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- ときな すすむ
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3 1 1.1,. De DeNA,, 1 3, 1 1.,,,, ,,. n 6. (p-q-r) p q r. NPB HP. 1.2 MLB Major League Baseball NPB Nippon Professional Baseball Organization QS, 6 6., 6 n (n = 0, 1, 2, ), n , 6 1 2/3., 7., 3 6., 9 1, 6 2.,, 2/3., 6 3,. 1 9, 10. 2,. 3, 6,. 1
4 1.4 ( 14 17), 2018, , 6 n,., NPB,. =, , 0.589,. 4, , , 1, 2, ,. 1: , : , x : µ k :k σ k :k D k :k D k :k 4. 5, , ,. 2
5 2018, 1 2, (µ).,, (x) (µ), D 1 = µ 1 x, D 2 = x µ 2.,,,.,. D 1 = µ 1 x σ 1, D 2 = x µ 2 σ NPB HP, µ 1 = 0.596, σ 1 = 0.043, µ 2 = 0.548, σ 2 = D 1 = D 2,. x, σ 1 + σ 2 0, µ 1, µ 2, σ 1, σ 2, µ 1 x σ 1 = x µ 2 σ 2 σ 2 (µ 1 x) = σ 1 (x µ 2 ) x(σ 1 + σ 2 ) = σ 2 µ 1 + σ 1 µ 2 x = σ 2µ 1 + σ 1 µ 2 σ 1 + σ 2 x = , 6, ( ) ,., 6., 6 n 6 n,. 1 ( ). 8 6 n,. 2 ( ). 6 n,. 6 +, -, ( ).,, ( ). 8 14:144, 15 :143 3
6 17 (n = 1)., (21-7-0), 28., = 0.196, 21 (21 + 7) = , 6 2,.,, 0 1, : 6 15 n (6 ) ( ) , n = 0, 1, 2, 3, n 4, n., (x 1 ) (x 2 ), (Y ),., Y n 6 n, Y n 6 n. Y 0 = a 1 x 1 + a 2 x 2 + a 0 Y 1 = b 1 x 1 + b 2 x 2 + b 0 Y 2 = c 1 x 1 + c 2 x 2 + c 0 Y 3 = d 1 x 1 + d 2 x 2 + d 0 Y 4 = e 1 x 1 + e 2 x 2 + e 0 a, b, c, d, e. 4.2,,.,,. x 1 x 2 r r = 0.12 (n = 0) 0.02 (n = 1) 0.08 (n = 2) 0.13 (n = 3) 0.13 (n 4) 4
7 , n, ( ). y y x 1 y x 2 y x 1 x 1 x 1 x 2 y x 2 x 1 x 2 x 2 V ar(y) Cov(y, x 1 ) Cov(y, x 2 ) Cov(y, x 1 ) V ar(x 1 ) Cov(x 1, x 2 ) Cov(y, x 2 ) Cov(x 1, x 2 ) V ar(x 2 ) 5.1 n = a 1, a 2 n = 0,. V ar(y) Cov(y, x 1 ) Cov(y, x 2 ) Cov(y, x 1 ) V ar(x 1 ) Cov(x 1, x 2 ) = Cov(y, x 2 ) Cov(x 1, x 2 ) V ar(x 2 ), V ar(y 0 ) Cov(Y 0, x 1 ) Cov(Y 0, x 2 ) Cov(Y 0, x 1 ) V ar(x 1 ) Cov(x 1, x 2 ) Cov(Y 0, x 2 ) Cov(x 1, x 2 ) V ar(x 2 ) Y n x 1, x 2, V ar(y 0 )., V ar(x 1 ), V ar(x 2 ), Cov(x 1, x 2 )., Cov(Y 0, x 1 ) Cov(Y 0, x 2 ). 1 ( ). Cov(x, ay + bz + c) = acov(x, y) + bcov(x, z) (a, b, c : const.). u := ay + bz + c Cov(x, u) = 1 n (x i x)(u i u) n = 1 n = 1 n i=1 n (x i x){(ay i + bz i + c) (ay + bz + c)} i=1 n (x i x){a(y i y) + b(z i z)} i=1 = a 1 n n i=1 (x i x)(y i y) + b 1 n = acov(x, y) + bcov(x, z) n (x i x)(z i z) i=1 5
8 , Y 0 = a 1 x 1 + a 2 x 2 + a 0 x 1 Cov(Y 0, x 1 ) Cov(Y 0, x 1 ) = Cov(a 1 x 1 + a 2 x 2 + a 0, x 1 ) = a 1 Cov(x 1, x 1 ) + a 2 Cov(x 2, x 1 ) = a 1 V ar(x 1 ) + a 2 Cov(x 1, x 2 ) = a a 2., Y 0 x 2 Cov(Y 0, x 2 ) Cov(Y 0, x 2 ) = Cov(a 1 x 1 + a 2 x 2 + a 0, x 2 ) = a 1 Cov(x 1, x 2 ) + a 2 V ar(x 2 ) = a a 2, (n = 0) V ar(y 0 ) a a a a a a a a , ,, 2. Cov(y, x 1 ) Cov(Y 0, x 1 ) Cov(y, x 2 ) Cov(Y 0, x 2 ) = a a = a a 2. 9 ( ) ( ) ( = ( ) 1 ( ) ( ) 1 ( ) ( = ( ) 1 ( ) ( ) a 1 = a 2 ( ) 1 ( ) = a 1 a 2 ) a 1 a 2 ), { a 1 = a 2 =
9 5.1.2 a 0 a 0 y, x 1, x 2 a 0 = y a 1 x 1 a 2 x 2. y = x 1 = x 2 = ,. a 0 = n = 1, 2, 3 n 4, n = n = 1, 2, 3 n n = 1 n = 1, V ar(y) Cov(y, x 1 ) Cov(y, x 2 ) Cov(y, x 1 ) V ar(x 1 ) Cov(x 1, x 2 ) = Cov(y, x 2 ) Cov(x 1, x 2 ) V ar(x 2 ) , b 0 = b 1 = b 2 = n = 2 n = 2, V ar(y) Cov(y, x 1 ) Cov(y, x 2 ) Cov(y, x 1 ) V ar(x 1 ) Cov(x 1, x 2 ) = Cov(y, x 2 ) Cov(x 1, x 2 ) V ar(x 2 ) , c 0 = c 1 = c 2 =
10 5.2.3 n = 3 n = 3, V ar(y) Cov(y, x 1 ) Cov(y, x 2 ) Cov(y, x 1 ) V ar(x 1 ) Cov(x 1, x 2 ) = Cov(y, x 2 ) Cov(x 1, x 2 ) V ar(x 2 ) , d 0 = d 1 = d 2 = n 4 n 4, V ar(y) Cov(y, x 1 ) Cov(y, x 2 ) Cov(y, x 1 ) V ar(x 1 ) Cov(x 1, x 2 ) = Cov(y, x 2 ) Cov(x 1, x 2 ) V ar(x 2 ) , e 0 = e 1 = e 2 = , n = 0, 1, 2, 3 Y 0 = x x Y 1 = x x Y 2 = x x Y 3 = x x Y 4 = x x
11 6.1, ( ) = ( ) ( ).,,.,, NPB HP. 10, n (n = 0), ( Y 0 ) = x x x 1 = 0.133, x 2 = 0.684, ( ) = 0.407, y = 0.444, ( ) < ( ), n = ,.,. n R, (n = 0) (n = 1) R = (n = 2) (n = 3) (n 4), n = 0, 1, 2 R 0.7, n = , n. 2, 2.. 2,,,., ,
12 7.2 n.. 2, 11,., 1, ,,, 16,,, 15,,, 14,,, µ, (n = 0) (n = 1) µ = (n = 2) (n = 3) (n 4) 7.3 2,. 5.3 Y n x 1.,, n Y , x 2 µ., (Y 0 =)0.565 = x (Y 1 =)0.565 = x (Y 2 =)0.565 = x (Y 3 =)0.565 = x (Y 4 =)0.565 = x , x 1, x 1 = (n = 0) (n = 1) (n = 2) (n = 3) (n 4) 37.,, (1), 2018.,, µ., µ, ( ) ( + ) , 4 10
13 4: ver. 1 n µ ( ) ( ) ,,. x 1 (n = 3).,. n , , n. 9 5, n 1 n 2 n 3 n 1 n 2 n 3,. Y n 6 n, Y n 6 n. Y 1 = x x Y 2 = x x Y 3 = x x Y 1 = x x Y 2 = x x Y 3 = x x , x 1, x 2 r 0.17 (n 1), (n 2) 0.40 (n 3) r = 0.25 (n 1) 0.02 (n 2) 0.05 (n 3)
14 , R (n 1) (n 2) (n 3) R = (n 1) (n 2) (n 3), n = 1, 2, 3 n 4. n 3, n 1, n 2 R, µ, (n 1) (n 2) µ (n 3) = (n 1) (n 2) (n 3)., x 1, (n 1) (n 2) (n 3) 103 x 1 = (n 1) (n 2) (n 3) (2)., n, n 143., n 2 n 3. n, n = 0 n 2 n = 2 n = 1 n = 1 n 1 n 2 n = 2 n 2 n 1 n = 3 n 3 n 2 n 4 ( ) n
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t χ 2 F Q t χ 2 F 1 2 µ, σ 2 N(µ, σ 2 ) f(x µ, σ 2 ) = 1 ( exp (x ) µ)2 2πσ 2 2σ 2 0, N(0, 1) (100 α) z(α) t χ 2 *1 2.1 t (i)x N(µ, σ 2 ) x µ σ N(0, 1
t χ F Q t χ F µ, σ N(µ, σ ) f(x µ, σ ) = ( exp (x ) µ) πσ σ 0, N(0, ) (00 α) z(α) t χ *. t (i)x N(µ, σ ) x µ σ N(0, ) (ii)x,, x N(µ, σ ) x = x+ +x N(µ, σ ) (iii) (i),(ii) z = x µ N(0, ) σ N(0, ) ( 9 97.
