ii 3.,. 4. F. (), ,,. 8.,. 1. (75%) (25%) =7 20, =7 21 (. ). 1.,, (). 3.,. 1. ().,.,.,.,.,. () (12 )., (), 0. 2., 1., 0,.
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1 24(2012) (1 C106) 4 11 (2 C206) ,,. 1., 2007 (). 2. P. G. Hoel, ,,.
2 ii 3.,. 4. F. (), ,,. 8.,. 1. (75%) (25%) =7 20, =7 21 (. ). 1.,, (). 3.,. 1. ().,.,.,.,.,. () (12 )., (), 0. 2., 1., 0,.
3 (1). (2).,. (3),,. 1.2 (), Ω., E. E Ω. (), E P (E) = E Ω..,.., Ω. 1.1 () ( ). ()., , 2 (A,K,Q,J). 1 10, /1024
4 , ? 1/221, 1/ , 1, , 3. [2/10] , 10. [2/10] 1.4 () ( ). 3, () 2.,. 1.,, 1 ().? 1.5 ( ) A,B 2. A 2/5, B 3/5. 4, A 3, B 2.? [.] 1.3 ( ) ( ) ( ) ( ) () ( ) ( ) () ().. ( ) () () ( ), ( ), ( ) ()
5 : Ω: () = (, F: () P : 2.1 (), Ω., E P (E) = E Ω,. 2.2 (Ω () ),,. 2.3 (Ω ) , (), ().
6 4 2 Ω E. P (E) = E Ω,..,,,, ,. 2 (, ). 2.2 E P (E), 3, P Ω., P (E) E. (i) 0 P (E) 1. (ii) P (Ω) = 1. (iii) [] E 1, E 2, F (, i j E i E j = ), ( ) P E n = P (E n ). n=1, 3 (Ω, F, P ). Ω,., E F E = F. a < b. n=1 2.3 AB 3. (, 3 3.) B A O :, ()?.
7 () ,. (1) ( A,K,Q,J,10) (2) (3) , 00001,..., (1) 9 1. (2) 9 2. (3) 0, 1,..., (4) 0, 1,..., () , 3. 4 A,B 2. A p, B q = 1 p. 5, A 3, B 2.? 5 2, P, 2 P. 1/3.
8
9 ,. () , ,. 1) 2) ()., x, y, z, t,...., 0 x 1, x 0 1.,,.,,.., X, Y, Z, T,....
10 X, X {1, 2, 3, 4, 5, 6}. P (X = 1) = P (X = 2) = = P (X = 6) = 1 6,. 3.7 L, X. X L/2 X L., x,, P (X = x) = 0, 3.6,. X, F (x) = P (X x), x R, X , 0 X. X. 6 2 X. X , (0 ). 2, X. X X 8 X
11 X, P (X x) = F (x) = x f(t)dt F (x) = f(x) f(x) X. (F (x).) 3.2 P (a X b) = b a f(x)dx 3.10 X X F (x). (1) x 1 x 2 F (x 1 ) F (x 2 ). (2) lim F (x) = 0, lim F (x) = 1. x x (3) lim ϵ +0 F (x + ϵ) = F (x). 3.4 f(x). (1) f(x) 0. (2) + f(x)dx = 1. () P (X = a i ) = p i () F (x) = P (X x) P (a X b) = b a f(x)dx
12
13 p n, X ( ) n P (X = k) = p k (1 p) n k, k = 0, 1, 2,.... k, B(n, p). 4.1 B(4, 1/2) B(4, 1/4) p, X P (X = k) = p(1 p) k, k = 0, 1, 2,.... p. (), p, ( 1 ) Y. P (Y = k) = p(1 p) k 1, k = 1, 2, X λ > 0, P (X = k) = λk k! e λ, k = 0, 1, 2,....
14 λ = 0.5, λ = 1, λ = (1) 1. (2) X {a 1, a 2,..., }, p i = P (X = a i ), p i 0, p i = 1 (p i = 0 a i, p i = 0 )., i m = i a i p i, σ 2 = i (a i m) 2 p i = i a 2 i p i m 2. X, X,, E[X], V[X].. (m) (σ 2 ) (2 ) B(1, p) p p(1 p) B(n, p) np np(1 p) ( p) (1 p)/p (1 p)/p 2 ( λ) λ λ 10 2, 2 100, 1 50, , ,,, ,.
15 f(x),. f(x) 0, + f(x) = f(x) = b a, a x b 0, 1) [a, b] 1. 2) L X, [L/2, L] λ > 0 { λe λx, x 0 f(x) = 0, x < () N(m, σ 2 ): m, σ 2 () { } 1 f(x) = exp (x m)2 2πσ 2 2σ 2 N(0, 1):, χ 2 -(), t-, F - ()
16 f(x) m = xf(x) dx, σ 2 = (x m) 2 f(x) dx = x 2 f(x) dx m 2. X, X,, E[X], V[X].. (m) (σ 2 ) [a, b] (a + b)/2 (b a) 2 /12 ( λ) 1/λ 1/λ 2 N(m, σ 2 ) m σ L 2, X. X,,,. 13 L 2, X. X,,,. 14,.
17 () 7 () {0, 1, 2,... } X, G(z) = z k P (X = k) k=0 X ( X )., E(X) = G (1), E(X 2 ) = G (1) + G (1), V(X) = G (1) + G (1) G (1) 2. 8,. 9, () X, () Y. X Y. 11,., ( ). + e x2 dx = π 12 1, X. X,,,. 13 O R 1, O X. X,,,.
18
19 A, B 2. P (A) > 0, A B P (B A) = P (A B) P (A) 5.1 () 10, ,,? E, F, P (E) = 1 3, P (F ) = 1 2, P (E F ) = P (E c ), P (E F c ), P ((E F c ) c ), P (E F ), P (E F c ), P (E F E F ) T, P (T m + n T m) = P (T n), m, n = 0, 1, 2,..., X, P (X a + b X a) = P (X b), a, b 0,.
20 Ω = A 1 A 2, A 1 A 2 =, B, P (A 1 B) = P (A 1 )P (B A 1 ) P (A 1 )P (B A 1 ) + P (A 2 )P (B A 2 ) (). 5.2, A B, 95%, 2% , A ? 14, A B, 95%, 100p %... p , 5, 2, 5.? (2,.) 16 2 E, F, P (E) = 1 3, P (F ) = 1 2, P (E F ) = P (E c ), P (E F c ), P ((E F c ) c ), P (E F ), P (E F c ), P (E F E F ) 17 () (1) 1., 6. (2) 1., 6.
21 19 第 6 章 正規分布 6.1 標準正規分布 N (0, 1): 標準正規分布 例 題 6.1 確率変数 Z の分布が標準正規分布である (このことを Z N (0, 1) と書く) とす る. 標準正規分布表を用いて, (1) 次の確率を求めよ: P (Z 1.15), P (Z 1.23), P ( Z < 2.4) (2) 次の等式が成り立つような a を求めよ. P (Z a) = 0.33, P (Z < a) = 0.75, P ( Z a) = 0.4 問 17 Z N (0, 1) とする. (1) 次の確率を求めよ: P (Z 1.82), P (Z 2.13), P ( Z > 1.5) (2) 次の等式が成り立つような a を求めよ. P (Z a) = 0.39, P (Z < a) = 0.91, P ( Z a) = 0.72 定 理 6.1 X N (m, σ 2 ) のとき, Z= X m N (0, 1) σ 例 題 6.2 X N (2, 52 ) のとき, P (X 3), P (X 0), P ( X 4) を求めよ. 問 18 (1) 確率変数 X が正規分布 N (20, 42 ) に従うとき, P (X > 17.8) を求めよ. (2) 確率変数 Y が正規分布 N ( 2, 52 ) に従うとき, P ( Y 1) を求めよ.
22 X N(50, 10 2 ), P (X > a) = a. ()., x = x 1 y = y 1, x = x 2 y = y 2, x 1 < x < x 2 y : y = y 2 y 1 x 2 x 1 (x x 1 ) + y B(100, 0.4) 6.2,. B(n, p) N(np, np(1 p)), 0 < p < 1, n , 225 (). 20 (1) 1000, 550. (2) 250, , 225.? 6.3 (),, ().,, n, x 1, x 2,..., x n 1 n n i=1 x i
23 ,.,,? 1 ( ). X. X.,, 1 X 1, 2 X 2,..., n X n., X 1, X 2,..., X n n.,., n, n. 6.3 m, σ 2, : X = 1 n ) X k N (m, σ2. n n k=1 6.4 n,. 18 X N(0, 1), Y = ax + b., a, b. 19 X N(0, 1), Y = X , 23.5 ( ) ?.
24 22 6 I(z) = 1 2π z 0 e x2 /2 dx z
25 ,? () 5 27 () () 10 (%) 05/21() 8: /21() 21: /26() 21: /27() 21: /22() 22: /21() 21: /22() 21: /25() 21: /22() 15: /26() 13: ,., 27, PM.,, PM 600, 200. ( ) : () 7.2 () m (), σ 2. X 1, X 2,..., X n : n ( (iid) )
26 24 7. X = 1 n n k=1 X k (1) : E( ( X) = m ) (2) : P X = m = 1 () lim n, X () (! X )., X,. () X = 1 n n ) X k N (m, σ2 n k=1 X m σ/ n N(0, 1), P ( z X ) m σ/ n z = 1 α z α α N α z z m 1 α [ X z σ n, X + z σ n ]. 90%(α = 0.1, z = 1.64) 95%(α = 0.05, z = 1.96) 99%(α = 0.01, z = 2.58)..
27 ,., 1 α m.,! g., 8g , 95% 1g? 21, 200, 2.2 g., 1.5 g., g?. [1.992, 2.408],,., U 2 = 1 n (X k n 1 X) 2 k=1, t- (). 7.3 E 2. E p. X 1, X 2,..., X n n ( p ) X i = { 1, i E, 0, i E n X = 1 n. ˆp. k=1 X k
28 () p ()., p(1 p) (7.2 )., σ 2 = p(1 p), 7.2, p 1 α [ ] p(1 p) p(1 p) ˆp z, ˆp + z n n 2 () p(1 p) ˆp(1 ˆp) ˆp p z ˆp p z n n, p 1 α, [ ] ˆp(1 ˆp) ˆp(1 ˆp) ˆp z, ˆp + z n n %, 0.54(1 0.54) 0.54 ± ± () %. 95%, 0.22(1 0.22) 0.22 ± ± , 95% 0.01,? % %. 23,, ( )
29 , 220.? (1), (2), (3) H 0 H T (), H 0, < α < 1 P (T W ) = α W R () H T t, W. t W. T, α., H 0 H 1. t W. T, α., H 0. (1),,. (2), ( ). (3),, 5%, 1%. (4) 2,. (5), H 0, (2 ). H 0,.
30 28 8 α α α W W W W 21 A Z N(0, 1), α = P ( Z z) = 1 1 2π z z e x2 /2 dx, z 0 z α α α -z z ,., , m. : H 0 : m = 120 H 1 : m > 120 m, σ 2 n X X = 1 n ) X k N (m, σ2 X m n n σ/ N(0, 1) n k= %., ?
31 ,. 1000, 545, N 35%., 37 %., H 0, 4. \ H 0 H 0 H 0 2 H 0 1, 1, 2. α: 1 = β: 2 θ θ β α c c , 215.? () 10,. p. H 0 : p = 1 2, H 1: p T. {T = 0, 1, 9, 10}., H 0 2 β p. H 0 P (2 T 8).
32
33 H 0 H T (), H 0, < α < 1 P (T W ) = α W R () H 1 ( α-, α ). 4. T t, W. t W. T, α., H 0 H 1. t W. T, α., H 0.., (,, ), (, t-, χ 2 -, F -),. 9.2 () m, σ 2 n, X = 1 n n ) X k N (m, σ2 n k=1 X m σ/ n N(0, 1) (. N(m, σ 2 ).) 9.1 (). 120,., , 2.4.
34 , m = 60 (g).,, m 50 70, σ = 3 ( )., 25,, m = 60? cm cm, 4.63 cm. [ 1% ] 31, 100 g 2g., 200, 2.2 g., 1.5 g. [ 5%] 9.3 () B(n, p) N(np, np(1 p)) np:, np(1 p): 9.2 () 400, 175.,. 32 8%., 175, , 38, 62. [ 5%] 9.4 () m, σ 2 n X 1,..., X n, U 2 = 1 n 1 n (X i X) 2, S 2 = 1 n i=1 n (X i X) 2 i=1,.
35 9.4. () U 2 E(U 2 ) = σ 2.,., n, S 2 U N(m, σ 2 ) n X 1,..., X n. X = 1 n n i=1 X i () U 2 = 1 n 1 n (X i X) 2 () i=1, T = X m U/ n t n 1 (n 1) t-,. n t- 1 n B ( n 2, 1 2) ( ) n t2 2 n = Γ( n+1 2 ) n Γ( n 2 )Γ( 1 2 ) ( ) n t2 2 n n n n (1) Γ. (2) B. B(x, y) = 1 0 Γ(x) = 0 t x 1 e t dt, x > 0. t x 1 (1 t) y 1 dt = Γ(x)Γ(y), x > 0, y > 0. Γ(x + y) (3) n = t- N(0, 1). (4), n 30 N(0, 1).
36 34 9 t P ( T t n (α)) = α n\α (kg), kg, g g, 10 2 g.,? 5%. 1% A , A. A. [ 5%]
37 Karl Pearson ( ) ( n ) x n 2 1 e x 2, x > 0, f n (x) = 2 n/2 Γ 2 0, x 0, n 2 (χ 2 -). (χ 2.), χ 2 n., Γ(t). n = n = n = n = n = χ 2 - (1) X 1, X 2,..., X n,, N(0, 1), n χ 2 n = i=1 n χ 2 -. (2) X 1, X 2,..., X n,, N(m, σ 2 )., χ 2 n 1 = 1 σ 2 n i=1 n 1 2. X 2 i (X i X) 2, X = 1 n n i=1 X i ()
38 A 1, A 2,..., A k k. n, X 1, X 2,..., X k. A 1 A 2 A k p 1 p 2 p k 1 X 1 X 2 X k n, p 1, p 2,..., p k m i = np i, χ 2 k 1 = k (X i m i ) 2 m i=1 i, m 1,..., m k (m i = np i 5), k , 120.? ,. 4 : 3 : 2 : 1.,? A O B AB , , 5.,, 5 1:1? [] : 0:5 1:4 2:3 3:2 4:1 5: , 45, 55.? (1) (2), 2.
39 A = {A 1,..., A r }, B = {B 1,..., B s }, ( Xij r s n X ) 2 i X j n n χ 2 = n i=1 j=1, n (X ij 5), (r 1)(s 1) 2. X i n X j n B 1 B 2 B s A 1 X 11 X 12 X 1s X 1 A 2 X 21 X 22 X 2s X 2. A r X r1 X r2 X rs X r X 1 X 2 X s n 10.2.? ? = 7 18 = ,. 4., (, ),. = 7 11 = ,. 6..,..
40 : P (χ 2 n χ 2 n(α)) = α α χ n α n\α (n = 1 ).
41 (x, y): x =, y = 11.2 x = 1 n σx 2 = 1 n n x i, ȳ = 1 n i=1 n (x i x) 2, σy 2 = 1 n i=1 n y i, i=1 n (y i ȳ) 2, (X, Y ), m = E[X], σ 2 = V[X] = E[(X m) 2 ] = E[X 2 ] E[X] (1) n = 205 x = ȳ = 50.9 σx 2 = = σy 2 = = (2) n = 917 x = ȳ = 63.8 σx 2 = = σy 2 = = i=1
42 x, y., BMI(),,,,. () x () y y = f(x)., 1,.,.,, (x i, y i ) (i = 1, 2,..., n) x, y 11.1 σ xy = 1 n n (x i x)(y i ȳ) = 1 n i=1 n x i y i xȳ i=1 r xy = σ xy σ x σ y x, y., x i = x i x σ x, ỹ i = y i ȳ σ y r xy 1. σ xỹ = r xy = r xỹ r xy > 0, r xy < 0., r xy > 0.8, r xy < (1) : σ xy = 19.96, r xy = 0.64 (2) : σ xy = 19.97, r xy = 0.43
43 (x 1, y 1 ),..., (x n, y n ), y = ax + b. y i = ax i + b + ϵ i, n n Q = ϵ 2 i = (y i ax i b) 2 i=1 a, b ()., n Q = (yi 2 + a 2 x 2 i + b 2 2ax i y i 2by i + 2abx i ) i=1 Q, i=1 = y 2 i + a 2 x 2 i + b 2 n 2a x i y i 2b y i + 2ab x i. Q a = 2a x 2 i 2 x i y i + 2b x i = 0, Q b = 2bn 2 y i + 2a x i = 0 a, b : a = σ xy σ 2 x, b = ȳ a x 11.3 (1) : y = 0.72x (2) : y = 0.69x 54.60
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ax 2 + bx + c = n 8 (n ) a n x n + a n 1 x n a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 4
20 20.0 ( ) 8 y = ax 2 + bx + c 443 ax 2 + bx + c = 0 20.1 20.1.1 n 8 (n ) a n x n + a n 1 x n 1 + + a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 444 ( a, b, c, d
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I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )
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4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
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... 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
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50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
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n=1 1 n 2 = π = π f(z) f(z) 2 f(z) = u(z) + iv(z) *1 f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x
n= n 2 = π2 6 3 2 28 + 4 + 9 + = π2 6 2 f(z) f(z) 2 f(z) = u(z) + iv(z) * f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x f x = i f y * u, v 3 3. 3 f(t) = u(t) + v(t) [, b] f(t)dt = u(t)dt
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1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1
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1 I 1.1 ± e = = - =1.602 10 19 C C MKA [m], [Kg] [s] [A] 1C 1A 1 MKA 1C 1C +q q +q q 1 1.1 r 1,2 q 1, q 2 r 12 2 q 1, q 2 2 F 12 = k q 1q 2 r 12 2 (1.1) k 2 k 2 ( r 1 r 2 ) ( r 2 r 1 ) q 1 q 2 (q 1 q 2
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1 1 1.1 *1 1. 1.3.1 n x 11,, x 1n Nµ 1, σ x 1,, x n Nµ, σ H 0 µ 1 = µ = µ H 1 µ 1 µ H 0, H 1 * σ σ 0, σ 1 *1 * H 0 H 0, H 1 H 1 1 H 0 µ, σ 0 H 1 µ 1, µ, σ 1 L 0 µ, σ x L 1 µ 1, µ, σ x x H 0 L 0 µ, σ 0
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waseda2010a-jukaiki1-main.dvi
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(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
5 Armitage x 1,, x n y i = 10x i + 3 y i = log x i {x i } {y i } 1.2 n i i x ij i j y ij, z ij i j 2 1 y = a x + b ( cm) x ij (i j )
5 Armitage. x,, x n y i = 0x i + 3 y i = log x i x i y i.2 n i i x ij i j y ij, z ij i j 2 y = a x + b 2 2. ( cm) x ij (i j ) (i) x, x 2 σ 2 x,, σ 2 x,2 σ x,, σ x,2 t t x * (ii) (i) m y ij = x ij /00 y
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)
(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](/thumbs/67/56612994.jpg "(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 ]
Z[i] Z[i] π 4,1 (x) π 4,3 (x) 1 x (x ) 2 log x π m,a (x) 1 x ϕ(m) log x 1.1 ( ). π(x) x (a, m) = 1 π m,a (x) x modm a 1 π m,a (x) 1 ϕ(m) π(x)
![Z[i] Z[i] π 4,1 (x) π 4,3 (x) 1 x (x ) 2 log x π m,a (x) 1 x ϕ(m) log x 1.1 ( ). π(x) x (a, m) = 1 π m,a (x) x modm a 1 π m,a (x) 1 ϕ(m) π(x)](/thumbs/94/118373946.jpg "Z[i] Z[i] π 4,1 (x) π 4,3 (x) 1 x (x ) 2 log x π m,a (x) 1 x ϕ(m) log x 1.1 ( ). π(x) x (a, m) = 1 π m,a (x) x modm a 1 π m,a (x) 1 ϕ(m) π(x)")
3 3 22 Z[i] Z[i] π 4, (x) π 4,3 (x) x (x ) 2 log x π m,a (x) x ϕ(m) log x. ( ). π(x) x (a, m) = π m,a (x) x modm a π m,a (x) ϕ(m) π(x) ϕ(m) x log x ϕ(m) m f(x) g(x) (x α) lim f(x)/g(x) = x α mod m (a,
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(1) 3 A B E e AE = e AB OE = OA + e AB = (1 35 e 0 1 15 ) e OE z 1 1 e E xy 5 1 1 5 e = 0 e = 5 OE = ( 2 0 0) E ( 2 0 0) (2) 3 E P Q k EQ = k EP E y 0 Q y P y k 2 M N M( 1 0 0) N(1 0 0) 4 P Q M N C EP
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