tokei01.dvi

Size: px
Start display at page:

Download "tokei01.dvi"

Transcription

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

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

3 1933 (Kol-. mogorov, A. N. ( )) 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

4 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

5 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, 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

6 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

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

8 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

9 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

10 1.1, ( ). ( ) ( ) ,. 2.,. 3.,. 4.. Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 13

11 4.,.,,,.,, 1,.,. Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 14

12 ,,, raw 2x [,9] 5 1 [1,19] 15 [2,29] 25 [3,39] 35 3 [4,49] 45 3 [5,59] 55 8 [6,69] [7,79] [8,89] 85 6 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 15

13 ,,, raw 2x [,9] 5 1 [1,19] 15 [2,29] 25 [3,39] 35 3 [4,49] 45 3 [5,59] 55 8 [6,69] [7,79] [8,89] 85 6 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 16

14 ,,, raw 2x Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 17

15 ,,, raw ( ) 2x Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 18

16 ,,, raw ( % ) 2x Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 19

17 5. :, mean x = 1 N x i =64.38 (19) N i=1 2x Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 2

18 (median) : (median). N Me = x( N +1 ) 2 N (2) Me = 1 2 (x(n 2 )+x(n 2 + 1)) = 67.5 (21) 2x Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 21

19 (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

20 (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 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 23

21 (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 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 24

22 : : 1 k 1 N x k i N i=1 k 1 N (x i x) k N i= (29) (3) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 25

23 , :, N (x σ 3 i x) 3 (31) N i=1 : N (x σ 4 i x) 4 N i= (32) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 26

24 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) (,) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 27

25 (covariance):1 2 s xy = 1 N (x i x)(y i y) = (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

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

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

28 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

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

30 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) Apr. - Jul., 26FY Dept. 2 of3 Mechanical 4 Engineering, Saga Univ., JAPAN 33

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

32 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

33 : 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

34 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

35 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

36 (-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

37 1,2,3,4 1. A 2. A p q =1 p N N A,2 Bernoulli Jakob Ars conjectandi Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 4

38 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

39 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

40 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

41 1., , , Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 44

42 1837 (Poisson, S.D., ) 2, (Bortkiewicz, L von, )., , (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

43 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

44 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

45 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

46 ( ),. 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)

47 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

48 2, N. 2. (demoiwe A., ,,. Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 51

49 ( ) 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

50 ( ) 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

51 ( ) 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

52 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

53 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 56

54 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 57

55 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

56 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

57 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

58 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 f(x, y), <x<, <y< f(x, y)dxdy =1 (9) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 61

59 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) x, y N(μ x,σx), 2 N(μ y,σy) gnuplot.4-1 Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN

60 11.. 2, Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 63

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

62 (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

63 ,,.,. 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

64 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)

65 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

66 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

67 , 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

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

69 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

70 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

71 ( ) (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

72 ( ) (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

73 ( ) 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 + = 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

74 ( ) 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

75 ( ) 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( 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

76 ( ) (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

77 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

78 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

79 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

80 , 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

81 (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

82 ( ) 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 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 ) = Γ( N+1 2 ) Γ( N NΓ( N N 2 1)Γ(3 2 ) 2 )Γ(1 2 )N Γ( N ) = N N 1 Γ( N+1 2 ) Γ( N NΓ( N N 2 1)Γ(3 2 ) 2 )Γ(1 2 )N Γ( N ) (142) (143) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 85

83 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

84 ( ) : 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 + ν 1 x) ν 2 1 ν 2 = A[ ν 1+ν 2 x ν1 ν 1 2 (1 + +1ν ν 2 ν 1 = A (ν 1 + ν 2 )+2ν ν 2 ν 1 = A (ν 1 + ν 2 )+2ν ) (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 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 + x) 1 ν 1 +ν 2 2 dx ν 2 (148) Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 87

85 ( ) 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 + 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 + x) ν 1 +ν 2 2 )dx ν 2 x ν 1 ν (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 + 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 + 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

86 ( ) 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 + ν 1 x) ν 1 +ν 2 2 dx ν 2 2A = x ν (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 ) x ν 1 ν 1 2 (1 + 2A ν 2 = ( ν 1 2 (ν 1 + ν 2 ) ν ) 2A ν 2 = ( ν 1 2 (ν 1 + ν 2 ) ν ) 2A 2 (ν 1 + ν 2 ) (ν ) 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 dx ν 2 x ν 1 2 (1 + ν 1 x) ν 1 +ν 2 2 dx ν 2 x ν (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

87 ( ) ν 2 E(x 2 2A ) = ( ν 1 2 (ν 1 + ν 2 ) ν ) x ν 1 ν 1 2 (1 + x) ν 1 +ν 2 2 dx ν 2 2A 2 (ν 1 + ν 2 ) (ν ) x ν (1 + ν 1 x) ν 1 +ν 2 2 dx ν 2 2A ν 2 = ( ν 1 2 (ν 1 + ν 2 ) ν ) x ν 1 ν 1 2 (1 + x) ν 1 +ν dx ν 2 2 (ν 1 + ν 2 ) (ν )E(x2 ) 2A ν 2 = ( ν 1 2 (ν 1 + ν 2 ) ν ) ν 2 ν (ν 1 + ν 2 ) (ν )E(x2 ) (157) 2 (1 + 2 (ν 1 + ν 2 ) (ν A ν 2 1))E(x2 )= ( ν 1 2 (ν 1 + ν 2 ) ν ) 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

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

89 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

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

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

92 : 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

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

A B P (A B) = P (A)P (B) (3) A B A B P (B A) A B A B P (A B) = P (B A)P (A) (4) P (B A) = P (A B) P (A) (5) P (A B) P (B A) P (A B) A B P

A B P (A B) = P (A)P (B) (3) A B A B P (B A) A B A B P (A B) = P (B A)P (A) (4) P (B A) = P (A B) P (A) (5) P (A B) P (B A) P (A B) A B P 1 1.1 (population) (sample) (event) (trial) Ω () 1 1 Ω 1.2 P 1. A A P (A) 0 1 0 P (A) 1 (1) 2. P 1 P 0 1 6 1 1 6 0 3. A B P (A B) = P (A) + P (B) (2) A B A B A 1 B 2 A B 1 2 1 2 1 1 2 2 3 1.3 A B P (A

More information

( 30 ) 30 4 5 1 4 1.1............................................... 4 1.............................................. 4 1..1.................................. 4 1.......................................

More information

Part () () Γ Part ,

Part () () Γ Part , Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35

More information

6.1 (P (P (P (P (P (P (, P (, P.

6.1 (P (P (P (P (P (P (, P (, P. (011 30 7 0 ( ( 3 ( 010 1 (P.3 1 1.1 (P.4.................. 1 1. (P.4............... 1 (P.15.1 (P.16................. (P.0............3 (P.18 3.4 (P.3............... 4 3 (P.9 4 3.1 (P.30........... 4 3.

More information

untitled

untitled 2 : n =1, 2,, 10000 0.5125 0.51 0.5075 0.505 0.5025 0.5 0.4975 0.495 0 2000 4000 6000 8000 10000 2 weak law of large numbers 1. X 1,X 2,,X n 2. µ = E(X i ),i=1, 2,,n 3. σi 2 = V (X i ) σ 2,i=1, 2,,n ɛ>0

More information

6.1 (P (P (P (P (P (P (, P (, P.101

6.1 (P (P (P (P (P (P (, P (, P.101 (008 0 3 7 ( ( ( 00 1 (P.3 1 1.1 (P.3.................. 1 1. (P.4............... 1 (P.15.1 (P.15................. (P.18............3 (P.17......... 3.4 (P................ 4 3 (P.7 4 3.1 ( P.7...........

More information

統計学のポイント整理

統計学のポイント整理 .. September 17, 2012 1 / 55 n! = n (n 1) (n 2) 1 0! = 1 10! = 10 9 8 1 = 3628800 n k np k np k = n! (n k)! (1) 5 3 5 P 3 = 5! = 5 4 3 = 60 (5 3)! n k n C k nc k = npk k! = n! k!(n k)! (2) 5 3 5C 3 = 5!

More information

201711grade1ouyou.pdf

201711grade1ouyou.pdf 2017 11 26 1 2 52 3 12 13 22 23 32 33 42 3 5 3 4 90 5 6 A 1 2 Web Web 3 4 1 2... 5 6 7 7 44 8 9 1 2 3 1 p p >2 2 A 1 2 0.6 0.4 0.52... (a) 0.6 0.4...... B 1 2 0.8-0.2 0.52..... (b) 0.6 0.52.... 1 A B 2

More information

I L01( Wed) : Time-stamp: Wed 07:38 JST hig e, ( ) L01 I(2017) 1 / 19

I L01( Wed) : Time-stamp: Wed 07:38 JST hig e,   ( ) L01 I(2017) 1 / 19 I L01(2017-09-20 Wed) : Time-stamp: 2017-09-20 Wed 07:38 JST hig e, http://hig3.net ( ) L01 I(2017) 1 / 19 ? 1? 2? ( ) L01 I(2017) 2 / 19 ?,,.,., 1..,. 1,2,.,.,. ( ) L01 I(2017) 3 / 19 ? I. M (3 ) II,

More information

2 1 Introduction

2 1 Introduction 1 24 11 26 1 E-mail: toyoizumi@waseda.jp 2 1 Introduction 5 1.1...................... 7 2 8 2.1................ 8 2.2....................... 8 2.3............................ 9 3 10 3.1.........................

More information

ii 3.,. 4. F. (), ,,. 8.,. 1. (75%) (25%) =7 20, =7 21 (. ). 1.,, (). 3.,. 1. ().,.,.,.,.,. () (12 )., (), 0. 2., 1., 0,.

ii 3.,. 4. F. (), ,,. 8.,. 1. (75%) (25%) =7 20, =7 21 (. ). 1.,, (). 3.,. 1. ().,.,.,.,.,. () (12 )., (), 0. 2., 1., 0,. 24(2012) (1 C106) 4 11 (2 C206) 4 12 http://www.math.is.tohoku.ac.jp/~obata,.,,,.. 1. 2. 3. 4. 5. 6. 7.,,. 1., 2007 (). 2. P. G. Hoel, 1995. 3... 1... 2.,,. ii 3.,. 4. F. (),.. 5... 6.. 7.,,. 8.,. 1. (75%)

More information

数理統計学Iノート

数理統計学Iノート I ver. 0/Apr/208 * (inferential statistics) *2 A, B *3 5.9 *4 *5 [6] [],.., 7 2004. [2].., 973. [3]. R (Wonderful R )., 9 206. [4]. ( )., 7 99. [5]. ( )., 8 992. [6],.., 989. [7]. - 30., 0 996. [4] [5]

More information

確率論と統計学の資料

確率論と統計学の資料 5 June 015 ii........................ 1 1 1.1...................... 1 1........................... 3 1.3... 4 6.1........................... 6................... 7 ii ii.3.................. 8.4..........................

More information

renshumondai-kaito.dvi

renshumondai-kaito.dvi 3 1 13 14 1.1 1 44.5 39.5 49.5 2 0.10 2 0.10 54.5 49.5 59.5 5 0.25 7 0.35 64.5 59.5 69.5 8 0.40 15 0.75 74.5 69.5 79.5 3 0.15 18 0.90 84.5 79.5 89.5 2 0.10 20 1.00 20 1.00 2 1.2 1 16.5 20.5 12.5 2 0.10

More information

..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A

..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A .. Laplace ). A... i),. ω i i ). {ω,..., ω } Ω,. ii) Ω. Ω. A ) r, A P A) P A) r... ).. Ω {,, 3, 4, 5, 6}. i i 6). A {, 4, 6} P A) P A) 3 6. ).. i, j i, j) ) Ω {i, j) i 6, j 6}., 36. A. A {i, j) i j }.

More information

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,. 9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,

More information

keisoku01.dvi

keisoku01.dvi 2.,, Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 4 Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 5 Mon, 2006, 401, SAGA, JAPAN Dept.

More information

( )/2 hara/lectures/lectures-j.html 2, {H} {T } S = {H, T } {(H, H), (H, T )} {(H, T ), (T, T )} {(H, H), (T, T )} {1

( )/2   hara/lectures/lectures-j.html 2, {H} {T } S = {H, T } {(H, H), (H, T )} {(H, T ), (T, T )} {(H, H), (T, T )} {1 ( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 1 2011 ( )/2 2 2011 4 1 2 1.1 1 2 1 2 3 4 5 1.1.1 sample space S S = {H, T } H T T H S = {(H, H), (H, T ), (T, H), (T, T )} (T, H) S

More information

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi) 0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()

More information

10:30 12:00 P.G. vs vs vs 2

10:30 12:00 P.G. vs vs vs 2 1 10:30 12:00 P.G. vs vs vs 2 LOGIT PROBIT TOBIT mean median mode CV 3 4 5 0.5 1000 6 45 7 P(A B) = P(A) + P(B) - P(A B) P(B A)=P(A B)/P(A) P(A B)=P(B A) P(A) P(A B) P(A) P(B A) P(B) P(A B) P(A) P(B) P(B

More information

ii 3.,. 4. F. ( ), ,,. 8.,. 1. (75% ) (25% ) =7 24, =7 25, =7 26 (. ). 1.,, ( ). 3.,...,.,.,.,.,. ( ) (1 2 )., ( ), 0., 1., 0,.

ii 3.,. 4. F. ( ), ,,. 8.,. 1. (75% ) (25% ) =7 24, =7 25, =7 26 (. ). 1.,, ( ). 3.,...,.,.,.,.,. ( ) (1 2 )., ( ), 0., 1., 0,. (1 C205) 4 10 (2 C206) 4 11 (2 B202) 4 12 25(2013) http://www.math.is.tohoku.ac.jp/~obata,.,,,..,,. 1. 2. 3. 4. 5. 6. 7. 8. 1., 2007 ( ).,. 2. P. G., 1995. 3. J. C., 1988. 1... 2.,,. ii 3.,. 4. F. ( ),..

More information

1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =

1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ = 1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A

More information

微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.

微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます.   このサンプルページの内容は, 初版 1 刷発行時のものです. 微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)

More information

I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google

I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google I4 - : April, 4 Version :. Kwhir, Tomoki TA (Kondo, Hirotk) Google http://www.mth.ngoy-u.c.jp/~kwhir/courses/4s-biseki.html pdf 4 4 4 4 8 e 5 5 9 etc. 5 6 6 6 9 n etc. 6 6 6 3 6 3 7 7 etc 7 4 7 7 8 5 59

More information

x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x

x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x [ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),

More information

r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B

r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B 1 1.1 1 r 1 m A r/m i) t ii) m i) t Bt; m) Bt; m) = A 1 + r ) mt m ii) Bt; m) Bt; m) = A 1 + r ) mt m { = A 1 + r ) m } rt r m n = m r m n Bt; m) Aert e lim 1 + 1 n 1.1) n!1 n) e a 1, a 2, a 3,... {a n

More information

waseda2010a-jukaiki1-main.dvi

waseda2010a-jukaiki1-main.dvi November, 2 Contents 6 2 8 3 3 3 32 32 33 5 34 34 6 35 35 7 4 R 2 7 4 4 9 42 42 2 43 44 2 5 : 2 5 5 23 52 52 23 53 53 23 54 24 6 24 6 6 26 62 62 26 63 t 27 7 27 7 7 28 72 72 28 73 36) 29 8 29 8 29 82 3

More information

June 2016 i (statistics) F Excel Numbers, OpenOffice/LibreOffice Calc ii *1 VAR STDEV 1 SPSS SAS R *2 R R R R *1 Excel, Numbers, Microsoft Office, Apple iwork, *2 R GNU GNU R iii URL http://ruby.kyoto-wu.ac.jp/statistics/training/

More information

( 28 ) ( ) ( ) 0 This note is c 2016, 2017 by Setsuo Taniguchi. It may be used for personal or classroom purposes, but not for commercial purp

( 28 ) ( ) ( ) 0 This note is c 2016, 2017 by Setsuo Taniguchi. It may be used for personal or classroom purposes, but not for commercial purp ( 28) ( ) ( 28 9 22 ) 0 This ote is c 2016, 2017 by Setsuo Taiguchi. It may be used for persoal or classroom purposes, but ot for commercial purposes. i (http://www.stat.go.jp/teacher/c2epi1.htm ) = statistics

More information

Microsoft Word - 信号処理3.doc

Microsoft Word - 信号処理3.doc Junji OHTSUBO 2012 FFT FFT SN sin cos x v ψ(x,t) = f (x vt) (1.1) t=0 (1.1) ψ(x,t) = A 0 cos{k(x vt) + φ} = A 0 cos(kx ωt + φ) (1.2) A 0 v=ω/k φ ω k 1.3 (1.2) (1.2) (1.2) (1.1) 1.1 c c = a + ib, a = Re[c],

More information

(pdf) (cdf) Matlab χ ( ) F t

(pdf) (cdf) Matlab χ ( ) F t (, ) (univariate) (bivariate) (multi-variate) Matlab Octave Matlab Matlab/Octave --...............3. (pdf) (cdf)...3.4....4.5....4.6....7.7. Matlab...8.7.....9.7.. χ ( )...0.7.3.....7.4. F....7.5. t-...3.8....4.8.....4.8.....5.8.3....6.8.4....8.8.5....8.8.6....8.9....9.9.....9.9.....0.9.3....0.9.4.....9.5.....0....3

More information

M3 x y f(x, y) (= x) (= y) x + y f(x, y) = x + y + *. f(x, y) π y f(x, y) x f(x + x, y) f(x, y) lim x x () f(x,y) x 3 -

M3 x y f(x, y) (= x) (= y) x + y f(x, y) = x + y + *. f(x, y) π y f(x, y) x f(x + x, y) f(x, y) lim x x () f(x,y) x 3 - M3............................................................................................ 3.3................................................... 3 6........................................... 6..........................................

More information

DVIOUT

DVIOUT A. A. A-- [ ] f(x) x = f 00 (x) f 0 () =0 f 00 () > 0= f(x) x = f 00 () < 0= f(x) x = A--2 [ ] f(x) D f 00 (x) > 0= y = f(x) f 00 (x) < 0= y = f(x) P (, f()) f 00 () =0 A--3 [ ] y = f(x) [, b] x = f (y)

More information

医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 第 2 版 1 刷発行時のものです.

医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます.   このサンプルページの内容は, 第 2 版 1 刷発行時のものです. 医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987

More information

1. (8) (1) (x + y) + (x + y) = 0 () (x + y ) 5xy = 0 (3) (x y + 3y 3 ) (x 3 + xy ) = 0 (4) x tan y x y + x = 0 (5) x = y + x + y (6) = x + y 1 x y 3 (

1. (8) (1) (x + y) + (x + y) = 0 () (x + y ) 5xy = 0 (3) (x y + 3y 3 ) (x 3 + xy ) = 0 (4) x tan y x y + x = 0 (5) x = y + x + y (6) = x + y 1 x y 3 ( 1 1.1 (1) (1 + x) + (1 + y) = 0 () x + y = 0 (3) xy = x (4) x(y + 3) + y(y + 3) = 0 (5) (a + y ) = x ax a (6) x y 1 + y x 1 = 0 (7) cos x + sin x cos y = 0 (8) = tan y tan x (9) = (y 1) tan x (10) (1 +

More information

st.dvi

st.dvi 9 3 5................................... 5............................. 5....................................... 5.................................. 7.........................................................................

More information

v er.1/ c /(21)

v er.1/ c /(21) 12 -- 1 1 2009 1 17 1-1 1-2 1-3 1-4 2 2 2 1-5 1 1-6 1 1-7 1-1 1-2 1-3 1-4 1-5 1-6 1-7 c 2011 1/(21) 12 -- 1 -- 1 1--1 1--1--1 1 2009 1 n n α { n } α α { n } lim n = α, n α n n ε n > N n α < ε N {1, 1,

More information

A

A A 2563 15 4 21 1 3 1.1................................................ 3 1.2............................................. 3 2 3 2.1......................................... 3 2.2............................................

More information

f(x) = f(x ) + α(x)(x x ) α(x) x = x. x = f (y), x = f (y ) y = f f (y) = f f (y ) + α(f (y))(f (y) f (y )) f (y) = f (y ) + α(f (y)) (y y ) ( (2) ) f

f(x) = f(x ) + α(x)(x x ) α(x) x = x. x = f (y), x = f (y ) y = f f (y) = f f (y ) + α(f (y))(f (y) f (y )) f (y) = f (y ) + α(f (y)) (y y ) ( (2) ) f 22 A 3,4 No.3 () (2) (3) (4), (5) (6) (7) (8) () n x = (x,, x n ), = (,, n ), x = ( (x i i ) 2 ) /2 f(x) R n f(x) = f() + i α i (x ) i + o( x ) α,, α n g(x) = o( x )) lim x g(x) x = y = f() + i α i(x )

More information

() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.

() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0. () 6 f(x) [, b] 6. Riemnn [, b] f(x) S f(x) [, b] (Riemnn) = x 0 < x < x < < x n = b. I = [, b] = {x,, x n } mx(x i x i ) =. i [x i, x i ] ξ i n (f) = f(ξ i )(x i x i ) i=. (ξ i ) (f) 0( ), ξ i, S, ε >

More information

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11

More information

(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y

(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y [ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)

More information

0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9

0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9 1-1. 1, 2, 3, 4, 5, 6, 7,, 100,, 1000, n, m m m n n 0 n, m m n 1-2. 0 m n m n 0 2 = 1.41421356 π = 3.141516 1-3. 1 0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c),

More information

y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a =

y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a = [ ] 9 IC. dx = 3x 4y dt dy dt = x y u xt = expλt u yt λ u u t = u u u + u = xt yt 6 3. u = x, y, z = x + y + z u u 9 s9 grad u ux, y, z = c c : grad u = u x i + u y j + u k i, j, k z x, y, z grad u v =

More information

Microsoft Word - 表紙.docx

Microsoft Word - 表紙.docx 黒住英司 [ 著 ] サピエンティア 計量経済学 訂正および練習問題解答 (206/2/2 版 ) 訂正 練習問題解答 3 .69, 3.8 4 (X i X)U i i i (X i μ x )U i ( X μx ) U i. i E [ ] (X i μ x )U i i E[(X i μ x )]E[U i ]0. i V [ ] (X i μ x )U i i 2 i j E [(X i

More information

2011 8 26 3 I 5 1 7 1.1 Markov................................ 7 2 Gau 13 2.1.................................. 13 2.2............................... 18 2.3............................ 23 3 Gau (Le vy

More information

1 1 [1] ( 2,625 [2] ( 2, ( ) /

1 1 [1] ( 2,625 [2] ( 2, ( ) / [] (,65 [] (,3 ( ) 67 84 76 7 8 6 7 65 68 7 75 73 68 7 73 7 7 59 67 68 65 75 56 6 58 /=45 /=45 6 65 63 3 4 3/=36 4/=8 66 7 68 7 7/=38 /=5 7 75 73 8 9 8/=364 9/=864 76 8 78 /=45 /=99 8 85 83 /=9 /= ( )

More information

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y No1 1 (1) 2 f(x) =1+x + x 2 + + x n, g(x) = 1 (n +1)xn + nx n+1 (1 x) 2 x 6= 1 f 0 (x) =g(x) y = f(x)g(x) y 0 = f 0 (x)g(x)+f(x)g 0 (x) 3 (1) y = x2 x +1 x (2) y = 1 g(x) y0 = g0 (x) {g(x)} 2 (2) y = µ

More information

II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2

II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2 II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh

More information

X G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2

More information

ohpmain.dvi

ohpmain.dvi fujisawa@ism.ac.jp 1 Contents 1. 2. 3. 4. γ- 2 1. 3 10 5.6, 5.7, 5.4, 5.5, 5.8, 5.5, 5.3, 5.6, 5.4, 5.2. 5.5 5.6 +5.7 +5.4 +5.5 +5.8 +5.5 +5.3 +5.6 +5.4 +5.2 =5.5. 10 outlier 5 5.6, 5.7, 5.4, 5.5, 5.8,

More information

W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)

W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2) 3 215 4 27 1 1 u u(x, t) u tt a 2 u xx, a > (1) D : {(x, t) : x, t } u (, t), u (, t), t (2) u(x, ) f(x), u(x, ) t 2, x (3) u(x, t) X(x)T (t) u (1) 1 T (t) a 2 T (t) X (x) X(x) α (2) T (t) αa 2 T (t) (4)

More information

i

i i 1 1 1.1..................................... 1 1.2........................................ 3 1.3.................................. 4 1.4..................................... 4 1.5......................................

More information

I, II 1, 2 ɛ-δ 100 A = A 4 : 6 = max{ A, } A A 10

I, II 1, 2 ɛ-δ 100 A = A 4 : 6 = max{ A, } A A 10 1 2007.4.13. A 3-312 tel: 092-726-4774, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ 1. 2. A 0. 1. 1. 2. 3. 2. ɛ-δ 1. ɛ-n

More information

N cos s s cos ψ e e e e 3 3 e e 3 e 3 e

N cos s s cos ψ e e e e 3 3 e e 3 e 3 e 3 3 5 5 5 3 3 7 5 33 5 33 9 5 8 > e > f U f U u u > u ue u e u ue u ue u e u e u u e u u e u N cos s s cos ψ e e e e 3 3 e e 3 e 3 e 3 > A A > A E A f A A f A [ ] f A A e > > A e[ ] > f A E A < < f ; >

More information

³ÎΨÏÀ

³ÎΨÏÀ 2017 12 12 Makoto Nakashima 2017 12 12 1 / 22 2.1. C, D π- C, D. A 1, A 2 C A 1 A 2 C A 3, A 4 D A 1 A 2 D Makoto Nakashima 2017 12 12 2 / 22 . (,, L p - ). Makoto Nakashima 2017 12 12 3 / 22 . (,, L p

More information

I, II 1, A = A 4 : 6 = max{ A, } A A 10 10%

I, II 1, A = A 4 : 6 = max{ A, } A A 10 10% 1 2006.4.17. A 3-312 tel: 092-726-4774, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ 1. 2. A 1. 1. 2. 3. 4. 5. 2. ɛ-δ 1. ɛ-n

More information

8 i, III,,,, III,, :!,,,, :!,,,,, 4:!,,,,,,!,,,, OK! 5:!,,,,,,,,,, OK 6:!, 0, 3:!,,,,! 7:!,,,,,, ii,,,,,, ( ),, :, ( ), ( ), :... : 3 ( )...,, () : ( )..., :,,, ( ), (,,, ),, (ϵ δ ), ( ), (ˆ ˆ;),,,,,,!,,,,.,,

More information

( 12 ( ( ( ( Levi-Civita grad div rot ( ( = 4 : 6 3 1 1.1 f(x n f (n (x, d n f(x (1.1 dxn f (2 (x f (x 1.1 f(x = e x f (n (x = e x d dx (fg = f g + fg (1.2 d dx d 2 dx (fg = f g + 2f g + fg 2... d n n

More information

I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )

I A A441 : April 15, 2013 Version : 1.1 I   Kawahira, Tomoki TA (Shigehiro, Yoshida ) I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17

More information

1 (1) () (3) I 0 3 I I d θ = L () dt θ L L θ I d θ = L = κθ (3) dt κ T I T = π κ (4) T I κ κ κ L l a θ L r δr δl L θ ϕ ϕ = rθ (5) l

1 (1) () (3) I 0 3 I I d θ = L () dt θ L L θ I d θ = L = κθ (3) dt κ T I T = π κ (4) T I κ κ κ L l a θ L r δr δl L θ ϕ ϕ = rθ (5) l 1 1 ϕ ϕ ϕ S F F = ϕ (1) S 1: F 1 1 (1) () (3) I 0 3 I I d θ = L () dt θ L L θ I d θ = L = κθ (3) dt κ T I T = π κ (4) T I κ κ κ L l a θ L r δr δl L θ ϕ ϕ = rθ (5) l : l r δr θ πrδr δf (1) (5) δf = ϕ πrδr

More information

Chap11.dvi

Chap11.dvi . () x 3 + dx () (x )(x ) dx + sin x sin x( + cos x) dx () x 3 3 x + + 3 x + 3 x x + x 3 + dx 3 x + dx 6 x x x + dx + 3 log x + 6 log x x + + 3 rctn ( ) dx x + 3 4 ( x 3 ) + C x () t x t tn x dx x. t x

More information

x i [, b], (i 0, 1, 2,, n),, [, b], [, b] [x 0, x 1 ] [x 1, x 2 ] [x n 1, x n ] ( 2 ). x 0 x 1 x 2 x 3 x n 1 x n b 2: [, b].,, (1) x 0, x 1, x 2,, x n

x i [, b], (i 0, 1, 2,, n),, [, b], [, b] [x 0, x 1 ] [x 1, x 2 ] [x n 1, x n ] ( 2 ). x 0 x 1 x 2 x 3 x n 1 x n b 2: [, b].,, (1) x 0, x 1, x 2,, x n 1, R f : R R,.,, b R < b, f(x) [, b] f(x)dx,, [, b] f(x) x ( ) ( 1 ). y y f(x) f(x)dx b x 1: f(x)dx, [, b] f(x) x ( ).,,,,,., f(x)dx,,,, f(x)dx. 1.1 Riemnn,, [, b] f(x) x., x 0 < x 1 < x 2 < < x n 1

More information

newmain.dvi

newmain.dvi 数論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published

More information

) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4

) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4 1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev

More information

.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z (g f(x = g(

.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z (g f(x = g( 06 5.. ( y = x x y 5 y 5 = (x y = x + ( y = x + y = x y.. ( Y = C + I = 50 + 0.5Y + 50 r r = 00 0.5Y ( L = M Y r = 00 r = 0.5Y 50 (3 00 0.5Y = 0.5Y 50 Y = 50, r = 5 .3. (x, x = (, u = = 4 (, x x = 4 x,

More information

(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t

(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t 6 6.1 6.1 (1 Z ( X = e Z, Y = Im Z ( Z = X + iy, i = 1 (2 Z E[ e Z ] < E[ Im Z ] < Z E[Z] = E[e Z] + ie[im Z] 6.2 Z E[Z] E[ Z ] : E[ Z ] < e Z Z, Im Z Z E[Z] α = E[Z], Z = Z Z 1 {Z } E[Z] = α = α [ α ]

More information

: , 2.0, 3.0, 2.0, (%) ( 2.

: , 2.0, 3.0, 2.0, (%) ( 2. 2017 1 2 1.1...................................... 2 1.2......................................... 4 1.3........................................... 10 1.4................................. 14 1.5..........................................

More information

基礎数学I

基礎数学I I & II ii ii........... 22................. 25 12............... 28.................. 28.................... 31............. 32.................. 34 3 1 9.................... 1....................... 1............

More information

2011de.dvi

2011de.dvi 211 ( 4 2 1. 3 1.1............................... 3 1.2 1- -......................... 13 1.3 2-1 -................... 19 1.4 3- -......................... 29 2. 37 2.1................................ 37

More information

2 1,2, , 2 ( ) (1) (2) (3) (4) Cameron and Trivedi(1998) , (1987) (1982) Agresti(2003)

2 1,2, , 2 ( ) (1) (2) (3) (4) Cameron and Trivedi(1998) , (1987) (1982) Agresti(2003) 3 1 1 1 2 1 2 1,2,3 1 0 50 3000, 2 ( ) 1 3 1 0 4 3 (1) (2) (3) (4) 1 1 1 2 3 Cameron and Trivedi(1998) 4 1974, (1987) (1982) Agresti(2003) 3 (1)-(4) AAA, AA+,A (1) (2) (3) (4) (5) (1)-(5) 1 2 5 3 5 (DI)

More information

i 6 3 ii 3 7 8 9 3 6 iii 5 8 5 3 7 8 v...................................................... 5.3....................... 7 3........................ 3.................3.......................... 8 3 35

More information

18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C

18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C 8 ( ) 8 5 4 I II III A B C( ),,, 5 I II A B ( ),, I II A B (8 ) 6 8 I II III A B C(8 ) n ( + x) n () n C + n C + + n C n = 7 n () 7 9 C : y = x x A(, 6) () A C () C P AP Q () () () 4 A(,, ) B(,, ) C(,,

More information

f(x) = x (1) f (1) (2) f (2) f(x) x = a y y = f(x) f (a) y = f(x) A(a, f(a)) f(a + h) f(x) = A f(a) A x (3, 3) O a a + h x 1 f(x) x = a

f(x) = x (1) f (1) (2) f (2) f(x) x = a y y = f(x) f (a) y = f(x) A(a, f(a)) f(a + h) f(x) = A f(a) A x (3, 3) O a a + h x 1 f(x) x = a 3 3.1 3.1.1 A f(a + h) f(a) f(x) lim f(x) x = a h 0 h f(x) x = a f 0 (a) f 0 (a) = lim h!0 f(a + h) f(a) h = lim x!a f(x) f(a) x a a + h = x h = x a h 0 x a 3.1 f(x) = x x = 3 f 0 (3) f (3) = lim h 0 (

More information

Untitled

Untitled II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j

More information

1

1 1 1 7 1.1.................................. 11 2 13 2.1............................ 13 2.2............................ 17 2.3.................................. 19 3 21 3.1.............................

More information

2 G(k) e ikx = (ik) n x n n! n=0 (k ) ( ) X n = ( i) n n k n G(k) k=0 F (k) ln G(k) = ln e ikx n κ n F (k) = F (k) (ik) n n= n! κ n κ n = ( i) n n k n

2 G(k) e ikx = (ik) n x n n! n=0 (k ) ( ) X n = ( i) n n k n G(k) k=0 F (k) ln G(k) = ln e ikx n κ n F (k) = F (k) (ik) n n= n! κ n κ n = ( i) n n k n . X {x, x 2, x 3,... x n } X X {, 2, 3, 4, 5, 6} X x i P i. 0 P i 2. n P i = 3. P (i ω) = i ω P i P 3 {x, x 2, x 3,... x n } ω P i = 6 X f(x) f(x) X n n f(x i )P i n x n i P i X n 2 G(k) e ikx = (ik) n

More information

y = f(x) y = f( + h) f(), x = h dy dx f () f (derivtive) (differentition) (velocity) p(t) =(x(t),y(t),z(t)) ( dp dx dt = dt, dy dt, dz ) dt f () > f x

y = f(x) y = f( + h) f(), x = h dy dx f () f (derivtive) (differentition) (velocity) p(t) =(x(t),y(t),z(t)) ( dp dx dt = dt, dy dt, dz ) dt f () > f x I 5 2 6 3 8 4 Riemnn 9 5 Tylor 8 6 26 7 3 8 34 f(x) x = A = h f( + h) f() h A (differentil coefficient) f f () y = f(x) y = f( + h) f(), x = h dy dx f () f (derivtive) (differentition) (velocity) p(t)

More information

1 Introduction 1 (1) (2) (3) () {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a, b] lim f n (x) f(x) (1) f(x)? (2) () f(x)? b lim a f n (x)dx = b

1 Introduction 1 (1) (2) (3) () {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a, b] lim f n (x) f(x) (1) f(x)? (2) () f(x)? b lim a f n (x)dx = b 1 Introduction 2 2.1 2.2 2.3 3 3.1 3.2 σ- 4 4.1 4.2 5 5.1 5.2 5.3 6 7 8. Fubini,,. 1 1 Introduction 1 (1) (2) (3) () {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a, b] lim f n (x) f(x) (1) f(x)?

More information

n ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................

More information

2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) =

2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) = 1 1 1.1 I R 1.1.1 c : I R 2 (i) c C (ii) t I c (t) (0, 0) c (t) c(i) c c(t) 1.1.2 (1) (2) (3) (1) r > 0 c : R R 2 : t (r cos t, r sin t) (2) C f : I R c : I R 2 : t (t, f(t)) (3) y = x c : R R 2 : t (t,

More information

simx simxdx, cosxdx, sixdx 6.3 px m m + pxfxdx = pxf x p xf xdx = pxf x p xf x + p xf xdx 7.4 a m.5 fx simxdx 8 fx fx simxdx = πb m 9 a fxdx = πa a =

simx simxdx, cosxdx, sixdx 6.3 px m m + pxfxdx = pxf x p xf xdx = pxf x p xf x + p xf xdx 7.4 a m.5 fx simxdx 8 fx fx simxdx = πb m 9 a fxdx = πa a = II 6 ishimori@phys.titech.ac.jp 6.. 5.4.. f Rx = f Lx = fx fx + lim = lim x x + x x f c = f x + x < c < x x x + lim x x fx fx x x = lim x x f c = f x x < c < x cosmx cosxdx = {cosm x + cosm + x} dx = [

More information

( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes )

( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes ) ( 3 7 4 ) 2 2 ) 8 2 954 2) 955 3) 5) J = σe 2 6) 955 7) 9) 955 Statistical-Mechanical Theory of Irreversible Processes 957 ) 3 4 2 A B H (t) = Ae iωt B(t) = B(ω)e iωt B(ω) = [ Φ R (ω) Φ R () ] iω Φ R (t)

More information

8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) Carathéodory 10.3 Fubini 1 Introduction 1 (1) (2) {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a

8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) Carathéodory 10.3 Fubini 1 Introduction 1 (1) (2) {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a % 100% 1 Introduction 2 (100%) 2.1 2.2 2.3 3 (100%) 3.1 3.2 σ- 4 (100%) 4.1 4.2 5 (100%) 5.1 5.2 5.3 6 (100%) 7 (40%) 8 Fubini (90%) 2007.11.5 1 8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) 10.1 10.2 Carathéodory

More information

2011 ( ) ( ) ( ),,.,,.,, ,.. (. ), 1. ( ). ( ) ( ). : obata/,.,. ( )

2011 ( ) ( ) ( ),,.,,.,, ,.. (. ), 1. ( ). ( ) ( ). :   obata/,.,. ( ) 2011 () () (),,.,,.,,. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.,.. (. ), 1. ( ). ()(). : www.math.is.tohoku.ac.jp/ obata/,.,. () obata@math.is.tohoku.ac.jp http://www.dais.is.tohoku.ac.jp/ amf/, (! 22 10.6; 23 10.20;

More information

untitled

untitled 3 3. (stochastic differential equations) { dx(t) =f(t, X)dt + G(t, X)dW (t), t [,T], (3.) X( )=X X(t) : [,T] R d (d ) f(t, X) : [,T] R d R d (drift term) G(t, X) : [,T] R d R d m (diffusion term) W (t)

More information

x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s

x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s ... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z

More information

( z = x 3 y + y ( z = cos(x y ( 8 ( s8.7 y = xe x ( 8 ( s83.8 ( ( + xdx ( cos 3 xdx t = sin x ( 8 ( s84 ( 8 ( s85. C : y = x + 4, l : y = x + a,

( z = x 3 y + y ( z = cos(x y ( 8 ( s8.7 y = xe x ( 8 ( s83.8 ( ( + xdx ( cos 3 xdx t = sin x ( 8 ( s84 ( 8 ( s85. C : y = x + 4, l : y = x + a, [ ] 8 IC. y d y dx = ( dy dx ( p = dy p y dx ( ( ( 8 ( s8. 3 A A = ( A ( A (3 A P A P AP.3 π y(x = { ( 8 ( s8 x ( π < x x ( < x π y(x π π O π x ( 8 ( s83.4 f (x, y, z grad(f ( ( ( f f f grad(f = i + j

More information

ii 3.,. 4. F. (), ,,. 8.,. 1. (75% ) (25% ) =9 7, =9 8 (. ). 1.,, (). 3.,. 1. ( ).,.,.,.,.,. ( ) (1 2 )., ( ), 0. 2., 1., 0,.

ii 3.,. 4. F. (), ,,. 8.,. 1. (75% ) (25% ) =9 7, =9 8 (. ). 1.,, (). 3.,. 1. ( ).,.,.,.,.,. ( ) (1 2 )., ( ), 0. 2., 1., 0,. 23(2011) (1 C104) 5 11 (2 C206) 5 12 http://www.math.is.tohoku.ac.jp/~obata,.,,,.. 1. 2. 3. 4. 5. 6. 7.,,. 1., 2007 ( ). 2. P. G. Hoel, 1995. 3... 1... 2.,,. ii 3.,. 4. F. (),.. 5.. 6.. 7.,,. 8.,. 1. (75%

More information

i

i i 3 4 4 7 5 6 3 ( ).. () 3 () (3) (4) /. 3. 4/3 7. /e 8. a > a, a = /, > a >. () a >, a =, > a > () a > b, a = b, a < b. c c n a n + b n + c n 3c n..... () /3 () + (3) / (4) /4 (5) m > n, a b >, m > n,

More information

B [ 0.1 ] x > 0 x 6= 1 f(x) µ 1 1 xn 1 + sin sin x 1 x 1 f(x) := lim. n x n (1) lim inf f(x) (2) lim sup f(x) x 1 0 x 1 0 (

B [ 0.1 ] x > 0 x 6= 1 f(x) µ 1 1 xn 1 + sin sin x 1 x 1 f(x) := lim. n x n (1) lim inf f(x) (2) lim sup f(x) x 1 0 x 1 0 ( . 28 4 14 [.1 ] x > x 6= 1 f(x) µ 1 1 xn 1 + sin + 2 + sin x 1 x 1 f(x) := lim. 1 + x n (1) lim inf f(x) (2) lim sup f(x) x 1 x 1 (3) lim inf x 1+ f(x) (4) lim sup f(x) x 1+ [.2 ] [, 1] Ω æ x (1) (2) nx(1

More information

211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,

More information

6. Euler x

6. Euler x ...............................................................................3......................................... 4.4................................... 5.5......................................

More information

D xy D (x, y) z = f(x, y) f D (2 ) (x, y, z) f R z = 1 x 2 y 2 {(x, y); x 2 +y 2 1} x 2 +y 2 +z 2 = 1 1 z (x, y) R 2 z = x 2 y

D xy D (x, y) z = f(x, y) f D (2 ) (x, y, z) f R z = 1 x 2 y 2 {(x, y); x 2 +y 2 1} x 2 +y 2 +z 2 = 1 1 z (x, y) R 2 z = x 2 y 5 5. 2 D xy D (x, y z = f(x, y f D (2 (x, y, z f R 2 5.. z = x 2 y 2 {(x, y; x 2 +y 2 } x 2 +y 2 +z 2 = z 5.2. (x, y R 2 z = x 2 y + 3 (2,,, (, 3,, 3 (,, 5.3 (. (3 ( (a, b, c A : (x, y, z P : (x, y, x

More information

TOP URL 1

TOP URL   1 TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7

More information

i

i 009 I 1 8 5 i 0 1 0.1..................................... 1 0.................................................. 1 0.3................................. 0.4........................................... 3

More information

入試の軌跡

入試の軌跡 4 y O x 4 Typed by L A TEX ε ) ) ) 6 4 ) 4 75 ) http://kumamoto.s.xrea.com/plan/.. PDF) Ctrl +L) Ctrl +) Ctrl + Ctrl + ) ) Alt + ) Alt + ) ESC. http://kumamoto.s.xrea.com/nyusi/kumadai kiseki ri i.pdf

More information

II R n k +1 v 0,, v k k v 1 v 0,, v k v v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ k σ dimσ = k 1.3. k

II R n k +1 v 0,, v k k v 1 v 0,, v k v v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ k σ dimσ = k 1.3. k II 231017 1 1.1. R n k +1 v 0,, v k k v 1 v 0,, v k v 0 1.2. v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ kσ dimσ = k 1.3. k σ {v 0,...,v k } {v i0,...,v il } l σ τ < τ τ σ 1.4.

More information

.1 z = e x +xy y z y 1 1 x 0 1 z x y α β γ z = αx + βy + γ (.1) ax + by + cz = d (.1') a, b, c, d x-y-z (a, b, c). x-y-z 3 (0,

.1 z = e x +xy y z y 1 1 x 0 1 z x y α β γ z = αx + βy + γ (.1) ax + by + cz = d (.1') a, b, c, d x-y-z (a, b, c). x-y-z 3 (0, .1.1 Y K L Y = K 1 3 L 3 L K K (K + ) 1 1 3 L 3 K 3 L 3 K 0 (K + K) 1 3 L 3 K 1 3 L 3 lim K 0 K = L (K + K) 1 3 K 1 3 3 lim K 0 K = 1 3 K 3 L 3 z = f(x, y) x y z x-y-z.1 z = e x +xy y 3 x-y ( ) z 0 f(x,

More information

18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α

18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α 18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α 2 ), ϕ(t) = B 1 cos(ω 1 t + α 1 ) + B 2 cos(ω 2 t

More information