1: *2 W, L 2 1 (WWL) 4 5 (WWL) W (WWL) L W (WWL) L L 1 2, 1 4, , 1 4 (cf. [4]) 2: 2 3 * , , = , 1

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1 I, A ( 3 ) : (1,2,6), (1,3,5), (1,4,4), (2,2,5), (2,3,4), (3,3,3) 10 : (1,3,6), (1,4,5), (2,2,6), (2,3,5), (2,4,4), (3,3,4) : = : = ( : 9 25/216, 10 27/216 = 1/8 ) 1.1 ( 2 ) (1), (2) (1) (2) (1) (2) (2) *1 1.2 ( ) *1 (cf. [7]) [7] ( ) 1

2 1: *2 W, L 2 1 (WWL) 4 5 (WWL) W (WWL) L W (WWL) L L 1 2, 1 4, , 1 4 (cf. [4]) 2: 2 3 * , , = , A B 5 (a), (b) A (1) (2) (a) A 4 2 (b) A *4 2.1 A, B P (A) > 0 A B P A (B) *5 P A (B) = P (A B). P (A) *2 (cf. [5]) p 90% [0.135, 0.983] ( ) *3 p n k f(p) = nc k p k (1 p) n k f(p) p p = k/n p ( ) *4 CNET JAPAN 2003/3/10 MS *5 P (B A) P A (B) A A A c ( ) 2

3 P A (B) A A B 2.1 ( ) A B 40 A, B A, B R R A 20/40 = 0.5 B 16/40 = 0.4 A B R R A B (M) A 18/30 = 0.6, B 7/10 = 0.7 R M R M M R F R F F A B (F ) A 2/10 = 0.2, B 9/30 = 0.3 B A (cf. [10]) *6 A, B A, B R P A (R) = = 0.5, P B(R) = = 0.4, P A(R) > P B (R). M B P A M (R) = = 0.6, P B M(R) = 2 10 = 0.2, P A M (R) < P B M (R), P A F (R) = 7 10 = 0.7, P B F (R) = 9 30 = 0.3, P A F (R) < P B F (R) P A ( ) A P A (U) = P A (A) = 1 (U ), P A ( ) = 0 B, C (B C = ) P A (B C) = P A (B) + P A (C) 2.2 ( ) 2 A, B P (A B) P (A B) = P (A)P A (B) *6 (cf. [1]) 3

4 2.3 ( ) A C 1, C 2,, C n U C 1 C 2 C n = U C i C j = (i j) P A (C i ) = P (C i )P Ci (A) P (C 1 )P C1 (A) + P (C 2 )P C2 (A) + + P (C n )P Cn (A) 1 B n = 2, C 1 = B, C 2 = B (B ) P (B)P B (A) P A (B) = P (B)P B (A) + P (B)P B (A) 2 : P (C i )P Ci (A) = P (C i A). P (C 1 )P C1 (A) + P (C 2 )P C2 (A) + + P (C n )P Cn (A) = P (C 1 A) + P (C 2 A) + + P (C n A) = P (A) 2 (C i A) (C j A) = (i j) C 1 C 2 C n = U (1) 2.2 (NG ) 60% (Spam) 40% (Ham) 80% NG 5% NG * 7 : NG A, S 60% P (S) = 0.6, P (S) = 0.4, 80% NG P S (A) = 0.8, 5% P S (A) = P A (S) P A (S) = P (S)P S (A) P (S)P S (A) + P (S)P S (A) = = = P (S), P A (S) 2.3 A A? 9% A % * 8 *7 ( p = 0.8) NG *8 NHK [9] 1, % 4

5 A 9% 91% : A, E A % P (A) = % P A (E) = P A (E) = 1. P E (A) P E (A) = P (A)P A (E) P (A)P A (E) + P (A)P A (E) = ( ) 0.09 = % * %, 2% 1% % * A,B A 3%, B 4% A B 4 : 5 1 A 2.3 M 1, M 2, M 3 60%, 30%, 10% 2%, 3%, 6% 1 M (3 ) 3 A, B, C 1 2 A B C A B A C 1/3 1/2 A : A, B, C A, B, C A, B, C P (A) = P (B) = P (C) = 1 3 F B *9 0.3% 3.2% ([9] ) *10 ( ) ( ) (cf. [2]) 5

6 A B, C P A (F ) = 1 2. B B P B (F ) = 0. C B P C (F ) = 1. P F (A) 1 P (A)P A (F ) P F (A) = P (A)P A (F ) + P (B)P B (F ) + P (C)P C (F ) = = (3, ) : A, B, C A B A, B, C A, B, C B S 3 P S (A) = 1/3, P S (C) = 2/3 * A, B, C, D, E 5 A (1), (2) C ( ) ( ) (1) B (2) B E 3 ([3] ) 2.6 ( 3 ) 3 A, B, C 2 1 A, B, C 1/4, 1/4, 1/2 A B C A B A : 2.4 P (A) = P (B) = 1 4, P (C) = 1 2 F B P A (F ) = 1 2, P B(F ) = 0, P C (F ) = 1. *11 [3] 1/2 6

7 P F (A) 1 P (A)P A (F ) P F (A) = P (A)P A (F ) + P (B)P B (F ) + P (C)P C (F ) = = A 1/3 A C 1 : 2 1 1/3 1/4 1/ A, B, C 1/4, 1/2, 1/4 B A A, B, C 1/2, 1/4, 1/ A, B, C, D, E 5 A, B, C, D, E 1/6, 1/6, 1/6, 1/4, 1/4 A (1), (2) A (1) B (2) B E 3 * x n x 1, x 2,, x n a. x = x 1 + x x n n ( ) x (1) x (2) x (n) = x ( n (1) 42, 38, 40, 44, 52 (2) 42, 38, 40, 44, 52, ) n ( ) x ( n 2 ) + x ( n 2 +1) n *12 I B ( ) [8] [6] 1903 H.G. IT 7

8 : (1) : x = = : 38 < 40 < 42 < 44 < (2) : x = = 69 6 : 38 < 40 < 42 < 44 < 52 < ( ) = (1) (2) (1) (2) ( ) * 13 * * ( ) * 16 * *14 *15 ( ) *16 ( ) 8

9 ( ) (%) 22.3 (24.2) 5.5 (6.0) 5.7 (6.2) 4.4 (4.8) 4.3 (4.7) 3.6 (3.9) 7.3 (7.9) (7.7) 9.7 (10.5) 5.1 (5.5) 7.0 (7.6) 10.1 (11.0) (100.0) ( ) (cf. [8]) b. x n x 1, x 2,, x n x (1) x (2) x (n) x (n) x (1) ( ) ( 3.2 ) n = 2m x (1), x (2),, x (m), x (m+1), x (m+2),, x (2m) n = 2m + 1 x (1), x (2),, x (m), x (m+2), x (m+3),, x (2m+1) n = 2m + 1 m 1 Q 1 3 Q 3 2 Q 2 ( ) Q 3 Q 1, 1 2 (Q 3 Q 1 ) Q 1 3 Q 3 (1) 65, 70, 47, 78, 92, 65, 89, 95, 59, (2) 65, 70, 47, 78, 92, 67, 89, 95, 59, 73 : (1) 47, 59, 65, 65, 70, 78, 89, 92, , 59, 65, 65. Q 1 = = , 89, 92, 95 Q 3 = = (2) 47, 59, 65, 65, 70, 73, 78, 89, 92, 95 Q 1 = 65, Q 3 = , 45,, 3 Q 3 : : Q , 73, 80, 78, 75 73, 75, 78, 78, 80 Q 3 = 78. 9

10 Q 1 m 3.2 Q 1 Q 3 Excel n (1, x (1) ), (2, x (2) ),..., (n, x (n) ) x (t), t y = f(t) f(t) = Q q = ( t t)x ( t ) + (t t )x ( t ), f(1 + q 4 (n 1)), q = 1, 3, t t t t 3.2 Q 3 (1) x (7) = 89, (2) 0.25x (7) x (8) = * 17 : 1. 1 Q 1 3 Q 3 2. Q (, ) , 26, , s 2 = 1 n { (x1 x) 2 + (x 2 x) (x n x) 2} s = 1 n {(x 1 x) 2 + (x 2 x) (x n x) 2 } x s 2 = x 2 x 2. x 2 x 2 x 1 2, x 2 2,, x n 2 : s 2 = 1 n (x 2 k 2xx k + x 2 ) = 1 n n n x 2 k 2x 1 n = x 2 2x x + 1 n nx = x2 x 2 n x k + 1 n n x ( ) *17 : I ( ) 10

11 : x, s x x 1 x s a z = (a 50)/10 % ( B ) 3.2 x x 1, x 2,, x m y y 1, y 2,, y n m + n z x, y, z x, y, z s 2 x, s 2 y, s 2 z (1) z = m m + n x + n m + n y (2) s z 2 = m m + n s x 2 + n m + n s y 2 mn + (x y)2 (m + n) 2 ( ) 3.2 x n x 1 x 2 x r f 1 f 2 f r n k x k f k x s 2 x = 1 n x k f k, s 2 = 1 n (x k x) 2 f k 3.3 x n 3.2 s 2 = x 2 x a b x n 3.2 y y k = ax k + b, k = 1, 2,, r, x x, y x, y s 2 x, s 2 y (1) y = ax + b (2) s 2 y = a 2 2 s x : x k y k = x k 5 10 x = 10y + 5, s x 2 = 10 2 s y 2 y = 1 ( ) = 5.9 x = y 2 = 1 30 ( ) = 38.5 s y 2 = y 2 y 2 = s x 2 =

12 , 45,, (1) x s 2 x (2) 1 Q 1 : (3) m x, y n x, y (x 1, y 1 ), (x 2, y 2 ),, (x n, y n ) x, y, x 1, x 2,, x n y 1, y 2,, y n x, y s x, s y x y s xy s xy = 1 n {(x 1 x)(y 1 y) + (x 2 x)(y 2 y) + + (x n x)(y n y)} x y r r = s xy s x s y s x > 0 s y > (1) r 1 r 1 (2) r = 1 n (3) r = 1 n : : (a 1 b 1 +a 2 b 2 + +a n b n ) 2 (a 2 1 +a a 2 n )(b 2 1 +b b 2 n ) a k = x k x, b k = y k y (1) c k b k = ca k * 18 c > 0 r = 1 y k y = c(x k x) c < 0 r = 1 y k y = c(x k x) (2), (3) 3.6 s xy = xy x y xy xy x 1 y 1, x 2 y 2,, x n y n, x y 2 x, y 2 x, y n *18 (a k t + b k ) 2 t 12

13 (cf. [8], p.60) (i) = ( = ): ( ) (ii) = ( = ): ( ) (iii) = ( = ): ( ) (iv) = : xy > 0 x y (x, y ) xy < 0 x y ( ) n (cf. : ) 3.4 (1) x, y ( ) (cf. [1], p.102) (2) ( ) x y x y ( ) (cf. [8]) y y i 2 x y y = a + bx (a, b ) ax i + b x i ax i + b y i O x i y = ax + b x 13

14 Q(a, b) = n {y i (ax i + b)} 2 a, b i=1 1 n Q(a, b) = 1 n n (yi 2 + a 2 x 2 i + b 2 2ax i y i 2by i + 2abx i ) i=1 = y 2 + a 2 x 2 + b 2 2axy 2by + 2abx = {b (y ax)} 2 + (x 2 x 2 )a 2 2(xy x y)a + y 2 y 2 = {b (y ax)} 2 + s 2 xa 2 2s xy a + s 2 y = {b (y ax)} 2 + s 2 x ( a s xy s 2 x ) 2 s 2 xy + s 2 y a = s xy, b = y ax = y s xy x y y = s xy (x x) s 2 x ( y x ) s 2 x ( ) s 2 x s 2 x = (cm) ([6] ) [1] :?,, [2] :,, [3] : 3,,, [4] :,, [5] : 2,, [6] :,, [7] (, ):,, [8],,, C.R. :,, [9] :,, [10] :,, ( 5 ) 4. ( 5 ) 4 (1)

15 ( 35 ) 24. ( 35 ) 24 (2) ( 35 ) 25 ( 35 ) 24 ( 35 ) < 0.5 < (a), (b) (a) (WWWWLL) W A L W A (WWWWLL) L L W A L L L B (b) (WWWLL) W W A W L W A W L L W A W L L L B L W W A (WWWLL) L W L W A L W L L B L L W W A L L W L B L L L B (1) (a) 1 ( 1 ) 2 ( 1 ) ( 1 2 ( 1 3 ( 1 ) 3 + = ). (b) = 2) (2) (a) ( 2 ) = 26 ( 3 2 ( 3 ) ). (b) 2 ( 3 ) 2 ( 2 ) = A, E P A (E) = 0.01, P A (E) = 0.02, P (A) = 0.01 P E (A) P E (A) = P (A)P A (E) P (A)P A (E) + P (A)P A (E) = 0.01 (1 0.01) 0.01 (1 0.01) = 1 3 P (A) = P E (A) = A, B A, B F P A (F ) = 0.03, P B (F ) = 0.04, P (A) = 4 9, P (B) = 5 9 P F (A) P F (A) = P (A F ) P (F ) = P (A)P A (F ) P (A)P A (F ) + P (B)P B (F ) = = A 1, A 2, A 3 M 1, M 2, M 3 F P (A 1 ) = 0.6, P (A 2 ) = 0.3, P (A 3 ) = 0.1, P A1 (F ) = 0.02, P A2 (F ) = 0.03, P A3 (F ) = 0.06 P F (A 3 ) P F (A 3 ) = P (A 3 )P A3 (F ) P (A 1 )P A1 (F ) + P (A 2 )P A2 (F ) + P (A 3 )P A3 (F ) = = A, B, C, D, E A, B, C, D, E P (A) = P (B) = P (C) = P (D) = P (E) = 1 5. (1) B S P A (S 1 ) = 1 4, P B(S 1 ) = 0, P C (S 1 ) = P D (S 1 ) = P E (S 1 ) = 1 3. P S1 (C) = P (C)P C (S 1 ) P (A)P A (S 1 ) + P (B)P B (S 1 ) + P (C)P C (S 1 ) + P (D)P D (S 1 ) + P (E)P E (S 1 ) =

16 (2) B, E S 2 (1) P A (S 2 ) = 1 4C 2 = 1 6, P B (S 2 ) = P E (S 2 ) = 0, P C (S 2 ) = P D (S 2 ) = 1 3C 2 = 1 3. P S 2 (C) = P A (F ) = 1 2, P B(F ) = 0, P C (F ) = 1. A, B, C 1/4, 1/2, 1/4 P (A) = P (C) = 1 4, P (B) = 1 2 P F (A) = /2, 1/4, 1/4 P F (A) = P (A) = P (B) = P (C) = 1 6, P (D) = P (E) = (1) P S1 (A) = 3 19, (2) P S 2 (A) = Q , 54, 56, 58 Q 1 = 54. m , 64, 65, 65, 65, 66, 67, 70, 70 m = = (1) (m + n)z = mx + ny (2) (m + n)s 2 z = (m + n)z 2 (m + n)z 2 = mx 2 + ny 2 1 (mx + ny)2 m + n ) ) = m(x 2 x 2 ) + n(y 2 y 2 ) + (m m2 x 2 + (n n2 y 2 2mn m + n m + n m + n x y = ms 2 x + ns 2 y + mn m + n (x y)2 3.3 s 2 = 1 n (x 2 k 2xx k + x 2 )f k = 1 n = x 2 2x x + x 2 = x 2 x (1) y = 1 y k f k = 1 (ax k + b)f k = a 1 n n n (2) s y 2 = 1 n (y k y) 2 f k = 1 n x k 2 f k 2x 1 n x k f k + x 2 1 n x k f k + b 1 n {ax k + b (ax + b)} 2 f k = 1 n 3.5 (1) x k y k = x k 5 10 f k f k = ax + b. a 2 (x k x) 2 f k = a 2 s 2 x. x = 10y + 5, s x 2 = 10 2 s y 2 y = 1 ( ) = 6.18 x = y 2 = 1 50 ( ) = s y 2 = y 2 y 2 = s x 2 = (2) Q , 54, 54, 56, 56, 59 Q 1 = 56. (3) , 70, 65, 68, , 64, 66, 65, 69, 61, 62, 65, 70, 61 m = = s xy = 1 n (x k y k xy k yx k + x y) = 1 n x k y k x 1 n y k y 1 n x k + 1 n x y n n n n n = xy x y y x + x y = xy x y. 16

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