tokei01.dvi

2. :,,,. :.... Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 4

3. (probability),, 1. : : n, α A, A a/n. :, p, p Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 5

1933 (Kol-. mogorov, A. N. (193-1989)) S, A ( ). p( ) 3, p S, p(a) A. 1 p(a) 2 p(s) =1 3 A 1, A 2,..., A n S (i j i, j A i A j = φ), p(a 1 A 2 A n )=p(a 1 )+p(a 2 )+...+ p(a n ) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 6

1 ( )., 1, 2, 3, 4, 5, 6 6,. (event). 1 2,3,4,5,6. 1 (complementary event). A, A Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 7

A + A = S 2,3 1. p(a)+p(a) =1 2. p(a) =1 p(a) 3. p(φ) = A 1, A 2,..., A n, 3. 4. p(a 1 A 2 A n )=p(a 1 )+p(a 2 )+ + p(a n ) (4) 1 p(a) 2 p(s) =1 3 A 1, A 2,..., A n S (i j i, j A i A j = φ), p(a 1 A 2 A n )=p(a 1 )+p(a 2 )+...+ p(a n ) (1) (2) (3) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 8

p(a) > A B 5. p(a B) p(b A) = p(a) (5) 6. p(a B) =p(b A)p(A) (6). 7. p(a 1 A 2 A n )=p(a 1 )p(a 2 ) p(a n ) (7) 1 p(a) 2 p(s) =1 3 A 1, A 2,..., A n S (i j i, j A i A j = φ), p(a 1 A 2 A n )=p(a 1 )+p(a 2 )+...+ p(a n ) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 9

8. p(b A) 9. p(s A) =1, B, C 1. p(b C A) =p(b A)+p(C A) (1) 8. 9. 1. 2,3, S p(b A). p(b) p(b A). (8) (9) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 1

11. p(a A) =1 (11) p(a B) =p(a B)p(B) (12) 12. p(a B) = p(a)p(b A) p(b) : (13) A 1 A 2 A n = S (14), B. 13. p(b) =p(a 1 )p(b A 1 )+p(a 2 )p(b A 2 )+ + p(a n )p(b A n ) (15) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 11

A 1 A 2 A n = S (16), B. 14. p(a i )p(b A i ) p(a i B) = p(a 1 )p(b A 1 )+p(a 2 )p(b A 2 )+ + p(a n )p(b A n ) (17) : (15) p(a i B) = p(a i)p(b A i ) p(b) (18). (15) p(b) p(a i ) p(b A i ), p(a i B), p(a i ), p(a i B). Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 12

1.1, ( ). ( ) ( ) 35 5 8 4 1.,. 2.,. 3.,. 4.. Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 13

4.,.,,,.,, 1,.,. Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 14

,,, raw 2x 77 64 7 64 78 66 7 64 41 37 85 53 34 69 78 67 61 57 68 69 39 74 84 74 69 5 86 56 65 66 69 61 76 65 87 85 78 79 54 7 82 49 75 71 58 75 52 45 53 [,9] 5 1 [1,19] 15 [2,29] 25 [3,39] 35 3 [4,49] 45 3 [5,59] 55 8 [6,69] 65 15 [7,79] 75 14 [8,89] 85 6 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 15

,,, raw 2x 77 64 7 64 78 66 7 64 41 37 85 53 34 69 78 67 61 57 68 69 39 74 84 74 69 5 86 56 65 66 69 61 76 65 87 85 78 79 54 7 82 49 75 71 58 75 52 45 53 [,9] 5 1 [1,19] 15 [2,29] 25 [3,39] 35 3 [4,49] 45 3 [5,59] 55 8 [6,69] 65 15 [7,79] 75 14 [8,89] 85 6 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 16

,,, raw 2x 77 64 7 64 78 66 7 64 41 37 85 53 34 69 78 67 61 57 68 69 39 74 84 74 69 5 86 56 65 66 69 61 76 65 87 85 78 79 54 7 82 49 75 71 58 75 52 45 53 16 14 12 1 8 6 4 2 5 15 25 35 45 55 65 75 85 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 17

,,, raw ( ) 2x 77 64 7 64 78 66 7 64 41 37 85 53 34 69 78 67 61 57 68 69 39 74 84 74 69 5 86 56 65 66 69 61 76 65 87 85 78 79 54 7 82 49 75 71 58 75 52 45 53 5 45 4 35 3 25 2 15 1 5 5 15 25 35 45 55 65 75 85 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 18

,,, raw ( % ) 2x 77 64 7 64 78 66 7 64 41 37 85 53 34 69 78 67 61 57 68 69 39 74 84 74 69 5 86 56 65 66 69 61 76 65 87 85 78 79 54 7 82 49 75 71 58 75 52 45 53 1 9 8 7 6 5 4 3 2 1 5 15 25 35 45 55 65 75 85 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 19

5. :, mean x = 1 N x i =64.38 (19) N i=1 2x 77 64 7 64 78 66 7 64 41 37 85 53 34 69 78 67 61 57 68 69 39 74 84 74 69 5 86 56 65 66 69 61 76 65 87 85 78 79 54 7 82 49 75 71 58 75 52 45 53 16 14 12 1 8 6 4 2 5 15 25 35 45 55 65 75 85 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 2

(median) : (median). N Me = x( N +1 ) 2 N (2) Me = 1 2 (x(n 2 )+x(n 2 + 1)) = 67.5 (21) 2x 77 64 7 64 78 66 7 64 41 37 85 53 34 69 78 67 61 57 68 69 39 74 84 74 69 5 86 56 65 66 69 61 76 65 87 85 78 79 54 16 14 12 1 8 6 4 7 82 49 75 71 58 75 52 45 53 2 5 15 25 35 45 55 65 75 85 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 21

(mode) : (mode). f m =max(f 1,f 2, f k ) (22) m, x m Mo = xm =65. (23) f m f m 1 Mo = a m + c (24) f m f m 1 + f m f m+1. c, a m. Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 22

(variance) : x x x 2. N 1 N (x i x) 2 =93.1 (25) N i=1 N ( ) x 1 N (x i x) 2 =95. (26) N 1 i=1 2x 77 64 7 64 78 66 7 64 41 37 85 53 34 69 78 67 61 57 68 69 39 74 84 74 69 5 86 56 65 66 69 61 76 65 87 85 78 79 54 7 82 49 75 71 58 75 52 45 53 16 14 12 1 8 6 4 2 5 15 25 35 45 55 65 75 85 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 23

(standard deviation) :. N 1 N (x i x) N 2 =9.65 (27) i=1 N ( ) x 1 N (x i x) N 1 2 =9.75 (28) 16 i=1 2x 77 64 7 64 78 66 7 64 41 37 85 53 34 69 78 67 61 57 68 69 39 74 84 74 69 5 86 56 65 66 69 61 76 65 87 85 78 79 54 7 82 49 75 71 58 75 52 45 53 14 12 1 8 6 4 2 5 15 25 35 45 55 65 75 85 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 24

: : 1 k 1 N x k i N i=1 k 1 N (x i x) k N i=1 16 14 12 1 8 6 4 2 (29) (3) 5 15 25 35 45 55 65 75 85 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 25

, :, 3 3 1 1 N (x σ 3 i x) 3 (31) N i=1 : 4 4 1 1 N (x σ 4 i x) 4 N i=1 16 14 12 1 8 6 4 2 (32) 5 15 25 35 45 55 65 75 85 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 26

6. 2x (, ) (73, 75) (79, 7) (71, 95) (64, 85) (87, 95) (82, 95) (71, 5) (58, 75) (19, 15) (87, 75) (74, 5) (79, 8) (74, 95) (63, 5) (74, 5) (66, 5) (75, 5) (58, 7) (85, 7) (84, 95) (93, 7) (2, ) (51, 15) (75, 75) (68, 85) (63, 75) (71, 2) (72, 7) (86, 7) (67, 95) (75, 5) (78, 95) (65, 5) (71, 9) (42, 75) (84, 7) (85, 75) (62, 75) (57, 95) (83, 6) (82, 7) (72, 15) (81, 95) (75, 5) (85, 75) (7, 64) (61, 3) (92, 95) (47, 4) (,) 1 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 1 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 27

(covariance):1 2 s xy = 1 N (x i x)(y i y) = 288.6 (33) N x = 1 N i=1 N x i =69.12, i=1 s xx = σ 2 x = 1 N y = 1 N N y i =64.68 (34) i=1 N (x i x) 2 = 326.3, s yy = σy 2 = 1 N i=1 N (y i y) 2 = 672.4(35) i=1 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 28

:,,, +1, -1 correlation coefficient.,. s xy = 288.6, s xx = 326.3, s yy = 672.4 (36) r = s xy sxx s yy =.616 (37). Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 29

7. (random variable). (discrete random variable) 1 continuous random variable) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 3

2 x ξ (x = ξ) p(x = ξ) x (a, b) (a, b], [a, b), [a, b] p x = a, x = b. Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 31

probability distribution (1),(2),(3) P (x = ξ) =p(x), p(x) (probability function p(x) (38) 1 prob1.txt.8.6.4.2 1 2 3 4 5 6 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 32

probability distribution f(x) f(x) p(a, b] = b a f(x)dx =1 (39) f(x)dx (4) x (a, b] f(x) x. 1 exp(-x**2).9.8.7.6.5.4.3.2.1 Apr. - Jul., 26FY -4-3 -2-1 1 Dept. 2 of3 Mechanical 4 Engineering, Saga Univ., JAPAN 33

, x = ξ. x ξ, F (ξ) = ξ f(x)dx (41) 1 norm(x).9.8.7.6.5.4.3.2.1-4 -3-2 -1 1 2 3 4 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 34

x, expectation. x E(x), μ.. N μ = E(x) = x i p i i=1, f(x) μ = E(x) = (42) xf(x)dx (43) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 35

: x 1. N σ 2 = V (x) = p i (x i μ) 2 (44) i=1, f(x) σ 2 = V (x) = : σ (x μ) 2 f(x)dx (45) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 36

3.1 1. E(ax + b) = ae(x)+ b (46) 2. V (ax + b) =a 2 V (x) (47) x f(x) E(ax + b) = V (ax + b) = = = (ax + b)f(x)dx = axf(x)dx + bf(x)dx = ae(x)+b (48) (ax + b E(ax + b)) 2 f(x)dx (ax + b (ae(x)+b)) 2 f(x)dx (ax ae(x)) 2 f(x)dx = a 2 (x E(x)) 2 f(x)dx = a 2 V (x) (49) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 37

8. (-1). x 1, p(x =1)=p 1, p(x =)=p 2 (= 1 p 1 ) (5) E(x) =p 1 1+p 2 =p 1 V (x) =p 1 (1 p1) 2 +(1 p1) ( p1) 2 = p 1 (1 p1) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 38

(-1). x 1, p(x =1)=p 1, p(x =)=p 2 (= 1 p 1 ) E(x) =p 1 1+p 2 =p 1 (51) V (x) =p 1 (1 p1) 2 +(1 p1) ( p1) 2 = p 1 (1 p1) (52) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 39

1,2,3,4 1. A 2. A p q =1 p 3. 4. N N A,2 Bernoulli Jakob 1654 175 1713 Ars conjectandi Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 4

3.2 N A x B(N,p,x) = N C x p x (1 p) N x (53). :N A x, NC x p x (1 p) N x.,, Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 41

B(N,p,x) = N C x p x (1 p) N x (54) : Np N x= xb(n,p,x) = = N x N C x p x (1 p) N x x= N N! x (N x)!x! px (1 p) N x x= = Np = Np N (N 1)! ((N 1) (x 1))!(x 1)! px (1 p) (N 1) (x 1) N B(N 1,p,x)=Np (55) x=1 x=1 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 42

B(N,p,x) = N C x p x (1 p) N x (56) : Np(1 p) N x 2 B(N,p,x) = x= = = N x 2 NC x p x (1 p) N x x= N N! (x(x 1) + x) (N x)!x! px (1 p) N x N N! N x(x 1) (N x)!x! px (1 p) N x N! + x (N x)!x! px (1 p) N x x= x= = p 2 N(N 1) (N 2)! ((N 2) (x 2))!(x 2)! px 2 (1 p) (N 2) (x 2) + Np x=2 = p 2 N(N 1) + Np (57) V (x) =p 2 N(N 1) + Np N 2 p 2 = Np 2 + Np = Np(1 p) (58) x= Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 43

1., 25.327. 26,4 3. 2.,1 3 5. Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 44

1837 (Poisson, S.D., 1781-184) 2,. 1898 (Bortkiewicz, L von, 1868 1931)., 1. 191, (Rutherford, E, and Geiger H.) Poisson : 2 Np μ, N, p Po(x) =e μμx x!, Poisson (59) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 45

2 Np μ, N, p Po(x) =e μμx x! : Np μ (1 p) N e μ. lim N N C x p x (1 p) N x = lim = lim N = μx x! e μ N N(N 1)(N 2) (N x +1) p x (1 p) N x x! Np(Np p)(np 2p) (Np (x 1)p) (1 p) N x x! (6) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 46

Po(x) =e μμx x! :E(x) =μ : x= :V (x) =μ : xe μμx x! = μ V (x) = = μ 2 x=2 x= x=1 (61) μ μ(x 1) e (x 1)! = μ (62) x 2 e μμx x! E2 (x) = (x(x 1) + x)e μμx x! μ2 x= μ μ(x 2) e (x 2)! + μ μ2 = μ 2 + μ μ 2 = μ (63) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 47

9. f(x) (a, b), (uniform distribution), U(a, b, x). : : f(x) = E(x) = V (x) = 1 (b a) b a b a a<x<b (64) x (b a) dx = a + b 2 x 2 (a + b)2 dx (b a) 4 = a + b 2 = (a b)2 12 (65) (66) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 48

( ),. x (exponential distribution).,. : f(x) =λe λx, λ >, x > (67) E(x) = xλe λx dx (68) :. f(x) =x, g (x) =λe λx. df(x)g(x) = f(x)g (x)+f (x)g(x) (69) dx. xe λx = xλe λx dx e λx dx (7),. E(x) = xλe λx dx = 1 λ Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 49 (71)

f(x) =λe λx, λ >, x > (72) V (x) = x 2 λe λx dx E 2 (x) = 1 λ 2 (73) :. f(x) =x 2, g (x) =λe λx. df(x)g(x) = f(x)g (x)+f (x)g(x) (74) dx. x 2 e λx = x 2 λe λx dx 2xe λx dx (75),. 2xe λx dx = x 2 λe λx dx (76),, 2/λ 2. V (x) =. x 2 λe λx dx E 2 (x) = 2 λ 2 1 λ 2 = 1 λ 2 (77) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 5

2, N. 2. (demoiwe A., 1667 1754 1773. 1816.,,. Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 51

( ) f(x) = 1 2πσ e (x μ)2 2σ 2, <x< (78) :μ :σ 2 N(μ, σ 2 ), 1 N(, 1). Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 52

( ) N(μ, σ 2 ) f(x) p(a <x b) = b a f(x)dx (79), f(x). N(, 1). N(μ, σ 2 ) N(, 1). ξ = x μ (8) σ, ξ N(, 1). Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 53

( ) 3.3 x, 1 y = a + bx. :b =, y = a y. x = y a b, f(x)dx =(1/( 2πσ)) exp( (x μ) 2 /(2σ 2 ))dx (81) (82) f(x)dx = 1 2πσ e (. g(y) = y a b μ)2 1 2σ 2 1 e (y (a+bμ))2 2(bσ) 2 2πbσ b dy = 1 y e ( 2πσ b a+bμ ) b 2 1 2σ 2 b dy = 1 e (y (a+bμ))2 2(bσ) 2 dy (83) 2πbσ y, y a + bμ, bσ. (84) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 54

1. 2 2,, 2.,., (x, y) 2. 2,. (73, 75) (79, 7) (71, 95) (64, 85) (87, 95) (82, 95) (71, 5) (58, 75) (19, 15) (87, 75) (74, 5) (79, 8) (74, 95) (63, 5) (74, 5) (66, 5) (75, 5) (58, 7) (85, 7) (84, 95) (93, 7) (2, ) (51, 15) (75, 75) (68, 85) (63, 75) (71, 2) (72, 7) (86, 7) (67, 95) (75, 5) (78, 95) (65, 5) (71, 9) (42, 75) (84, 7) (85, 75) (62, 75) (57, 95) (83, 6) (82, 7) (72, 15) (81, 95) (75, 5) (85, 75) (7, 64) (61, 3) (92, 95) (47, 4) (,) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 55

2 1 2 3 4 5 6 7 8 9.2......... 1..2........ 2.2......... 3.......... 4.....2...2.. 5..2.2.....4..2 6....2..6..4.4.2 7...2...12.2.8.2.8 8.......2.14.8. 9........2..2 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 56

1 2 3 4 5 6 7 8 9.2..........2 1..2.........2 2.2..........2 3........... 4.....2...2...4 5..2.2.....4..2.1 6....2..6..4.4.2.18 7...2...12.2.8.2.8.34 8.......2.14.8..24 9........2..2.4.4.4.4.2.2.18.4.34.14.14 1 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 57

2 P (x, y) p(x, y) =p x (y)p y (x), x,y =, 1, 2, 3, 4, 5, 6, 7, 8, 9 (85) x, y (independent) (dependent). p(x, y) p x (y) x p y (x) y Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 58

2 ( ) 3 N x, y p(x, y) = N! x!y!(n x y)! px 1p x 2(1 p 1 p 2 ) N x y (86) p 1, p 2. x, y 3 k multinomial distribution Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 59

2 ( ) 3 p y (x) = = N p(x, y) = y= N! x!(n x!) px 1 N y= N x y= N! x!y!(n x y)! px 1p x 2(1 p 1 p 2 ) N x y (N x)! y!(n x y)! py 1 (1 p 1 p 2 ) N x y = N C x p x 1(1 p 1 ) N x (87) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 6

2 ( ) x, y 2 (x, y) xy p(x (a, b] y (c, d]) (88) b d a c f(x, y)dxdy (89), f(x, y) join tprobability density function 1. 2. f(x, y), <x<, <y< f(x, y)dxdy =1 (9) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 61

2 2,2. x, y μ x, μ y, σ x, σ y f(x, y) = 1 2πσ x σ y 1 ρ 2 x,y 1 exp ( 2(1 ρ 2 x,y) ((x μ x) 2 ρ x,y x, y σ 2 x + (y μ y) 2 σ 2 y 2ρ x,y(x μ x )(y μ y ) )) (91) σ x σ y exp(-x**2/2-y**2/8).8.6.4.2 x, y N(μ x,σx), 2 N(μ y,σy) 2..8.9 1.5.6.7.1.2.3 gnuplot.4-1 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 62-5 5-1 -5 5 1

11.. 2, Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 63

( )(complete enumeration),.. (sample suevey),, Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 64

(population) (finite population) (sample) sampling 1. (sampling with replacement):,. 2. sampling without replacement,. 3. purposive selection 4. random sampling 2 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 65

,,.,. x, (x 1,x 2,x 3,...,x n ). x 1,x 2,x 3,...,x n...,,. Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 66

x i (= 1, 2,...,N) T (x 1,x 2,...,x N ) ( ) (statistics x (distribution of population) T (sampling distribution distribution of sample) : x = 1 N : s xx = 1 N N i=1 x i N (x i x) 2 i=1. Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 67 (92) (93)

12. x = 1 N x i n i=1 (94) x E(x) =μ N ( ), N. 4.1: μ σ 2 N (x 1,x 2,...x N ). x E(x) =μ (95) V (x) = σ2 N. (96) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 68

1: x 1, x 2,...x N,. E(x 1 )=E(x 2 )=...= E(x N )=μ (97)., E(x) = 1 N N E(x i )=μ i=1 (98) 2: x 1, x 2,...x N,. V (x 1 + x 2 )=V (x 1 )+V (x 2 ) (99). V (αx 1 )=α 2 V (x 1 ) (1). V ( 1 N N i=1 x i )= 1 N 2 N V (x i )= 1 N σ2 (11) i=1 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 69

, x μ 1. (law of largenumbers). 4.2: x μ, σ 2. ɛ δ, ɛ>, <δ<1. N >σ 2 /(ɛ 2 δ) p( x μ <ɛ) 1 δ (12) lim p( x μ <ɛ)=1 (13) N. Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 7

,,.. Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 71

x μ, σ 2 k(> ) p( x μ kσ) 1 (14) k 2 :. N σ 2 = (x i μ) 2 p(x i )= (x i μ) 2 p(x i )+ (x i μ) 2 p(x i ) i=1 x k μ kσ x k μ kσ x k μ <kσ (x i μ) 2 p(x i )= x k μ kσ x i μ 2 p(x i ) x k μ kσ k 2 σ 2 p(x i )=k 2 σ 2 p( x μ kσ) (15) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 72

x. Ex = μ, V (x) = 1 N σ2 (16) ɛ = k N σ p( x μ ɛ) σ2 Nɛ 2 (17), N. lim p( x μ <ɛ)=1 (18) N Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 73

( ) (central limit theorem) N, x μ σ 2 /N N(μ, σ 2 /N ) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 74

( ) (central limit theorem) z N = x μ σ/ N (19) N b 1 p(z N (a, b]) e x2 /2 dx (11) 2π a (convergence in law), (convergence in distribution) z N (asymptotically) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 75

( ) x 1, x 2,...x N,. y i = x i μ (111) σ, 1. y i. M yi (t) =1+μ 1 t + μ 2 2! t2 + μ 3 3! t3 + =1+ 1 2 t2 + μ 3 3! t3 + (112) z N = x μ σ/ N = 1 σ/ N 1 N N (x i μ) = 1 N i=1 N i=1 x i μ σ = 1 N N i=1 y i (113) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 76

( ) x i i.i.d M xi +x j (t) =M xi (t)m xy (t) (114) M xi (t) =E(e tx i ) (115) M axi (t) =E(e tax i )=M xi (at) (116). z N M zn (t) = Π N i=1 (M y i / N (t)) = ΠN i=1 M y i (t/ N)) = Π N i=1 (1 + 1 t 2 2 N + μ 3 t 3 3! N N + ) = (1+ 1 t 2 2 N + μ 3 t 3 3! N N + )N (117) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 77

( ) M x(t) = E(e tx )= = 1 2π e tx 1 e x2 /2 dx = 1 2π 2π e (x t)2 /2+t 2 /2 dx = e t2 /2 1 2π e x2 /2+tx dx e (x t)2 /2 dx = e t2 /2 (118) (117), (118) t 2 t 3 log(m zn (t)) t 2 /2=Nlog(1 + 1 2 N + μ 3 3! N N + ) t2 /2 (119) t 2 t 3 u = 1 2 N + μ 3 3! N N + (12), N u<1. (119) log(1 + u) =u u 2 /2+u 3 /3... (121) t 2 t 3 N( 1 2 N + μ 3 3! N N + ) t2 /2 t 2 t 3 N( 1 2 N + μ 3 3! N N + )2 /2+N( 1 2 N + μ 3 3! t 2 t 3 N N + )3 /3+... (122) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 78

( ) (122) t 3 N( μ 3 3! N N + ) t 2 t 3 t 2 N( 1 2 N + μ 3 3! N N + )2 /2+N( 1 2 N + μ 3 3! N N + )3 /3+... (123). N t 3 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 79

13. 2 N(, 1) (z 1,z 2,...z N ) x = N i=1 z 2 i f(x) = (124) 1 2 n/2 Γ(N/2) xn/2 1 e x/2 (125), x (d.f.)(degrees of freedom) N 2 (χ 2 -distribution) χ 2 N. Γ(m) Γ(m) =. e x x m 1 dx 1. Γ(1) = 1, Γ( 1 2 )= π 2. Γ(m +1)=mΓ(m) (126) 3. m Γ(m) =m! Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 8

2 : xf(x)dx = = 1 x N/2 1 e x/2 dx 2 N/2 Γ(N/2) 2(N/2) x N/2 e x/2 dx = N (127) 2 N/2+1 Γ(N/2+1) : x 2 f(x)dx N 2 1 = x 2 x N/2 1 e x/2 dx N 2 2 N/2 Γ(N/2) = 4(N/2)(N/2+1) x N/2+1 e x/2 dx N 2 2 N/2+2 Γ(N/2+2) = 2N (128) 2 N α χ 2 N (α) p(x χ 2 N (α)) = α (129) N 2 x χ 2 N (α) α χ 2 N (α) 1α% Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 81

x N(, 1), y ν 2. 2 x, y t = x y ν (d.f.)ν (t-distribution), t ν. t f(t) = Γ(ν+1 2 ) νπγ( ν 2 (13) )(1 + t2 ν ) ν+1 2, t (, + ) (131) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 82

, N(μ, σ 2 ) x μ σ/ N σ μ σ 2 s xx x μ sxx / N (132) (133) y = Ns xx σ 2 = 1 σ 2 N (x i x) 2 (134) i=1 χ 2 (N 1) ˆx = x μ σ/ N N(, 1), σ (135) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 83

(d.f.)n (t-distribution), t N. t f(t) = Γ( N+1 2 ) t2 + ) N+1 2, t (, + ) (136) NΓ( N 2 )Γ(1 2 )(1 N : : 1+ x2 N = 1 t, Γ( N+1 2 ) NΓ( N 2 )Γ(1 2 ) = A (137). x = N( 1 t 1), 2x N dx = 1 t2dt, dx = N 2 (1 t 1) 1/2 t 2 dt (138) x (, ) t (, 1] (139) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 84

( ) E[x 2 ] = A x 2 (1 + x2 = AN N, α = N 2 Γ(α)Γ(β) Γ(α + β) = AN N 1 1 ) N+1 N 1 2 dx = AN N 1 t N 2 2 (1 t) 1/2 dt = AN N 1, β =3/2 t N+1 2 2 ( 1 t 1)1/2 dt 1 t (N 2 1) 1 (1 t) 3/2 1 dt (14) t α 1 (1 t) β 1 dt (141) t (N 2 1) 1 (1 t) 3/2 1 dt = AN N Γ(N 2 1)Γ(3 2 ) Γ( N 2 + 1 2 ) = Γ( N+1 2 ) Γ( N NΓ( N N 2 1)Γ(3 2 ) 2 )Γ(1 2 )N Γ( N 2 + 1 2 ) = N N 1 Γ( N+1 2 ) Γ( N NΓ( N N 2 1)Γ(3 2 ) 2 )Γ(1 2 )N Γ( N 2 + 1 2 ) (142) (143) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 85

x 1, x 2, χ 2 (ν 1 ), χ 2 (ν 2 ), F = x 1 ν 1 x 2 ν 2, (ν 1,ν 2 ) F F (ν 1,ν 2 ). (144).,. f(x) = Γ(ν 1+ν 2 2 ) Γ( ν 1 2 )Γ( ν 2 2 ) (ν 1 ν 2 ) ν1 2 x ν1 2 2 (1 + ν 1 ν 2 x) ν 1 +ν 2 2 (145) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 86

( ) : E(x) = xf(x)dx = A = Γ(ν 1+ν 2 ) 2 Γ( ν 1 2 )Γ( ν 2 2 ) (ν 1 ) ν1 2 ν 2 x Γ(ν 1+ν 2 2 ) Γ( ν 1 2 )Γ( ν 2 E(x) = Axx ν 1 2 2 (1 + ν 1 x) ν 2 1 ν 2 = A[ ν 1+ν 2 x ν1 ν 1 2 (1 + +1ν 2 1 2 ν 2 ν 1 = A (ν 1 + ν 2 )+2ν 1 2 2 ν 2 ν 1 = A (ν 1 + ν 2 )+2ν 1 2 2 ) (ν 1 ν 1 +ν 2 2 dx = A x) 1 ν 1 +ν 2 ν 2 ν 2 ) ν1 2 x ν1 2 2 (1 + ν 1 ν 2 x) x ν 1 2 (1 + ν 1 x) ν 1 +ν 2 2 dx ν 2 2 ] A 1 ν 1+ν 2 ν 1 +1ν 2 1 2 ν 1 +ν 2 2 dx (146) ν 2 x ν 1 2 1(1 + ν 1 x) 1 ν 1 +ν 2 2 dx ν 2 x ν 1 2 1(1 + ν 1 x) ν 1 +ν 2 2 (1 + ν 1 x)dx ν 2 ν 2 (147) x ν 1 ν 1 2 1 (1 + x) 1 ν 1 +ν 2 2 dx ν 2 (148) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 87

( ) ν 2 E(x) = A (ν 1 + ν 2 )+2 ( (1 + E(x) = ν 2 = A (ν 1 + ν 2 )+2 (ν 1 ν 2 x ν 1 2 1 (1 + ν 1 ν 2 x) ν 1 +ν 2 2 ν 1 xdx ν 2 x ν 1 ν 1 2 (1 + x) ν 1 +ν 2 2 dx ν 2 A x ν 1 2 (1 + ν 1 x) ν 1 +ν 2 2 dx = A ν 2 (ν 1 + ν 2 )+2 ν 2 A (ν 1 + ν 2 )+2 ν 2 ν 1 ν 2 A(1 + ν 1 )x ν1 ν 1 2 (1 + x) ν 1 +ν 2 2 dx = A ν 2 ν 2 (ν 1 + ν 2 )+2 ν 1 (ν 1 + ν 2 )+2 ) Ax ν 1 2 (1 + ν 1 x) ν 1 +ν 2 ν 2 x ν 1 ν 1 2 1 (1 + x) ν 1 +ν 2 2 )dx ν 2 x ν 1 ν 1 2 1 (1 + x) ν 1 +ν 2 2 )dx(149) ν 2 x ν 1 2 (1 + ν 1 x) ν 1 +ν 2 2 dx ν 2 x ν 1 1 2 (1 + ν 1 x) ν 1 +ν 2 2 dx (15) ν 2 ν 2 Ax ν 1 2 (1 + ν 1 x) ν 1 +ν 2 2 dx = ν 2 (ν 1 + ν 2 )+2 2 dx = ν 2 (ν 1 +ν 2 )+2 ν 2 (ν 1 +ν 2 )+2 = ν 2 ν 2 2 ν 2 x ν 1 1 2 (1 + ν 1 x) ν 1 +ν 2 2 dx (151) ν 2 (152) (153) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 88

( ) E(x 2 )= x 2 f(x)dx = A = Γ(ν 1+ν 2 2 ) Γ( ν 1 2 )Γ( ν 2 2 ) (ν 1 ) ν1 2 ν 2 E(x 2 ) = x 2 f(x)dx = ν 2 x 2 Γ(ν 1+ν 2 ) 2 Γ( ν 1 2 )Γ( ν 2 2 ) (ν 1 ν 2 ) ν1 2 x ν1 2 2 (1 + ν 1 x 2 Ax ν 1 2 2 (1 + ν 1 x) ν 1 +ν 2 2 dx ν 2 2A = x ν1 +1 2 (1 + ν 1 x) ν 1 +ν 2 2 (ν 1 + ν 2 ) ν 1 ν 2 2A ν 2 ( ν 1 2 (ν 1 + ν 2 ) ν 1 2 +1) x ν 1 ν 1 2 (1 + 2A ν 2 = ( ν 1 2 (ν 1 + ν 2 ) ν 1 2 +1) 2A ν 2 = ( ν 1 2 (ν 1 + ν 2 ) ν 1 2 +1) 2A 2 (ν 1 + ν 2 ) (ν 1 2 +1) 2 +1 x) ν 1 +ν 2 ν 2 ν 2 x) 2 +1 dx x ν 1 ν 1 2 (1 + x) ν 1 +ν 2 2 +1 dx ν 2 x ν 1 2 (1 + ν 1 x) ν 1 +ν 2 2 dx ν 2 x ν 1 +1 2 (1 + ν 1 x) ν 1 +ν 2 2 dx ν 2 ν 1 +ν 2 2 dx (154) (155) (156) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 89

( ) ν 2 E(x 2 2A ) = ( ν 1 2 (ν 1 + ν 2 ) ν 1 2 +1) x ν 1 ν 1 2 (1 + x) ν 1 +ν 2 2 dx ν 2 2A 2 (ν 1 + ν 2 ) (ν 1 2 +1) x ν 1 +1 2 (1 + ν 1 x) ν 1 +ν 2 2 dx ν 2 2A ν 2 = ( ν 1 2 (ν 1 + ν 2 ) ν 1 2 +1) x ν 1 ν 1 2 (1 + x) ν 1 +ν 2 2 2 dx ν 2 2 (ν 1 + ν 2 ) (ν 1 2 +1)E(x2 ) 2A ν 2 = ( ν 1 2 (ν 1 + ν 2 ) ν 1 2 +1) ν 2 ν 2 2 2 2 (ν 1 + ν 2 ) (ν 1 2 +1)E(x2 ) (157) 2 (1 + 2 (ν 1 + ν 2 ) (ν 1 2 + 2A ν 2 1))E(x2 )= ( ν 1 2 (ν 1 + ν 2 ) ν 1 2 +1) ν 2 ν 2 2 V (x 2 )=E(x 2 ) (E(x)) 2 = 2ν2 2(ν 1 + ν 2 2) ν 1 (ν 2 2) 2 (ν 2 4) (158) (159) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 9

14.,,,,,,,. Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 91

μ, σ, σ 2 ρ ( x ) x, s xx, s xx r (x 1,x 2,...,x N ), N iid independent and identically distributed random samples Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 92

k, k k. ˆμ = x, ˆσ 2 = s xx (16) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 93

15. Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 94

: unbiasedness : unbiased estimator,,,, : x 1, x 2,...x N,. E(x 1 )=E(x 2 )=...= E(x N )=μ (161)., E(x) = 1 N E(x i )=μ (162) N i=1 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 95

Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 96