 n np=80, npq=6 X N(80, 6) yy` a b(n) n III. 69

70 z O f(z) z P(0 Z z) f(z)=;;::;;;e -;:: z p ' z 0.0 0. 0. 0. 0.4 0.5 0.6 0.7 0.8 0.9.0....4.5.6.7.8.9.0....4.5.6.7.8.9.0....4 0.00 0.0000 0.098 0.079 0.79 0.554 0.95 0.57 0.580 0.88 0.59 0.4 0.64 0.849 0.40 0.49 0.4 0.445 0.4554 0.464 0.47 0.477 0.48 0.486 0.489 0.498 0.498 0.495 0.4965 0.4974 0.498 0.4987 0.4990 0.499 0.4995 0.4997 0.0 0.0040 0.048 0.08 0.7 0.59 0.950 0.9 0.6 0.90 0.86 0.48 0.665 0.869 0.4049 0.407 0.445 0.446 0.4564 0.4649 0.479 0.4778 0.486 0.4864 0.4896 0.490 0.4940 0.4955 0.4966 0.4975 0.498 0.4987 0.499 0.499 0.4995 0.4997 0.0 0.0080 0.0478 0.087 0.55 0.68 0.985 0.4 0.64 0.99 0. 0.46 0.686 0.888 0.4066 0.4 0.457 0.4474 0.457 0.4656 0.476 0.478 0.480 0.4868 0.4898 0.49 0.494 0.4956 0.4967 0.4976 0.498 0.4987 0.499 0.4994 0.4996 0.4997 0.0 0.00 0.057 0.090 0.9 0.664 0.09 0.57 0.67 0.967 0.8 0.485 0.708 0.907 0.408 0.46 0.470 0.4484 0.458 0.4664 0.47 0.4788 0.484 0.487 0.490 0.495 0.494 0.4957 0.4968 0.4977 0.498 0.4988 0.499 0.4994 0.4996 0.4997 0.04 0.060 0.0557 0.0948 0. 0.700 0.054 0.89 0.704 0.995 0.64 0.508 0.79 0.95 0.4099 0.45 0.48 0.4495 0.459 0.467 0.478 0.479 0.488 0.4875 0.4904 0.497 0.4945 0.4959 0.4969 0.4977 0.4984 0.4988 0.499 0.4994 0.4996 0.4997 0.05 0.099 0.0596 0.0987 0.68 0.76 0.088 0.4 0.74 0.0 0.89 0.5 0.749 0.944 0.45 0.465 0.494 0.4505 0.4599 0.4678 0.4744 0.4798 0.484 0.4878 0.4906 0.499 0.4946 0.4960 0.4970 0.4978 0.4984 0.4989 0.499 0.4994 0.4996 0.4997 0.06 0.09 0.066 0.06 0.406 0.77 0. 0.454 0.764 0.05 0.5 0.554 0.770 0.96 0.4 0.479 0.4406 0.455 0.4608 0.4686 0.4750 0.480 0.4846 0.488 0.4909 0.49 0.4948 0.496 0.497 0.4979 0.4985 0.4989 0.499 0.4994 0.4996 0.4997 0.07 0.079 0.0675 0.064 0.44 0.808 0.57 0.486 0.794 0.078 0.40 0.577 0.790 0.980 0.447 0.49 0.448 0.455 0.466 0.469 0.4756 0.4808 0.4850 0.4884 0.49 0.49 0.4949 0.496 0.497 0.4979 0.4985 0.4989 0.499 0.4995 0.4996 0.4997 0.08 0.09 0.074 0.0 0.480 0.844 0.90 0.57 0.8 0.06 0.65 0.599 0.80 0.997 0.46 0.406 0.449 0.455 0.465 0.4699 0.476 0.48 0.4854 0.4887 0.49 0.494 0.495 0.496 0.497 0.4980 0.4986 0.4990 0.499 0.4995 0.4996 0.4997 0.09 0.059 0.075 0.4 0.57 0.879 0.4 0.549 0.85 0. 0.89 0.6 0.80 0.405 0.477 0.49 0.444 0.4545 0.46 0.4706 0.4767 0.487 0.4857 0.4890 0.496 0.496 0.495 0.4964 0.4974 0.498 0.4986 0.4990 0.499 0.4995 0.4997 0.4998

7 97 4 0 4 8 6 6 77 79 9 7 9 9 58 59 5 57 7 7 58 94 9 94 8 7 4 74 9 77 7 6 6 4 7 9 7 9 9 05 9 95 95 «P 5 n! 7 «P «C 9 «H S(n, k) 7 P(n, k) 8 P(A) 59 P(B A) 74 P(X=x) 95 E(X) 97 V(X) 99 r(x) 00 B(n, p) 0 N(m, r ) 4 N(0, ) 6 X 9 S 9 S 9 p^ 7

Gan, L. Is the Zipf law spurious in explanining city-size distributions?, Economics Letter 9, 56-6, 006 Martin, G., The Colossal Book of Mathematics, W. W. Norton & Company, 00 Tomas, G. & Kevin, M., An Introduction to Biostatistics, Waveland Press, 008 http://www.kosis.kr http://www.fifa.com http://www.archery.or.kr http://www.sportsanalyst.or.kr http://www.census.go.kr http://www.yonhapnews.co.kr http://www.zdnet.co.kr http://www.actuary.or.kr http://www-history.mcs.st-and.ac.uk/mathematicians/pascal.html 7

I. 05 6 4 0 4 6 06 0 5 6! 4! 4! P 0 07 7 M, O, N, S, T, E, R 00 70 04 P P P 08 6 I.

09 4 0,, 4 A D B A D 8 a, a, b, b, b, c, c, c C A B 5 4 A, B, C, D 4 B A A B C D

6 n «P :«P =4: 9 5 a, b, c, d, e abcde edcba bcdea «P =56 «P 7 0 8 X={a, b, c, d}, Y={,,, 4, 5, 6} f:x Y f(a)+f(b)=9 f 8 I.

6 9 9 5 A B C C B A Z Y 00 6 6 0, 0,,,, 7 P 4 9 W, E, L, L, B, E, I, N, G N G 7 P 4

8 9 6 0 5 0 0 9 A Y A B B A A x (i=,, y, 9) (x, x, y, xª) x -,, x +x +y+xª=0 I. 5

I. 05 6 0 Cº C ºC 0 A={,,, 4, 5, 6, 7} 4 06 A={,,, 4, 5, 6} 5 0 5 4 07 9 5 04 08 H H (x-)fi (x+y) 6

09 (x-) x 78 0 Cº+ C + C +y+ C 4 7 ºCº- ºC + ºC -y+ ºC º C + C + C +y+ C 5 X={,,, 4, 5} Y={a, b, c, d} n 6 n r «C = «C««C«=5 «H =8 ºH =ª C I. 7

7 6 4 8 x+y+z+w=8 {x - } x x 9 (a+b+c) +(+x)+(+x) +y+(+x) x 0 7 4 (a+b) 8

5 7 7 (+x) (-x)fi x 6 4 8 (x+)«x 9, 0, n 9 l, m, n n l, n m n ;K+) C μc«= μc«i. 9

II. 0 5 5 A, B, 7 C 05 55 45 B A, B, C 06 4 0 A, B 0 5 A, B, C, D, E E 07 7 04 000 7 08 4 0 0

09 A, B A a B b x +ax+b=0 X={a, b, c} Y={,,, 4, 5} f f f(a) f(b) f(c) 0 60 60 4 4 5 8 4 4 II.

9 A, B P(A)=, 6 0 4 0 5 P(B)=, P(AÇ'BÇ )= 4 6 P(A'B) 0 a, b 0.6 ab 7 A, B, C A B 0.6, B C 0.5, C A 0.5 A A A B C 8 5 4 5

A={,,, 4, 5, 6, 7} a, b, c a+b_c 5 4 A 4 B A B 6 4 a, a, a, a a a a a 0 9 4 A 8 5 B II.

II. 0 600 04 A, B P(A)=, P(A;B)= P(B) 4 60 40 00 0 70 00 90 0 600 05 A, 4 B, 6 C A B B C 0 A, B P(A)=, 4 P(B)=, P(A;B)= 5 5 P(A B) P(B A) 06 0.7, 0.8 A, B 0 5, 07 5 4

08 A, B P(A)=, 4 P(B)=, P(B A)= P(BÇ AÇ ) 4 7 09 AB 5 % AB % AB 0., 0.4 0.6 0 4 0 A, B, C 0 %, 0 %, 50 % 0.5 %, %, % A B II. 5

4 A, B, A, B 7 7 0. 5 0 0 0 8 P 6 P P - - - 0 6 P(A)>0, P(B)>0 A, B A B P(A B)=P(A) P(B AÇ )=P(B) P(B A)=P(A B) P(BÇ A)=P(B) 9 6

0 0.8, 0. 0 % k k+ 6 A, B A, B k A, B 5,,, 4, 5 5 A B 5 0.99 log =0.0, log =0.477 II. 7

III. 0 04 X 5-x P(X=x)= (x=,,, 4) 0 X 00 6 05 X X X 0 X x+ P(X=x)= (x=0,,, ) k X P(X=x) 4 5 0. 0. 0. 0. 0. k 0 X 06 X 4 X X- P(X<) 8

07 X f(x) (0 x ) P(0 X ) [0~] z P(0 Z z) 0.5 0.95.0 0.4 f(x) f(x).5.0 0.4 0.477.0 0.4987 O x 0 Z N(0, ) P( Z ) P(Z.5) P(Zæ) 08 X f(x)=-x (0 x ) P{0 X } X N(60, 4) P(6 X 64) 4 09 X B{6, 4 } X B{00, 5 } X X III. 9

0 0 5 6 n,,, y, n n X X X X X X 4 4 X X 7 X X a, b X 0 P(X=x) a b ;4!; P(5 X 6) 8 X, Y a, b 5 X P(X ) X - - 0 Y=aX+b E(X)=0, V(X)=6 a>0 P(X=x) a a-;9!; ;!; a E(Y)=0, V(Y)= 0

9 4 X 5X+ 0 % 0 0 4D 0 0 0 X X X f(x)(0 x ) f(x) a f(x) 4 A, B, C 0 %, 0 %, 0 % 40 % O x 00 A a P( X ) X X III.

5 X N(m, r ) X 8 80 g, 5 g f(x) 70 g 90 g 400 x=0 P(0 Z )=0.48 m r r m 6 N(m, 9) X P(Xæ6)=0.08 m 9 40 % 50 50 7 60 0 68 74 80 50 5 60

0 A, B 7 4 A B 0.6 X X 0 0 0 y, n n X X 5 X 5 g g g 500 7% P(0 Z.5)=0.4, P(0 Z )=0.48 III.

III. 0 TV 04 00 40 PC PC p^ 05 0. 0 p^ 0 00, 5 44 X 0 N(0, 6) 9 X 06 900 49 95 % 99 % 4

07 00 5 0 4 kg, kg 9 95 % 99 % 7 kg 08 0 0 (X, X, X ) N(50, 5) n X P(49 X 5)=0.9544 n (,, ) (,, ) 09 6,,,,, 6 4 X 0.5 00 0. 0. III. 5

p 49 p^ P( p^-p a"çp^q^)=0.95 a q^=-p^) 6 400 p 95 % [0.0706, 0.94] p^ 4 64 75 4 95 % 7 n % 99 % 0.0 n 5 5 99 % n 6

8 5 g, 4 g g 7 g 6 (5-a) g (5+a) g a 0 m n X f(m)=p{x.96_ } 'n f(0)+f(0.9).055 n 9 m 0 n X X -m 0.95 n 0 % 95 % 0.0 III. 7

I 0 0 0 05 06 07 08 09 0 8 0 6 04 60 0 0 60 0 400 5040 0 64 7 560 p. 6 =6 =7 6-7=89 4 A B D _=9 A C D _=4 9+4= 5 A 4 B A C A, B D B, C 6 7 8 9 4 =48 4 «P =«P 4n(n-)=n(n-)(n-) næ 4=n- n=6 (n+)(n+)n(n-)=56n(n-) næ (n+)(n+)=56 n=6 n=-9 n=6 P P 4 4! P _ P _4!=6!!!_!_=7 f(a)+f(b)=9 f(a), f(b) (f(a), f(b)) (, 6), (4, 5), (5, 4), (6, ) f(c), f(d) 4 P 4_ P =48 a 4! ba! bca! bcda! 4!+!+!+!= bcdea 4 8

0 6 5 4 (-)!=!!!_!= 8 4 8! =0080 4 Pª=0077696 n Z Y ««æ00 næ8 8 N G 8 N G! 8!!_! _!=060 8!!_! 7 8 5! =0!_! 5 5! =0!_! 0+0=40 7 x 7-x P x-(7-x)=x-7 x-7= x= 7 P 7!!_4! =5 6 5 A P B 4!!_! =6 A Q B _=0 A B 6+0=6 B C! =! 5!!_! A P B Q C 9,,! 6_!=6 A A!! 6_=48!_!_!_!=48 I. 9

0!_!_!=8 48_8=84 5 (5-)!=4! 5 5! 4!_5!=880 a=, b=7 c=0 (x, x, y, xª) - 7 9! =6!_7! a=, b= c=4 (x, x, y, xª) -, 4 9! =60!_!_4!, (x, x, y, xª) 6+60=96 B Q P P, Q A B A! =46 6!_5! A P B! 7! =05! 4!_! A Q B 6! 4! =80!_!! A P, Q B! 4!! =4!! 46-05-80+4=0 -,, x (i=,, y, 9) a, b, c a, b, c 0 9 a+b+c=9, -a+b+c=0 5a+b=7 a, b a=, b=7 a=, b= p.6 0 56 0 0 04 0 5 60 56 0 05 06 07 8 5 5 08 xfi -80x +80x -40x +0x- 8x +6x y+6x y +96xy +6y 09 0-79 (+x) = Cº+ C x+ C x +y+ C x x= = Cº+ C + C +y+ C 56 (+x) = ºCº+ ºC x+ ºC x +y+ ºC º x 0

4 5 x=- 0= ºCº- ºC + ºC -y+ ºC º (+x) = Cº+ C x+ C x +y+ C x yy x=, x=- = Cº+ C + C +y+ C 0= Cº- C + C -y+ C - =( C + C +y+ C ) = C + C +y+ C 048 «C = «C «= «C«5=n- n-5=n- n 4, 6 «C«=«C =5 (n+)(n+) =5 n=-7 n=4 næ0 n=4 yy yy n «C =78 n(n-) =78 n=- n= næ n= C _ C _ C _ =05! X 5 C 4! C _4!=40 6 «H =«C =8 (n+)n =8 n=-8 n=7 næ n=7 ºH =ª C =ª Cª ª Cª=ª C 9=r+ r=8 7 4 H = C =5 8 H = C =65 H = C =5 9 (a+b+c) a b cω x+y+z=7(x, y, z ) H =ªC =6 0 S(7, ) 7 (6, ), (5, ), (4, ) (6, ) C _ C =7 (5, ) C _ C = (4, ) C _ C =5 S(7, )=7++5=6 P(, 4) 4 =8+++=7+++ =6+++=6+++ =5+4++=5+++ =5+++=4+4++ =4+++=4+++ =+++ P(, 4)= I.

4 5 6 {x - x }8 C (x ) {- }r = C (-) x fl x x fl =x r=4 x C _(-) =70 +(+x)+(+x) +y+(+x) (+x) - = = (+x)- (+x) x =(+) C =65 = ºCº_ + ºC _ _ +y+ ºCª + ºC º_ ºC º_ ºC º_ 7,, C _ C _ C _ =70! C _ C = 70 =60 4 a, b, c a b, c C _ C _ C =6 b a, c C _ C _ C =4 c a, b (+x) - x C _ C _ C = b c a,, 6+4+=7 7 (+x) C (x) = C x (-x)fi Cß(-x)ß= Cß(-)ßxß C _ Cß_ _(-)ß_x ±ß x r+s= r=0, s= Cº_ C _ _(-) =0 r=, s= C _ C _ _(-) =-40 r=, s=0 C _ Cº_ _(-) =4,, x 0-40+4=-6 8 (x+)«x 9, 0, «C, «Cª, «C º _«Cª=«C +«C º _n! n! n! = + 9!(n-9)! 8!(n-8)! 0!(n-0)! næ0 n -7n+=0 n=4 n= 9 (+x) = Cº+ C x+ C x +y+ C x (+x)μ =μcº+μc x+μc x +y+μcμxμ (+x) (+x)μ =( Cº+ C x+ C x +y+ C x ) _(μcº+μc x+μc x +y+μcμxμ ) (+x) (+x)μ x«;k+) C μc«(+x) ±μ x«μc«(+x) (+x)μ =(+x) ±μ n n ;K+) C μc«= μc«

II 0 0 0 04 05 06 07 08 {5, 7, 8, 9, 0,,, 4, 5} B C 5 7 000 0 7 0 5 8 5 6 p.0 7 7! 5 5!! 5!_! 5!_! = 7! 7!_4!!_4! = 7! 5 5 (5-)!=4! (-)!=!!!_! = 4! 6 09 6 =6 a, b a -b<0 a <b a, b (a, b) f P =5 f(a) f(b) f(c) f H =5 5 7 = 5 5 (, ), (, ), (, ), (, 4), (, 5), (, 6), (, ), (, ), (, 4), (, 5), 4 7 C =5 (, 6), (, 4), (, 5), (, 6), (4, 6) 5 5 5 = 6 0 60= _ _5 60 4 =4 60= 5 60 C _ C + C _ C =0 0 6 = 5 7 = = 4 5 x ÆC = 4 C 4 II.

6 7 8 9 x(x-) = 8 4 x -x-4=0 x=-6 x=7 xæ x=7 7 4 A 0 B P(A'B)=P(A)+P(B)-P(A;B) C 9 = + - ºC ºC ºC 0 9 = + - 90 90 90 7 = 90 A B 0.5_0.6=0.5 A C (-0.5)_(-0.5)=0.75 0.5+0.75=0.55 C _ C ªC 40 = 84 C 0 = ªC 84 40 0 50 5 + = = 84 84 84 4 5 P(AÇ 'BÇ )=P((A;B)Ç )= 6 5 P(A;B)=- = 6 6 P(A'B)=P(A)+P(B)-P(A;B) = + - 4 6 = 7 0 ab a, b (-0.6)_(-0.6)=0.6-0.6=0.84 C =0 6_0=60 60 = 0 8 - = 5 H =56 H = = 56 8 5 - = 8 8 a, b, c P =0 a+b_c (a, b, c)!=6 P _ P =4 P _ P =4 P =4 ~ 6+4+4+4 0 78 = = 0 5 4

4 _4= a, b (a, b) (, ), (4, ), (4, ), (9, ), (9, ), (9, 5), (9, 8) 7 7 7 5 - = 0 0 0 04 05 6 9 5 56 8 p.4 6 A B B C 5 06 0.56 5 C _ C C _ C = 5 07 08 5 6 P(A;B) P(B A)= = P(A) - = 5 5 P(A;B)= P(A)= P(A'B)=P(A)+P(B)-P(A;B) 6 a a a a 0 a, a, a, a 5 P(BÇ AÇ )= = + - = 4 P(AÇ ;BÇ ) P(AÇ ) 4 A, 5 B P(A)={ } 5, P(B)={ } 6 = P((A'B)Ç ) P(AÇ ) -;@; 4 = = 9 -;4!; P(A;B)={ } a a a a 0 P(A'B)=P(A)+P(B)-P(A;B) ={ } 5 +{ } -{ } 6 = 5 6 09 0 AB A B P(A)=0.5, P(A;B)=0. P(A;B) 0. P(B A)= = = P(A) 0.5 5 A B 5 - = 0 6 6 C P(A)= = ºC II. 5

C C _ C P(A;B)= + ºC ºC 8 9 = + = = 45 45 45 5 P(B A)= _ = 4 7 4 5 = = 5 A B P(A)=0.6, P(AÇ )=0.4 P(B A)=0., P(B AÇ )=0.4 P(A;B) P(A) P(B)=P(B;A)+P(B;AÇ ) =P(A)P(B A)+P(AÇ )P(B AÇ ) =0.6_0.+0.4_0.4=0.4 A, B, C A B C E P(A)=0., P(B)=0., P(C)=0.5 P(E A)=0.005, P(E B)=0.0 P(E C)=0.0 P(E)=P(E;A)+P(E;B)+P(E;C) =P(A)P(E A)+P(B)P(E B) +P(C)P(E C) =0._0.005+0._0.0+0.5_0.0 =0.04 P(E;(A'B)) P((A'B) E)= P(E) P(E;A)+P(E;B) = P(E) 0.004 = = 0.04 7 4 5 6 A B 6 _ + _ = 7 7 x A B P(A;B)=P(A)P(B) 0 80+x 50 = 80+x x=45 0 x 0+x 0 0 50 50 0+x 80+x 0+x 80+x 45 P(B;A) P(A B)= P(B) P(B)P(A) = P(B) =P(A) P(B;AÇ ) P(B AÇ )= P(AÇ ) P(B;AÇ )=P(B)-P(A;B) =P(B)-P(A)P(B) =P(B){-P(A)} =P(B)P(AÇ ) P(B)P(AÇ ) P(B AÇ )= =P(B) P(AÇ ) P(A B)=P(A) P(A;B) P(B A)= P(A) P(A)P(B) = P(A) =P(B) P(A)+P(B) 6

P(BÇ A)= P(A;BÇ ) P(A) P(A;BÇ )=P(A)-P(A;B) P(A;B) 0.4 4 P(B A)= = = P(A) 0. =P(A)-P(A)P(B) =P(A){-P(B)} =P(A)P(BÇ ) P(A)P(BÇ ) P(BÇ A)= P(A) =P(BÇ ) A, B 5 P(A)= = 5 5 P(B A)= = 6 4 7 C (0.) (0.7)=0.89 P(A;B)=P(A)P(B A) = _ = 5 4 0 C (0.) =0.07 k P(A)=, P(B)= k 6 0.89+0.07=0.6,,, 4, 5 8 6 P k P(A) P(B) P(A)P(B) P(A;B) ;!; ;6!; ; ; 0 5 C { } { } = 6 ;!; ;!; ;!; ;!; ;6!; ;4!; ;6!; ;6!; 9 4 5 ;!; ;!; ;@; ;6%; ;!; ; ; ;!; ;!; C { } { }_ = 7 A, B k, 4 0 A B P(B)=0., P(BÇ )=0.7 P(A B)=0.8, P(A BÇ )=0. P(A)=P(A;B)+P(A;BÇ ) =P(B)P(A B)+P(BÇ )P(A BÇ ) n -«Cº{ } «-{ } «æ0.99, { } «0.0 5 5 n(log -log 5) - næ 5 - -0. =0._0.8+0.7_0. næ9.009y =0. 0 II. 7

III 0 0 04 05 06 07 08 09 0 0.96 0 P(X=x)= (x=,,, 4, 5, 6) 6 E(X-)=7, r(x-)=' 4 p.8 E(X)=4, r(x)=' 0.574 0.9 0.08 0.857 N(0, 6) X 5, 6, 7, 8, 9, 0 C C P(X=5)=, P(X=6)= ºC ºC C C P(X=7)=, P(X=8)= ºC ºC C ªC P(X=9)=, P(X=0)= ºC ºC Æ C P(X=x)= (x=5, 6, 7, 8, 9, 0) ºC 4 5 6 X 4, 5, 6, 7, 8 P(X=4)= Cº{ }0 { }4 = 6 4 P(X=5)= C { }{ } = 6 6 P(X=6)= C { } { } = 6 4 P(X=7)= C { } { }= 6 P(X=8)= C { }4 { }0 = 6 P(X=x)= CÆ { }4 (x=4, 5, 6, 7, 8) P(5 X 6)=P(X=5)+P(X=6) 4 6 5 = + = 6 6 8 a+a- + +a = 9 9a +8a-7=0 a= a=- aæ0 a= P(X )=P(- X ) =-P(X=-)= X,, y, n n P(X=x)= (x=,, y, n) n E(X)=_ +_ +y+n_ n n n = (++y+n) n n(n+) = _ n = n+ 7 8

7 V(X)= _ + _ +y+n _ n n n a+b+ 4 n+ -{ } n+ = ( + +y+n )-{ } n n(n+)(n+) = _ - n 6 = a+b= n - = 4 yy` E(X)=0_a+_b+_ =b+ 4 E(X )=0 _a+ _b+ _ 4 (n+) 4 9 X 0,, X E(X)=0_ +_ +_ = 5 5 5 Cº_ C P(X=0)= = C 5 C _ C P(X=)= = C 5 C _ Cº P(X=)= = C 5 X P(X=x) r(x)=æ = 5 0 5 'å0 5 5 V(X)=0 _ + _ + _ - = 5 5 5 5 5 r(5x+)=5r(x)='å0 =b+ 0 V(X)=b+-{b+ } =-b + b æ0 X b=0 b=0 a= 4 4 X X f(x)= (0 x 0) 0 f(x) ;::; 0 P(Xæ0)=0_ = 0 O f(x)=;::; 0 0 0 x 8 E(Y)=aE(X)+b=0a+b 0a+b=0 yy` f(x) x V(Y)=a V(X)=6a 6a = a= 4 a= a= P( X )= _{ + }_= a>0 a= 4 a= b=-5 4 X X B{0, } 5 III. 9

P(X ) =P(X=0)+P(X=)+P(X=) 4 4 = ºCº{ }0 { } 0 + ºC { }{ }9 5 5 5 5 4 + ºC { } { }8 5 5 0 4 = =0.678 5 8 80-74 = 60-50 = 5 X X N(80, 5) X B{0, 0 E(X)=0_ = V(X)=0 = 40 9 } P(70 X 90) 70-80 90-80 =P{ Z } 5 5 =P(- Z ) =_P(0 Z )=0.96 400 4 A 80 4 = X 00 5 B{00, 4 5 } 4 E(X)=00_ =80 5 4 V(X)=00 =6 5 5 r(x)='å6=4 9 400_0.96=04 50 X 40 X n=50, p= = 00 5 B{50, } 5 npæ5, nqæ5 n np=50_ =60 5 5 6 x=m ' pr P(Xæ6)=P{Zæ }=0.08 6-m P{0 Z }=0.5-0.08 =0.477 P(0 Z )=0.477 6-m 6-m = m=57 0 npq=50 =6 5 5 X N(60, 6) 50 75 75-60 P(Xæ75)=P{Zæ } 6 =P(Zæ.5) =0.5-P(0 Z.5) =0.006 X 4, 5, 6, 7 7 68-60 = 4 0 5 P(X=4)=0.6 +0.4 P(X=5)=( C _0.6 _0.4)_0.6 +( C _0.6_0.4 )_0.4 40

P(X=6)=( C _0.6 _0.4 )_0.6 +( C _0.6 _0.4 )_0.4 P(X=7)=( C _0.6 _0.4 )_0.6 +( C _0.6 _0.4 )_0.4 X P(X=x) =Æ C _0.6 _0.4 +Æ C _0.6 _0.4 =Æ C (0.6 _0.4 +0.6 _0.4 ) (x=4, 5, 6, 7) +++y+n= X X P(X=x) 4 E(X)=_ +_ n(n+) n(n+) n +y+n_ n(n+) = ( + +y+n ) n(n+) n(n+)(n+) = _ n(n+) 6 n+ = E(X)=5 n+ =5 n=7 n(n+) y n n(n+) 4 n(n+) 4 4 V(X)= _ + _ +y+7 _ -5 56 56 56 = y n n(n+) E(Y)=(0+a)E(X)-0a =0-8a E(Y)=0 0-8a=0 a=.5.5 X -5 P(X )=P{Z } =P(Z -) =0.5-P(0 Z ) =0.0 500 Y Y n=500, p=0.0 B(500, 0.0) npæ5, nqæ5 n np=500_0.0=50 npq=500_0.0_0.98=49 Y N(50, 49) c c-50 P(Y c) 0.07 P{Z } 0.07 7 P{Zæ 50-c 7 } 0.07 P(Zæ.5)=0.5-P(0 Z.5) =0.5-0.4 =0.07 50-c æ.5 c 9.5 7 9 X X B{0, } E(X)= 5 Y a Y=0X-a(0-X) =(0+a)X-0a p.4 0 0 5 E(X )=00, V(X )=, r(x )= 5 44 III. 4

0 04 05 06 07 08 09 0 6 N{0, } 9 0 E(p^)=0., V(p^)=0.008 [48.869, 49.] [48.88, 49.7] [0.6, 0.784] [0.0968, 0.0] = 0 0 0 000 = 0 9 8 X X X P(X=x) 70 6 E(X)=_ +_ +_ = 6 6 6 6 V(X)= _ + _ + _ -{ } 6 6 6 6 7 = 6 4 X 7 6 V(X )= = 4 7 X X N(4, 9) X N(4, ) 7-4 P(X æ7)=p{zæ } =P(Zæ) =0.5-P(0 Z ) =0.00 X N(50, 5) 6 X N{50, 6 5 } n P(49 X 5)=0.9544 49-50 5-50 P Z =0.9544 'n 'n P{- Z }=0.9544 5 5 'n P{0 Z }=0.477 5 P(0 X )=0.477 'n 5 5 'n = n=00 00 p^ p=0.5 npæ5, nqæ5 n p^ 0.5_0.75 N{0.5, } 00 P(0. p^ 0.) 0.-0.5 0.-0.5 =P Z æ =P(- Z ) =_P(0 Z ) =0.9544 P( p^-p aøπp^q^)=0.95 aøπp^q^ P Z =0.95 ª º æ p^q^ 49 P( Z 7a)=0.95 P(-7a Z 7a)=0.95 P(0 Z 7a)=0.475 P(0 Z.96)=0.475 7a=.96 0.5_0.75 00 a=0.8 4 n=64 x =75 s=4 95 % 4 4 [75-.96_, 75+.96_ ] '6å4 '6å4 [74.0, 75.98] 5 'n æ 0.5_0.75 00 4

5 r=5 99 % 9 X N(m, 0) 5 0 _.58_ X N{m, } 'n n næ665.64 666 6 0.94-0.0706=0.0588 _.96_ æ p^(-p^) =0.0588 400 p^(-p^)=0.09 p^=0. p^=0.9 95 % [0.0706, 0.94] _.58_æ næ579.8464 p^=0. 7 p^=0.0 99 % 0.0 580 8 X X N(5, 6) P( X 7) -5 7-5 =P{ Z } 4 4 =P(-0.5 Z 0.5) 6 6 X N{5, } 6 P(5-a X 5+a) -a a =P Z ;@; ;@; 0.0_0.98 n =P{- a Z a} P(-0.5 Z 0.5)=P{- a Z a} 0.5= a a= 0 P{ X -m }æ0.95 ;!; P Z æ0.95 æ : nº: P( Z.96)=0.95 ;!; æ.96 næ5.664 æ : nº: n 54 X N(m, 4) X N{m, f(m)=p{x.96_ } 'n }.96_-m 'n =P Z 'n m'n =P{Z.96- } f(0)=p(z.96)=0.9750 f(0)+f(0.9).055 0.9750+f(0.9).055 f(0.9) 0.0505 f(0.9)=p(z.96-0.45'n) =P(Zæ0.45'n-.96) P(Zæ.64)=0.5-0.4495=0.0505 0.45'n-.96æ.64 næ64 n 64 n p^=0. p^-p.96æ 0._0.9 n 4 n 0.0 0._0.9.96æ 0.0 næ864.6 n 865 III. 4