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2 Web http://mybook-pub-site.sakura.ne.jp/statistics Multivariate/index.html

1 2 1 2.1................................. 1 2.2 (............... 8 2.2.1................... 9 2.2.2.................. 12 2.2.3.............. 14 2.2.4.................... 17 2.3 (................. 20 2.3.1................... 21 2.3.2.................... 23 2.4........................ 27 29

1 2 2.1 2.1: RAND() (.xlsx) RAND() 2.1 Microsoft Excel 2007 Sheet1 RAND().xlsx Excel Excel URL(http://www.kyoritsu-pub.co.jp/bookdetail/9784320122680) B3 = RAND() (2.1)

2 2 2 2 RAND Enter RAND() B3 2.2 12 RAND() 10 2.3 (1) B3 Shift B12 (2) (3) (4) (5) 2.4 10 RAND() F9 F9 2.2: RAND() (.xlsx) RAND() x 0 < x < 1 (0, 1) 0 x 1 [0, 1] 0 < x 1 (0, 1] RAND() (0, 1) 27

2.1. 3 2.3: 2.4: (.xlsx) RAND() =17 15 RAND() [1] Excel Excel VBA(Visual Basic for Application) Excel VBA???? IX, IY, IZ [1, 30268], [1, 30306], [1, 30322]

4 2 IX = MOD(171 IX, 30269) (2.2) IY = MOD(172 IY, 30307) (2.3) IZ = MOD(170 IZ, 30323) (2.4) RAN DOM = MOD(IX/30269 + IY/30307 + IZ/30323, 1) (2.5) 30269, 30307, 30323 MOD(I, J) I J (2.2) 1 30268 2.5 Excel B5 10000 B6 (2.2) B5 B30272 1 30268 B30273 10000 B 5 30272 D 2.6 IX B5 Shift Ctrl (2) (3) (7) IX (4) D5 IX 2.5 E D E5 E30271 D G5 E 1 30267 30268 IX (2.3) (2.4) 1 30306, 1 30322 IX, IY, IZ 3 7 [2]. 1 30268, 30306, 30322 (=6953607871644) (2.5) IX, IY, IZ 1 IX/30269, IY/30307, IZ/30323 1 0 < IX/30269 + IY/30307 + IZ/30323 < 3 (2.6)

2.1. 5 2.5: RAND() IX (IX.xlsx) 2.6: IX (0, 3) 27 IX/30269 + IY/30307 + IZ/30323 1 (0, 1), (1, 2), (2, 3) (0, 1) x, y a, b, c, d 0 < a, c < x, 0 < b, d < y a x + b y = 1 + c x + d y (2.7)

6 2 c/x + d/y a/x + b/y 1 (b d) x y = x (a c) (2.8) x, y b d < y 0 0 (2.5) RANDOM (0, 1) 27 2.7 (a) (2.2) (2.4) IX/30269+IY/30307+IZ/30323 (0, 3] (0, 0.0001], (0.0001, 0.0002] 0.0001 30000 IX/30269 + IY/30307 + IZ/30323 (0, 1], (1, 2], (2, 3] 3 (0, 1] 2.7 (b) (2.5) RANDOM RAND() (0, 1) 7 1 RAND() 2.8 (2.5) RANDOM (2.2) (2.5) 1000 1 i R 1i R 1i R 1i = 1 nv 2 e n (r 1j r)(r ij r) (2.9) j=1 r ij i j r, v 2 e n = 1000 v 2 e v 2 e = 1 n 2 1 n i=1 n (r ij r) 2 (2.10) 1 1 1 0.1 7 RAND() j=1 Excel RAND() 2.1, 2.2 RAND() Excel RAND() (2.2) (2.4) IX, IY, IZ RAND() Excel =RAND() (0, 1)

2.1. 7 2.7: RAND() 2.8: RAND() 2.9 2.10 1 RAND() 1000 B4 D4 1 IF(a, b, c) a b c B7 A7 0.0101 0 0.0101 IF 0.010101 0 0.010101 1 A7 [0.0101, 0.010101] 1 1000 A1009 1 1 1000 0.989 95% [0.925, 1.053] Excel RAND()

8 2 2.9:.xlsx 2.10: 2 2.2 ( p p 1 6 {2, 4, 6}

2.2. ( 9 2.2.1 10 RAND() RAND() 0 1 RAND() 2.11 B7 B16 = INT(2 RAND()) (2.11) 2 RAND() (0, 2) INT() 1 0 B7 B16 10 B4 = COUNT(B7 : B16) (2.12) B7 B16 C19 = COUNTIF($B$7 : $B$16, B19)/$B$4 (2.13) B7 B16 B19 B7, B16, B4 $B$7 $B$16 $B$4 $ (2.13) C20 B7, B16, B4 B19 $ (2.13) B19 B20 C20 2.11 0.5 2.12 (1) (2) (3) (4)2-D 2.13 2.14 (1) (2) (3) ( B19 B20 (4)(5) 2.11

10 2 2.11: (n=10)( n=10.xlsx) 2.12: (n=10) 2.15 1000 0.5 0.5 10 1000 0.5

2.2. ( 11 2.13: (n=10) 2.14: (n=10) 2.11 2.15 2.16 2.17 B7 B16 INT(2*RAND()) B7 B1006 B4 COUNT(B7:B16) COUNT(B7:B1006) n

12 2 F19, F20 2.18 0 2.15 2.15: (n=1000)( n=1000.xlsx) 2.16: (n=1000) 2.2.2 Excel 1 6 1/6 2.19

2.2. ( 13 2.17: (n=1000) 2.18: (n=1000) = INT(6 RAND()) + 1 (2.14) 6*RAND() (0, 6) INT() 1 1 6 B7 B5006 B4 F19 F24 5000 1/6

14 2 2.19: (n=5000)( n=5000.xlsx) 2.2.3 f(x) = 1 2πσ exp ( ) (x µ)2 2σ 2 (2.15) µ σ µ = 0, σ = 1 f(x) = 1 ) exp ( x2 2π 2 F (x) f() F (x) = x (2.16) f(y)dy (2.17) 2.20 x B4 x -4 B5 = ROUND(B4 + 0.1, 1) (2.18)

2.2. ( 15 ROUND(a, b) a b+1 B4 0.1 2 2 0.1 15 10 0.1 2 0.1 (10) = 0.0001100110011 (2) (2.19) (i) i 2 10 0.000110011 (2) = 2 4 1 + 2 5 1 + 2 8 1 + 2 9 1 + (10) = 0.099609375 + (10) (2.20) 2 0.1 (10) 2.20: (.xlsx) 2.20 C4 = NORMDIST(B4, 0, 1, FALSE) (2.21) NORMDIST(a, b, c, d) d FALSE b, c x = a b = 0, c = 1 (2.21) B4

16 2 2.21: 2.22: 2.21 (1) (2) (3) (4) NORMDIST(x, 0, 1,TRUE) 2.22

2.2. ( 17 (2.17) F (x) f(x) (a) x = 1.81 x x = 1.81 (b) F (x) 2.2.4 2.23: (n=5000) n=5000.xlsx 2.23 = NORMSINV(RAND()) (2.22)

18 2 NORMSINV() RAND() (0, 1) 2.24 RAND() F (x) NORMSINV() x RAND() (0, 1) 0.5 RAND() NORMSINV() 0 0 x RAND() 0 1 NORMSINV() x 2.24: 2.23 = ROUND(NORMSINV(RAND()), 0) (2.23)

2.2. ( 19 2.25 G19 = NORMSDIST(E19 + 0.5) NORMSDIST(E19 0.5) (2.24) 3 ± 0.5 NORMSDIST(x) x G19 4 2.26 (1) (2) (3) (4)-1234... (5) 4 2.27 (1) (2) (3) (4) (5) 2.25 Excel.xlsx 2.25:.xlsx

20 2 2.26: 2.27: 2.3 ( x n n q x = q/n n

2.3. ( 21 2.3.1 Excel 2.28 n = 2 q q/n 1 i q i x i = q i /n 2.28 N = 5000 x i, i = 1, 2,, N x i AVERAGE() B10 AVERAGE(B7:B8) B7, B8 i j x ij i x i n x i = 1 n j=1 x ij (2.25) B12 5000 x x x = 1 N N x i (2.26) i=1 C12 ve 2 VAR(B10:GJI10) v 2 e v 2 e = 1 N 1 N (x i x) 2 (2.27) i=1 2.29 5000 C15 B10 BJI10 5000 B15 (=0) C4 (=5000) 0 NORMDIST(x, µ, σ,true) (2.17) f() (2.15) µ, σ B12, C12 5000 x ve 2 ve 2 NORMDIST(x + 0.25, x, ve,true)-normdist(x 2 0.25, x, ve,true) 2 x ± 0.25 TRUE NORMDIST(x, µ, σ,true) x FALSE x 2.30 x i = 0 1 x i = 0.5

22 2 2.28:, n=2 n=2.xlsx 2.29: /, n=2 n=2.xlsx 2.30: / n=2 n 2.31 2.33 n=10 5000 x i n=10

2.3. ( 23 2.31:, n=10 n=10.xlsx 2.32: /, n=10 n=10.xlsx 2.3.2 2.34 2.36 f(x) = λe λx (x 0) (2.28)

24 2 2.33: / n=10 [0, x] F (x) F (x) = x λe λx 0 = [ e λx] x 0 = 1 e λx (2.29) 2.24 F (x) RAND() (0, 1) F (x) RAND() = 1 e λx (2.30) x = 1 ln(1 RAND()) (2.31) λ 1-RAND() (0, 1) RAND() (0, 1) x = 1 ln(rand()) (2.32) λ 2.34 B7 (2.32) B4 λ = 2 B9 2 5000 1 1 2.35 C14 B9 GJI9 5000 0 2.36 0 [0, 0.1) 2.37 2.39 n=4 5000 4 2

2.3. ( 25 2.34:.xlsx 2.35:, n=10.xlsx 2.36: 0 1 0.1 2.39 n=4 2.40 n=100 5000

26 2 2.37:, n=4 n=4.xlsx 2.38: /, n=4 n=4.xlsx 2.39: / n=4

2.4. 27 2.40: / n=100 n=100.xlsx 2.4 x i, i = 1, 2,, n ve 2 s2 ve 2 1 n = (x i x) 2 n 1 i=1 s 2 = 1 n (x i x) 2 (2.33) n i=1 ve 2 s2 n 1 n Excel VAR() (2.27) ve 2 s2 2.41 B9 B18 µ = 0 σ 2 = 4 10 µ = 0 σ 2 = 4 B9 B18 C9 C18 µ = 0 x i 2 (x i µ) 2 B20 C9 C18 B21 B22

28 2 10 5000 B24 C26 σ 2 = 4 B26 5000 2.42 20 21 22 2.41:.xlsx 2.42:

29 [1] B. A. Wichmann and I. D. Hill, Algorithm AS 183: An Efficient and Portable Pseudo-Random Number Generator. Journal of the Royal Statistical Society. Series C (Applied Statistics), Vol. 31, No. 2, pp. 188-190, 1982. [2] B. A. Wichmann and I. D. Hill, Correction: Algorithm AS 183: An Efficient and Portable Pseudo-Random Number Generator. Journal of the Royal Statistical Society. Series C (Applied Statistics), Vol. 33, No. 1, p. 123, 1984.

30 (0, 1), 2 [0, 1], 2 2, 15 AVERAGE(), 21 COUNTIF(), 9 COUNT(), 9 Excel VBA, 3 INT(), 9 NORMDIST(), 15, 16, 21 NORMSDIST(), 19 NORMSINV(), 18 RAND(), 1 RAND(), 4 ROUND(), 15 VAR(), 21, 2, 2, 14, 18, 14, 2, 18, 2, 2, 8, 10, 9, 12, 1, 2, 8, 6, 8, 15, 8, 14, 27, 1, 8, 12, 2, 21, 8, 2 $, 9, 2, 19, 14, 18, 17, 14, 27, 27

31, 6, 21, 27, 21, 14, 21, 9, 19, 27, 27, 27, 3, 15, 2, 9, 2, 2

32 3, 2012 http://www.kyoritsu-pub.co.jp/bookdetail/9784320122680 Web