ii 3.,. 4. F. (), ,,. 8.,. 1. (75% ) (25% ) =9 7, =9 8 (. ). 1.,, (). 3.,. 1. ( ).,.,.,.,.,. ( ) (1 2 )., ( ), 0. 2., 1., 0,.
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1 23(2011) (1 C104) 5 11 (2 C206) ,,. 1., 2007 ( ). 2. P. G. Hoel, ,,.
2 ii 3.,. 4. F. (), ,,. 8.,. 1. (75% ) (25% ) =9 7, =9 8 (. ). 1.,, (). 3.,. 1. ( ).,.,.,.,.,. ( ) (1 2 )., ( ), 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). 1.3 () , 1, 2. 2/10
4 , / , ? 1/221, 1/ , 10. 2/ () ( ). 3, ( ) 2.,. 1.,, 1 ( ).? 1.8 ( ) 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 ,.,.,. (). X, Y, Z , , , 1) 2) 3.2 (Ω, F, P ). Ω X : Ω R, x R,. {X x} F..
10 8 3 X, F (x) = P (X x), x R, 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). ( ) P (X = a i ) = p i ( ) P (a X b) = f(x). F (x) = x 3.2 f(x). (1) f(x) 0. b a f(x)dx f(t)dt f(x) = F (x) (2) + f(x)dx = X.
11 p n, X ( ) n P (X = k) = p k (1 p) n k, k = 0, 1, 2,.... k, B(n, p) 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, Y., Y., Y. 8 2 () X, () Y. X, Y. 9,,, ( ).
![第 3 章 確率変数 10 3.4 3.4.1 連続分布の例 一様分布 1, a x b f (x) = b a 0, その他 1) 区間 [a, b] からどの点も同等な確からしさで 1 点を選ぶときのモデルとして現れる. 2) 長さ L の棒をランダムに折ってできる長いほうの断片の長さ X は, [L/2, L] 上の一様 分布に従う. 3.4.2 指数分布 λ > 0 を定数として f (x) = ( λe λx, x 0 0, x<0 ランダム到着の待ち時間をモデル化するときに現れる.](/docs-images/93/113738911/images/12-0.jpg "3.4.3 正規分布 (ガウス分布) N (m, σ 2 ): 平均 m, 分散 σ 2 の正規分布 (またはガウス分布) ½ ¾ 1 (x m)2 f (x) = exp 2σ 2 2πσ 2 N (0, 1): 標準正規分布 他に, 数理統計学で頻出なものとして, χ2 -(カイスクエア) 分布, t-分布, F -分布 例 題 3.")
12 第 3 章 確率変数 連続分布の例 一様分布 1, a x b f (x) = b a 0, その他 1) 区間 [a, b] からどの点も同等な確からしさで 1 点を選ぶときのモデルとして現れる. 2) 長さ L の棒をランダムに折ってできる長いほうの断片の長さ X は, [L/2, L] 上の一様 分布に従う 指数分布 λ > 0 を定数として f (x) = ( λe λx, x 0 0, x<0 ランダム到着の待ち時間をモデル化するときに現れる 正規分布 (ガウス分布) N (m, σ 2 ): 平均 m, 分散 σ 2 の正規分布 (またはガウス分布) ½ ¾ 1 (x m)2 f (x) = exp 2σ 2 2πσ 2 N (0, 1): 標準正規分布 他に, 数理統計学で頻出なものとして, χ2 -(カイスクエア) 分布, t-分布, F -分布 例 題 3.11 単位円の内部から 1 点をランダムに選んだとき, その点と中心との距離 X は連 続型の確率変数になる. この X の確率密度関数を求めよ.
13 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. f(x),. f(x) 0, + f(x) = 1 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 ) (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 [a, b] (a + b)/2 (b a) 2 /12 ( λ) 1/λ 1/λ 2 N(m, σ 2 ) m σ 2
14 ( ) {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. 11,. 12,. 13,. 14,., ( ). + e x2 dx = π 15 L 2, X. X,,,. 16 1, X,,,. 17 O R 1, O X. X,,, () X, () Y. X, Y.
15 , ,,? 4.2 A, B 2. P (A) > 0, A B P (B A) = P (A B) P (A) 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,.
16 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 ). (). 4.1, A B, 95%, 2%... 19, A B, 95%, 100p %... p , 5, 2, 5.? (2,.) 21 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 ) 22 () (1) 1., 6. (2) 1., 6.
17 N(0, 1): 5.1 Z N(0, 1)., (1). P (Z 1.15), P (Z 1.23), P ( Z < 2.4) (2) a. 5.1 X N(m, σ 2 ), P (Z a) = 0.33, P (Z < a) = 0.75, P ( Z a) = 0.4 Z = X m σ 5.2 X N(2, 5 2 ),. N(0, 1) P (X 3), P (X 0), P ( X 4) 23 (1) X N(20, 4 2 ), P (X > 17.8). (2) Y N( 2, 5 2 ), P ( Y 1). 24 X N(50, 10 2 ), a, b. P (X a) = 0.33, P (X > b) = X N(0, 1), Y = ax + b., a, b.
18 B(100, 0.4),. B(n, p) N(np, np(1 p)), 0 < p < 1, n. 5.3 [ ]. (1) 400, 215. (2) [] 400, 225.? 26 (1) 1000, 550. (2) 250, X N(0, 1), Y = X 2.
19 I(z) = 1 2π z 0 e x2 /2 dx z
20
21 A, B, P (A B) = P (A)P (B). (P (A) = 0, P (B) = 0.) P (A) > 0, A, B P (B A) = P (B) , A B A,B 2, A B? (1),. (2),. A 1, A 2,..., A i1, A i2,..., A in, P (A i1 A in ) = P (A i1 ) P (A in ). 6.2 (Ω, F, P ), X, Y 2., P ({X x} {Y y}) = P (X x)p (Y y), x, y R, X, Y., X, Y, X {a 1, a 2,... }, Y {b 1, b 2,... },. P ({X = a i } {Y = b j }) = P (X = a i )P (Y = b j ),
22 , X = (), Y = ( ). 6.4 L > 0. Ω = {(x, y) ; 0 x L, 0 y L}, 1, x X y Y. X 1, X 2,..., P (X 1 x 1,, X n x n ) = P (X 1 x 1 ) P (X n x n )., X i x i X i = x i. ( ) X 1, X 2,..., X n, E(X 1 X 2... X n ) = E(X 1 )E(X 2 ) E(X n ) () X 1, X 2,..., X n, V(X 1 + X X n ) = V(X 1 ) + V(X 2 ) + + V(X n ) X, Y ( ), α, E(X + Y ) = E(X) + E(Y ), E(αX) = αe(x), V(αX) = α 2 V(X). 6.3 X 1, X 2,..., X n,... : (iid)., (1), n ( = 1, = 0) X n. (2) n X n. (3), n X n. X = 1 n.. n k=1 X k 6.5 ( ),, ( )., n X n, X 1, X 2,... iid,., (X k ), X 1, X 2,....,.
23 X 1, X 2,..., m, σ 2. n ( n ) X = 1 n X k n. k= n, X m., (1) () ϵ > 0, ( ) lim P 1 n X k m n n ϵ = 0. k=1 (2) () P ( lim n 1 n ) n X k = m = 1. k=1 6.6 () X 1, X 2,..., n X = 1 n X k n n. k=1
24 : X = 1 n n k=1 X k N ) (m, σ2. n : Z k = X k m σ, lim P n ( a 1 n ) n Z k b = 1 b e x2 /2 dx, a < b. 2π k=1 a 6.7 ( ),. ( ) = x m σ, A, B, A c B (A c A ). 29. (1) 1 2 X. X (2) 2 Y. Y. 30, 4% ,. 31 () X, m, σ 2., ϵ > 0, P ( X m ϵ) σ2 ϵ 2. [,.]
25 ,? ( ) 6 19 ( ) () 10 (%) 06/19( ) 21: /17( ) 08: /19( ) 20: /19( ) 21: /17( ) 21: /16( ) 21: /16( ) 22: /15( ) 21: /18( ) 21: /13( ) 21: ,. 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? 32, 200, 2.2 g., 1.5 g., g?. [1.992, 2.408] 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 7.4 ( ) 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
28 () ˆp p z p(1 p) n ˆp p z ˆp(1 ˆp) n, p 1 α, [ ] ˆp(1 ˆp) ˆp(1 ˆp) ˆp z, ˆp + z n n %, 0.54 ± (1 0.54) ± ( ) %. 95%, 0.22(1 0.22) 0.22 ± ± , 95% 0.01,? % %. 34,,.,,., U 2 = 1 n (X k n 1 X) 2 k=1, t- ().
29 , 220.? ( H 0 ). (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,.
30 28 8 (5), H 0, (2 ). H 0,. α α α W W W W 8.3 ( ) Z N(0, 1), α = P ( Z z) = 1 1 2π z, [0, ) z α (0, 1]. z e x2 /2 dx, z 0, z α α α -z z 8.2 A B 400 (), A 220, B 180. A B? %. 400, 235.?
31 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 , 220.? 2. 35? (1) 100, 55. (2) 1000, ,. 100, 40, ,,., 2 4, 4.,. ( )
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 ).) , 175., , ( 1 ) 2.
34 , m = 60 (g).,, m 50 70, σ = 3 ( )., 25,, m = 60? 9.4 (). 120,., , () 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,. 9.1 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) B, Γ. ( ) n t2 2 n = Γ( n+1 2 ) n Γ( n 2 )Γ( 1 2 ) ( ) n t2 2 n
35 9.3. () 33 n n n t P ( T t n (α)) = α n\α () n = t- N(0, 1)., n 30 N(0, 1).
36 (kg), kg, g g, 10 2 g.,? 5%. 1% % %. [0.175 ± ] g., 8g. 1 95%., 95% 1g? [156 ± ] , 40, 60. [ 5%] cm cm, 4.63 cm. [ 1% ] 42, 100 g 2g., 200, 2.2 g., 1.5 g. [ 5%] A , A. A. [ 5%]
37 Karl Pearson ( ) X 1, X 2,..., X n,, N(0, 1)., n χ 2 = i=1 n 2 (χ 2 -). (χ 2.), 1 ( n ) x n 2 1 e x 2, x > 0, f n (x) = 2 n/2 Γ 2 0, x 0,. X 2 i n = n = n = n = n = 10.1 X 1, X 2,..., X n,, N(m, σ 2 )., χ 2 = 1 n (X σ 2 i X) 2 i=1
38 n 1 2., X X = 1 n X i () n 10.2 i=1. A 1, A 2,..., A k k. n, X 1, X 2,..., X k. A 1 A 2 A k X 1 X 2 X k n, p 1, p 2,..., p k m i = np i, χ 2 = k (X i m i ) 2, m 1,..., m k, k 1 2. i=1 10.1, 120.? m i , , 5.,, 5 1:1? : 0:5 1:4 2:3 3:2 4:1 5: ,. 4 : 3 : 2 : 1.,? A O B AB , 45, 55.? (1) (2), 2.
39 : P (χ 2 n χ 2 n(α)) = α α χ n α n\α (n = 1 ).
40 = 8 10 = ,.. obata,, = 8 3 = 8 4 (, ),.,, = 9 7 = 9 8, obata/lecture/lect-j.html 8 31.,.
41 William Feller ( ) B, A. A B 1, B 1. n X n, P (X n = +1) = P (X n = 1) = 1 2, X 1, X 2,.... n A : n S n = k=1 X k n = (1). (2) S n.
42 n, (+1) A n, ( 1) B n, n = A n + B n, S n = A n B n,, ( P lim n A n = S n + n 2, B n = S n n 2 A n n = 1 ) ( B n = P lim 2 n n = 1 ) = 1. 2, S n N(0, n), X 1, X 2,..., X n n., m = E(X 1 ) = 0, σ 2 = V[X 1 ] = 1., 1 ) ( n S n N (m, σ2 = N 0, 1 ) n n., S n N(0, n). 46 () P ( S ).
43 {S n } 2n ( 0) () P (L 2n = 2k) = L 2n = max{0 m 2n ; S m = 0} ( 2k k )( 2n 2k n k ) ( 1 4 ) n, k = 0, 1, 2,..., n, (11.1). (.) 11.1: : L 100, : 11.2 () L 2n, ( ) 1 lim P n 2n L 2n a = 1 a dx = 2 π x(1 x) π arcsin a, 0 a 1, (11.2). 0 0 a < b 1. P (2an L 2n 2bn). [2an, 2bn] 2k 1, 2k 2, P (2an L 2n 2bn) = k 2 k=k 1 P (L 2n = 2k) = k 2 k=k 1 np (L 2n = 2k) 1 n (11.3). 11.1, ( )( ) ( ) n 2k 2n 2k 1 np (L 2n = 2k) = n k n k 4 n π k(n k)
44 42 11, (11.3), P (2an L 2n 2bn) k 2 k=k 1 n π k(n k) k2 1 n =, n,., b a dx π x(1 x) 1 ( k=k 1 k π n 1 k n ) 1 n b lim P (2an L 2n 2bn) = n a dx π x(1 x) (11.4). (..) n! ( n ) n 2πn e b n, a n b n lim = 1., n n a n, n. P ( ) ( ) 1 1 2n L 2n 0.1 = P 2n L 2n k 2n, (k 1, S k 1 ) (k.s k ) x k H 2n, H 2n L 2n ( 11.1).
45 (x, y): x =, y = 12.2 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
46 (x i, y j ) (i = 1, 2,..., n) x, y 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 r xy > 0, r xy < 0., r xy > 0.8, r xy < 0.2. (X, Y ), Cov[X, Y ] = E[(X E[X])(X E[Y ])] = E[XY ] E[X]E[Y ], [ ] r XY = Cov[X, Y ] X E[X] = E X E[Y ] V[X] V[Y ] V[X] V(Y ) 12.2 (1) : σ xy = 19.96, ρ xy = 0.64 (2) : σ xy = 19.97, ρ xy = 0.43
47 , +. ( )? 1) 2) m = 1 n σ 2 = 1 n n i=1 x i n x 2 i m 2 3) 2 (x 1, x 2 ), (x 2, x 3 ),... i=1 σ1 2 = 1 n 1 (x i m)(x i+1 m) n 1 i=1 4) 5) r = σ2 1 σ
ii 3.,. 4. F. ( ), ,,. 8.,. 1. (75% ) (25% ) =7 24, =7 25, =7 26 (. ). 1.,, ( ). 3.,...,.,.,.,.,. ( ) (1 2 )., ( ), 0., 1., 0,.
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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%)
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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
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