(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. ( ),.. 5... 6.. 7.,,. 8.,. 1. (75% ) (25% ). 60. 2. =7 24, =7 25, =7 26 (. ). 1.,, ( ). 3.,...,.,.,.,.,. ( ) (1 2 )., ( ), 0., 1., 0,.
1 1 1.1 (1). (2).,. (3),,. 1.2 ( ), Ω., E. E Ω. ( ), E P (E) = E Ω..,.., Ω. 1.1 ( ) ( 1654 1705). ( ).,.. 1.2 52 2, 2 (A,K,Q,J). 1.1 10, 1. [1023/1024] 1.2 52 2, 2. 4 4? [1/221, 1/270725]
2 1 1.3 ( ) 10 2. 2, 1, 2. 1.3 10 2. 3, 3. [2/10] 1.4 10 2. 10, 10. [2/10] 1.4 ( ) A,B 2. A 2/5, B 3/5. 3, 10000. A 2, B 1.? [.] 1.5 ( ) ( ). 3, ( ) 2.,. 1.,, 1 ( ).? 1.5 A,B 2. 1.4. 4, 10000. A 3, B 2,? 1.3 (1501 1576) (1564 1642) (1623 1662) (1601 1665) ( ) (1654 1705) (1749 1827) ( ) ( ).. (1903 1989) ( ) ( ) (1886 1971), (1894 1964), (1915 2008) ( )
3 2 2.1 3 : Ω: ( ) = (, F: ( ) P : 2.1 ( ), Ω., E P (E) = E Ω,. 2.2 (Ω ( ) ),,., P (X = k) = λk k! e λ, k = 0, 1, 2,...,., λ > 0. λ. 2.3 (Ω ). 2. 1, ( ), 0. 2.1 2, 3.
4 2 2.2, 30cm 40cm, 5cm. 2.4 ( )?, 3 1 : 2 : 3. 10 40 100 Ω E. P (E) = E Ω,..,,,,.... 2.3 2 1 10.,. 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,. ω Ω n=1
2.3. 5 E F (E = F. a < b.) Ω E c E F, E 1 E n E F, E 1 E n E F = 2.3 AB 3. (, 3 3.) B A O [ ], ( )?. 1 2 ( ) 1 52 5,. (1) ( A,K,Q,J,10) (2) (3) 2 0 9 5 00000, 00001,..., 99999 1. (1) 9 1.
6 2 (2) 9 2. (3) 0, 1,..., 9 1 2. (4) 0, 1,..., 9 2 1. 3 ( ) 1 2 3 4 5 +, 3. 4 A,B 2. A p, B q = 1 p. 5, 10000. A 3, B 2.? 5 1 P, 2 P. 1/3. 6 λ.? [ e λ e λ ] 7 A,B,C,. A,B,C.
7 3 3.1,. ( ). (1) 1, 0. (2). (3) 5. (4). (5) 1, ( )., x, y, z, t,...., 0 x 1, x 0 1.,,.,,.., X, Y, Z, T,....,,,,.. 3.2 ( ) 3.1 X, X {1, 2, 3, 4, 5, 6}., P (X = 1) = P (X = 2) = = P (X = 6) = 1 6
8 3. X,, X ( ). X {a 1, a 2,..., a i,... }, P (X = a i ) = p i, i = 1, 2,...,, X., ( ) X., p i 0, p i = 1. 3.1 X. (1) 1, 0 X. (2) 2 X. i 3.3 3.2 L, X, X. X L/2 X L., x,, P (X = x) = 0, 3.1.,. X ( ), F (x) = P (X x), x R, X.. ( 3.2 ), 0, x L/2, 2x L F (x) =, L/2 x L, L 1, x L,
3.4. ( ) 9 3.3 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 ( ) 3.1 X, P (X x) = F (x) = x f(t)dt F (x) = f(x) f(x) X. (F (x).), P (a X b) = b a f(x)dx. f(x),. 3.4 ( 3.2 ) L, X. X. 3.2, f(x). (1) f(x) 0. (2) + f(x)dx = 1. 3.2 X. 3.3 1, X. X. 3.4 O R 1, O X. X.
11 4 4.1 4.1.1 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). 4.1.2 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,.... 4.1.3 X λ > 0, P (X = k) = λk k! e λ, k = 0, 1, 2,.... 4.2 λ = 0.5, λ = 1, λ = 2.
12 4 4.3 4.1 1 3. (1) 1. (2) 1 5. 4.1.4 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].. 4.4 2, 2 100, 1 50, 2 10. 1,. (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 4.2. 4.3 1000 1. 1,.
4.2. 13 8 ( ) {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.,. 9 2 ( ) X, ( ) Y. X Y. 4.2 f(x). 4.2.1 [a, b] 1. 1 f(x) = b a, a x b 0, 4.2.2 λ > 0 f(x) = { λe λx, x 0 0, x < 0. 4.2.3 ( ) N(m, σ 2 ): m, σ 2 ( ) { } 1 f(x) = exp (x m)2 2πσ 2 2σ 2 N(0, 1):
14 4, χ 2 -( ), t-, F - ( ) 4.2.4 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 2 4.4,.,, ( ). + e x2 dx = π 4.5 L 2, X. X,,,. 10 1, X. X,,,. 11 O R 1, O X. X,,,.
15 5 5.1 A, B 2. P (A) > 0, A B P (B A) = P (A B) P (A) 5.1 ( ) 10, 2. 2 1,,?. 5.1 2 E, F, P (E) = 1 3, P (F ) = 1 2, P (E F ) = 1 4.. P (E c ), P (E F c ), P ((E F c ) c ), P (E F ), P (E F c ), P (E F E F ) 5.2 5.2.1 T, P (T m + n T m) = P (T n), m, n = 0, 1, 2,...,. 5.2.2 X, P (X a + b X a) = P (X b), a, b 0,.
16 5 5.3 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 500 2. B, 95%, 2%... 5.2 5.2, A 1000 2? 12, A 500 2. B, 95%, 100p %... p. 13 5 2., 5, 2, 5.? (2,.) 14 2 E, F, P (E) = 1 3, P (F ) = 1 2, P (E F ) = 2 3.. P (E c ), P (E F c ), P ((E F c ) c ), P (E F ), P (E F c ), P (E F E F ) 15 ( ) 1 10 10. 1 2. 4. (1) 1., 6. (2) 1., 6.
17 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 6.1 Z N (0, 1) に対して, 確率 P (Z 1.82), P (Z 2.13), P ( Z > 1.5) を求めよ. 問 6.2 Z N (0, 1) とするとき, 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 (標準化) 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 6.3 (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. 問 6.4 X N (50, 102 ) のとき, P (X > a) = 0.985 を満たす a を求めよ.
18 6 ( )., 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 1 6.2 B(100, 0.4) 6.2,. B(n, p) N(np, np(1 p)), 0 < p < 1, n. 6.3 400, 225 ( ). 6.5 (1) 1000, 550. (2) 250, 1 30. 400, 225.? 6.3 ( ),, ( ).,, n, x 1, x 2,..., x n 1 n n i=1 x i
6.3. 19,.,,? 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,. 16 X N(0, 1), P (X x) = 1 2π x e t2 /2 dt, Y = ax + b., a 0, b. (a ) 17 44.5, 23.5 ( 22 10 ). 25 32.?. 2500 2000 1500 1000 500 0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110
20 6 I(z) = 1 2π z 0 e x2 /2 dx z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359 0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753 0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141 0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517 0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879 0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224 0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549 0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852 0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133 0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389 1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621 1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830 1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015 1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177 1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319 1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441 1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545 1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633 1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706 1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767 2.0 0.4773 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817 2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857 2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890 2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916 2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936 2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952 2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964 2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974 2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981 2.9 0.4981 0.4982 0.4983 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986 3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990
21 7 7.1.,? 2013 5 6 ( ) 5 12 ( ) ( ) 10 (%) 13/05/06( ) 21:00-54 20.9 13/05/07( ) 8:00-15 20.4 13/05/12( ) 20:00-45 15.0 13/05/09( ) 22:00-54 14.9 13 13/05/11( ) 21:00-130 14.6 13/05/11( ) 21:00-141 13.7 35 13/05/11( ) 21:00-54 12.9 13/05/08( ) 22:00-69 12.2 13/05/06( ) 14:04-146 12.1 13/05/07( ) 22:00-54 11.5 32,., 27, PM.,, PM 600, 200. (. 2013.5 ) : ( ) 7.2 E 2. ( E p). X 1, X 2,..., X n n ( p ) { 1, i E, X i = 0, i E
22 7 ˆp = 1 n, ˆp ( ) (! ˆp )., ˆp,. n k=1, n X 1, X 2,..., X n, X = 1 n.,. n k=1 X k X k 7.3 ( ) m, σ 2, : X = 1 n ) X k N (m, σ2 n n p [ = p, = σ 2 = p(1 p)]., n ) ˆp N (p, σ2 ˆp p n σ/ N(0, 1). n k=1, Z N(0, 1) ( ) P ( z Z z) = 1 α z 1.00 1.64 1.96 2.00 2.58 3.00 3.29 α 0.317 0.100 0.050 0.045 0.010 0.003 0.001 1 α 0.683 0.900 0.950 0.955 0.990 0.997 0.999 N α z z
7.3. 23 p 1 α [ ] ˆp(1 ˆp) ˆp(1 ˆp) ˆp z, ˆp + z n n. 90%(α = 0.1, z = 1.64) 95%(α = 0.05, z = 1.96) 99%(α = 0.01, z = 2.58).. 1 1.,, 1 α p.,! 2 ( ) ˆp p z p(1 p) n ˆp p z ˆp(1 ˆp) n 7.1 ( ) 600 21%. 95%, 0.21(1 0.21) 0.21 ± 1.96 0.21 ± 0.033 600 7.2, 95% 0.01,? 90%? 18 100, 12.. [ ] 90%, 0.12 ± 1.64 0.12(1 0.12) 100 0.12 ± 0.053 19,,. 20, 2013 3, 0.3 93.9 %. 112 6250 ( 25 5 17 ) 95 %.
25 8 8.1 ( ),,. n ( ) X n, X n, X 1, X 2,... (iid)., (X k ), X 1, X 2,....,. X 1, X 2,..., n = 1, 2, 3,... x 1, x 2,..., x n 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. 8.1 X 1, X 2,..., X n. (1) [ ] E(X 1 X 2 X n ) = E(X 1 )E(X 2 ) E(X n ) (2) [ ] V(X 1 + X 2 + + 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). 8.1 (1) 2 X. X. (2) 2 Y. Y. 8.2 X 1, X 2,... (, ). m, σ 2. X = 1 n X k n. k=1
26 8 X m : E[ X] = m ( n, X m : P lim n ) X = m = 1 ( ) 8.2 ( ) X 1, X 2,..., X = 1 n X k n. n k=1 :, X = 1 n ) X k N (m, σ2 n n n., k=1 X m σ/ n = 1 n n k=1 X k m σ N(0, 1) n.,, ( ) lim P a 1 n X k m b = 1 b e x2 /2 dx, a < b. n n σ 2π k=1 a 8.3 ( ) m ( ), σ 2 X 1, X 2,..., X n : n ( (iid) ) 1 n : X = X k n k=1 m 1 α [ X z σ n, X + z σ n ], z N(0, 1) α
8.4. ( ) 27. 90%(α = 0.1, z = 1.64) 95%(α = 0.05, z = 1.96) 99%(α = 0.01, z = 2.58). 8.3 1. 40 156g., 8g. 1. 8.4 8.3, 95% 1g? 21, 200, 2.2 g., 1.5 g., g?. [1.992, 2.408] 8.4 ( ) 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,. (,, ) 8.2 U 2 : E(U 2 ) = σ 2.,., n, S 2 U 2. 8.3 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-,.
28 8 n t- 1 n B ( n 2, 1 2) ( ) n+1 1 + t2 2 n = Γ( n+1 2 ) n Γ( n 2 )Γ( 1 2 ) ( ) n+1 1 + t2 2 n n n n (1) Γ. Γ(x) = 0 t x 1 e t dt, x > 0. (2) B. B(x, y) = 1 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). m 1 α [ X t U n, X + t U n ], t t n 1 α 8.5 8,. 90%. 32.5 31.8 33.0 32.4 32.2 31.3 32.9 32.1 22,. 95%. 23 42 33 29 34 41 30 36 34 28
8.4. ( ) 29 t P ( T t n (α)) = α n\α 0.100 0.050 0.020 0.010 1 6.314 12.706 31.821 63.657 2 2.920 4.303 6.965 9.925 3 2.353 3.182 4.541 5.841 4 2.132 2.776 3.747 4.604 5 2.015 2.571 3.365 4.032 6 1.943 2.447 3.143 3.707 7 1.895 2.365 2.998 3.499 8 1.860 2.306 2.896 3.355 9 1.833 2.262 2.821 3.250 10 1.812 2.228 2.764 3.169 11 1.796 2.201 2.718 3.106 12 1.782 2.179 2.681 3.055 13 1.771 2.160 2.650 3.012 14 1.761 2.145 2.624 2.977 15 1.753 2.131 2.602 2.947 16 1.746 2.120 2.583 2.921 17 1.740 2.110 2.567 2.898 18 1.734 2.101 2.552 2.878 19 1.729 2.093 2.539 2.861 20 1.725 2.086 2.528 2.845 21 1.721 2.080 2.518 2.831 22 1.717 2.074 2.508 2.819 23 1.714 2.069 2.500 2.807 24 1.711 2.064 2.492 2.797 25 1.708 2.060 2.485 2.787 26 1.706 2.056 2.479 2.779 27 1.703 2.052 2.473 2.771 28 1.701 2.048 2.467 2.763 29 1.699 2.045 2.462 2.756 30 1.697 2.042 2.457 2.750 1.645 1.960 2.326 2.576 23, 4% 1000 1050,. 24 ( ) m, σ, ( ) = 50 + 10 x m σ,., 20 80,. 25 17.5%. 2000. 90%.
31 18 100, 12.. 21, 200, 2.2 g., 1.5 g., g?. 22,. 95%. 23 42 33 29 34 41 30 36 34 28 23, 4% 1000 1050,. 4 A,B 2. A p, B q = 1 p. 5, 10000. A 3, B 2.? 10 1, X. X,,,. 5.1 2 E, F, P (E) = 1 3, P (F ) = 1 2, P (E F ) = 1 4.. P (E c ), P (E F c ), P ((E F c ) c ), P (E F ), P (E F c ), P (E F E F ) 12, A 500 2. B, 95%, 100p %... p. 6.3 (1) X N(20, 4 2 ), P (X > 17.8). (2) Y N( 2, 5 2 ), P ( Y 1). 6.4 X N(50, 10 2 ), P (X > a) = 0.985 a.
33 9 9.1 Sir Ronald Aylmer Fisher (1890 1962) 9.1 400, 220.? (1), (2), (3). 9.2 1. H 0 H 1. 2. T ( ), H 0,. 3. 0 < α < 1 P (T W ) = α W R ( ) H 1. 4. 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,.
34 9 α α α W W W W 9.2 60%., 400 235.? 9.3 N 35%., 37 %.,. 1000. Z N(0, 1), α = P ( Z z) = 1 1 2π z z e x2 /2 dx, z 0 z 1.00 1.64 1.96 2.00 2.58 3.00 3.29 α 0.317 0.100 0.050 0.045 0.010 0.003 0.001 1 α 0.683 0.900 0.950 0.955 0.990 0.997 0.999 α -z z 9.3 2 H 0, 4. \ H 0 H 0 H 0 2 H 0 1, 1, 2. α: 1 = β: 2
9.3. 2 35 9.4 400, 215.? 2. θ θ β α c c 26 A. 100 58.. 27 100,. 1000, 545, 455.. 28, 2013 3, 0.3 93.9 %. 112 6250 ( 25 5 17 ),,. 29 ( ) 23 22 19 18 538 271 550 742 559 847 560 439 512 535 520 562 529 971 532 235 1 050 806 1 071 304 1 089 818 1 092 674 30 ( ) 10,. p. H 0 : p = 1 2, H 1: p 1 2. 10 T. {T = 0, 1, 9, 10}., H 0 2 β p. H 0 P (2 T 8).
37 10 10.1 William Sealy Gosset (1876 1937) 1. H 0 H 1. 2. T ( ), H 0,. 3. 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 - ),. 10.2 ( ) m, σ 2 n, X = 1 n n ) X k N (m, σ2 n k=1 X m σ/ n N(0, 1),, n (. N(m, σ 2 ) ). 10.1 ( ), m = 60 (g).,, m 50 70, σ = 3 ( )., 25,, 61.43. m = 60?
38 10 10.2 ( ). 120,., 16 121.2., 2.4.. m. : H 0 : m = 120 H 1 : m > 120 10.3 ( ) B(n, p) N(np, np(1 p)) np:, np(1 p):, ˆp : ( ) p(1 p) ˆp N p, n p:, n: 10.3 ( ) 400, 175.,. 10.4 ( ) m, σ 2 n X 1,..., X n, U 2 = 1 n (X i n 1 X) 2,. X, i=1 T = X m U/ n t n 1 (n 1) t- 10.4 10 (kg), 53.2 61.5 48.1 51.3 55.7 47.2 54.5 57.9 53.8 49.2. 50kg,
10.4. ( ) 39 31 1000 200 157.7 cm. 158.6 cm, 4.63 cm. [ 1% ] 32, 100 g 2g., 200, 2.2 g., 1.5 g. [ 5% ] 33 8%., 175, 25.. 34. 100, 38, 62. [ 5% ] 35 500g 120 498g, 10 2 g.,? 5%. 1%. 36 66. A 10. 78 72 65 86 58 64 76 88 74 59, 72 66 A. A. [ 5% ]
41 11 11.1 (x, y): x =, y = 11.2 m x = 1 n σx 2 = 1 n n x i, m y = 1 n i=1 n (x i m x ) 2, σy 2 = 1 n i=1 n y i, i=1 n (y i m y ) 2, (X, Y ), m X = E[X], σx 2 = V[X] = E[(X m X ) 2 ] = E[X 2 ] E[X] 2 11.1 (1) n = 205 m x = 157.9 m y = 50.9 σx 2 = 27.83 = 5.28 2 σy 2 = 34.43 = 5.87 2 (2) n = 917 m x = 171.6 m y = 63.8 σx 2 = 28.94 = 5.38 2 σy 2 = 75.69 = 8.70 2 i=1
42 11 11.3 2 x, y., BMI( ),,,,. ( ) x ( ) y y = f(x)., 1,.,.,,. 11.4 2 (x i, y i ) (i = 1, 2,..., n) x, y 11.1 σ xy = 1 n n (x i m x )(y i m y ) = 1 n i=1 r xy = σ xy σ x σ y n x i y i m x m y i=1 (X, Y ), σ XY = E[(X m X )(Y m Y )] = E[XY ] E[X]E[Y ], r XY = σ XY σ X σ Y r XY = σ XY σ X σ Y [ X mx = E σ X Y m Y σ Y ]. X, Y., X = X m X σ X, Ỹ = Y m Y σ Y r XY = σ XỸ = r XỸ
11.5. 43 11.2 1 r xy 1. r xy > 0, r xy < 0., r xy > 0.8, r xy < 0.2. 11.2 (1) : σ xy = 19.96, r xy = 0.64 (2) : σ xy = 19.97, r xy = 0.43 11.5 (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
44 11 a, b : a = σ xy σ 2 x, b = ȳ a x 11.3 (1) : y = 0.72x 62.79 (2) : y = 0.69x 54.60 37 X, Y, a > 0, r XY = r ax,y. 38 2 ( ) X, ( ) Y. (1) X. (2) Y. (3) X, Y. (4) X, Y. 1. = 7 24 = 7 25 = 7 26 2... 3. 1,. 4. ( ),. 5..,.
45 12 2 Karl Pearson (1857 1936) 12.1 2 1 ( 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 ( )
46 12 2 12.2. 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. 12.1 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 1 2. 12.1, 120.? 1 2 3 4 5 6 24 18 16 22 23 17 120 12.1,. 4 : 3 : 2 : 1.,? A O B AB 47 23 21 9 100 39 150, 5 3868, 5.,, 5 1:1? [ ] : 0:5 1:4 2:3 3:2 4:1 5:0 92 603 1137 1254 657 125 3868 40, 45, 55.? (1) (2), 2.
12.3. 47 12.3 12.2 2 A = {A 1,..., A r }, B = {B 1,..., B s }, χ 2 = n r i=1 s j=1 ( Xij n X i n X i n X j n ) 2 X j n, n (X ij 5), (r 1)(s 1) 2. 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 12.2.? 21 102 123 28 49 77 49 151 200 41 1.? 24 25 35 36 2 37 155 78 270 2 3 24 59 25 108 3 29 56 77 162 90 270 180 540.
48 12 2 : P (χ 2 n χ 2 n(α)) = α α χ n α n\α 0.995 0.99 0.975 0.95 0.05 0.025 0.01 0.005 1 0.0 4 393 0.0 3 157 0.0 3 982 0.0 2 393 3.841 5.024 6.635 7.879 2 0.010 0.020 0.051 0.103 5.991 7.378 9.210 10.597 3 0.072 0.115 0.216 0.352 7.815 9.348 11.345 12.838 4 0.207 0.297 0.484 0.711 9.488 11.143 13.277 14.860 5 0.412 0.554 0.831 1.145 11.070 12.833 15.086 16.750 6 0.676 0.872 1.237 1.635 12.592 14.449 16.812 18.548 7 0.989 1.239 1.690 2.167 14.067 16.013 18.475 20.278 8 1.344 1.646 2.180 2.733 15.507 17.535 20.090 21.955 9 1.735 2.088 2.700 3.325 16.919 19.023 21.666 23.589 10 2.156 2.558 3.247 3.940 18.307 20.483 23.209 25.188 11 2.603 3.053 3.816 4.575 19.675 21.920 24.725 26.757 12 3.074 3.571 4.404 5.226 21.026 23.337 26.217 28.300 13 3.565 4.107 5.009 5.892 22.362 24.736 27.688 29.819 14 4.075 4.660 5.629 6.571 23.685 26.119 29.141 31.319 15 4.601 5.229 6.262 7.261 24.996 27.488 30.578 32.801 16 5.142 5.812 6.908 7.962 26.296 28.845 32.000 34.267 17 5.697 6.408 7.564 8.672 27.587 30.191 33.409 35.718 18 6.265 7.015 8.231 9.390 28.869 31.526 34.805 37.156 19 6.844 7.633 8.907 10.117 30.144 32.852 36.191 38.582 20 7.434 8.260 9.591 10.851 31.410 34.170 37.566 39.997 21 8.034 8.897 10.283 11.591 32.671 35.479 38.932 41.401 22 8.643 9.542 10.982 12.338 33.924 36.781 40.289 42.796 23 9.260 10.196 11.689 13.091 35.172 38.076 41.638 44.181 24 9.886 10.856 12.401 13.848 36.415 39.364 42.980 45.559 25 10.520 11.524 13.120 14.611 37.652 40.646 44.314 46.928 26 11.160 12.198 13.844 15.379 38.885 41.923 45.642 48.290 27 11.808 12.879 14.573 16.151 40.113 43.195 46.963 49.645 28 12.461 13.565 15.308 16.928 41.337 44.461 48.278 50.993 29 13.121 14.256 16.047 17.708 42.557 45.722 49.588 52.336 30 13.787 14.953 16.791 18.493 43.773 46.979 50.892 53.672 40 20.707 22.164 24.433 26.509 55.758 59.342 63.691 66.766 50 27.991 29.707 32.357 34.764 67.505 71.420 76.154 79.490 60 35.534 37.485 40.482 43.188 79.082 83.298 88.379 91.952 70 43.275 45.442 48.758 51.739 90.531 95.023 100.425 104.215 80 51.172 53.540 57.153 60.391 101.879 106.629 112.329 116.321 90 59.196 61.754 65.647 69.126 113.145 118.136 124.116 128.299 100 67.328 70.065 74.222 77.929 124.342 129.561 135.807 140.169 4 (n = 1 ).
49 13 13.1 William Feller (1906 1970) 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 = 50 100. (1). (2) S n.
50 13 13.2 15 15 15 10 10 10 5 5 5 0-5 20 40 60 80 100 0-5 20 40 60 80 100 0-5 20 40 60 80 100-10 -10-10 -15-15 -15 13.3 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,. 13.1 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). 42 ( ) P ( S 100 15).
13.4. 51 13.4 {S n } 2n ( 0) ( ) L 2n = max{0 m 2n ; S m = 0}. 13.2. P (L 2n = 2k) = ( 2k k )( 2n 2k n k ) ( 1 4 ) n, k = 0, 1, 2,..., n, (13.1). (.) 13.1: : L 100, : 13.3 ( ) L 2n, ( ) 1 lim P n 2n L 2n a = 1 a dx = 2 π x(1 x) π arcsin a, 0 a 1, (13.2). 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 (13.3). 13.2, ( )( ) ( ) n 2k 2n 2k 1 np (L 2n = 2k) = n k n k 4 n π k(n k)
52 13, (13.3), 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) b lim P (2an L 2n 2bn) = n a 1 ( k=k 1 k π dx n π x(1 x) 1 k n ) 1 n (13.4). (..) 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 0.9 0.21 1 k 2n, (k 1, S k 1 ) (k.s k ) x k H 2n, H 2n L 2n ( 13.2).
( 1, ) (2013.07.24) [1] [6]. [7] [8] 1. (100 ).. (,, ).,. [1] ( ) 10 2 (1 ). 3, 1, 2, 3. (10 ) [2] ( ) O R 1, O X. X F (x) = P (X x).,, X,,. (20 ) [3] ( ). (10 ) (1) X N(4, 5 2 ) P (X 5.6). (2), 10%., 65, 8.,. [4] ( ) 100 A. 600 22.1%,, 95% 22.1 ± 3.3%. (20 ) (1)?. (2) 95%, 22.1% ± 0.33%,?. [5] ( ), A 100 4. B, 90%, 5%.. (%, 1 ). (10 )
[6] ( ) 400 222.? 5%. 1%? 2. (10 ) [7] ( ) 2 X, Y.,, X = Y. (20 ) (1) P (X = 4). (2) P (Y 2 X 5). (3) X E[X]. (4) X, Y σ XY = E[(X E[X])(Y E[Y ])]. [8] ( ) 400, 215. 5% 2., 2 5%. (20 ) P = 1 z e x2 /2 dx 2π z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359 0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753 0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141 0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517 0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879 0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224 0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549 0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852 0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133 0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389 1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621 1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830 1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015 1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177 1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319 1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441 1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545 1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633 1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706 1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767 2.0 0.4773 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817 2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857 2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890 2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916 2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936 2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952 2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964 2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974 2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981 2.9 0.4981 0.4982 0.4983 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986 3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990 0
(2013.07.24 ) [1] 1 A, 2 B, 3 C.,,.. 1, P (A) = 2 10 P (A c ) = 8 10. P (B A) = 1 9, P (B Ac ) = 2 9, P (B) = P (B A)P (A) + P (B A c )P (A c ) = 1 9, 2, 2 10 + 2 9 8 10 = 18 90 = 2 10 = P (A)., P (C A B) = 0, P (C A B c ) = P (C A c B) = 1 8, P (C Ac B c ) = 2 8 P (C) = P (C A B)P (A B) + P (C A B c )P (A B c ) + P (C A c B)P (A c B) + P (C A c B c )P (A c B c ) = 1 8 2 10 8 9 + 1 8 8 10 2 9 + 2 8 8 10 7 9 = 144 720 = 2 10 = P (A),. [2] x X 0 X R, x < 0 F (x) = 0, x R F (x) = 1., 0 x R. X x 1 O x, O x., F (x) = P (X x) = πx2 πr 2 = x2 R 2., 0, x 0, x F (x) = 2 R 2. 0 x R, 1, x R.
: 2x f(x) = R 2. 0 x R, 0,., σ 2 = R 0 m = x 2 f(x)dx m 2 = R 0 R 0 xf(x)dx = 2 3 R 2x 3 ( ) R 2 dx m2 = R2 2 2 2 3 R = 1 18 R2 [3] (1) X N(4, 5 2 ) Z = X 4 N(0, 1), 5 ( X 4 P (X 5.6) = P 5.6 4 ) = P (Z 0.32) = 0.5 + 0.1255 = 0.6255 5 5 (2) X N(65, 8 2 ), P (X a) = 0.1 a., Z N(0, 1), P (Z 1.28) = 0.1., Z = X 65 8 1.28 X 75.2, 76. [4] (1). (2) 95% X ± 1.96 σ n, 1/10 100. [5] A P (A) = 4 100, P (Ac ) = 96 100. B,,, P (B A) = 0.9 P (B A c ) = 0.05 P (A)P (B A) P (A B) = P (A)P (B A) + P (A c )P (B A c ) = 4 100 4 100 0.9 96 0.9 + 100 0.05 = 3.6 = 0.428 43%. 3.6 + 4.8
[6] H 0 : p = 1 2, H 1 : p 1, α = 0.05. 400 2 X, X B(400, 1/2) N(200, 10 2 ). H 1 W : x 200 10 1.96 x = 222 W., 5% H 0,. α = 0.01, W : x 200 10 2.58, x = 222 W., 1% H 0,. [7] 2 X, Y. 1 i, 2 j,, Ω = {ω = (i, j) ; i, j {1, 2, 3, 4, 5, 6}}, P ({ω}) = P ({(i, j)}) = 1 36. X, Y Ω, X : Ω ω = (i, j) max{i, j}, Y : Ω ω = (i, j) min{i, j},. X, Y,. i\j 1 2 3 4 5 6 1 1 2 3 4 5 6 2 2 2 3 4 5 6 3 3 3 3 4 5 6 4 4 4 4 4 5 6 5 5 5 5 5 5 6 6 6 6 6 6 6 6 X(i, j) Y (i, j) (1) {X = 4} 7, i\j 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 2 2 2 3 1 2 3 3 3 3 4 1 2 3 4 4 4 5 1 2 3 4 5 5 6 1 2 3 4 5 6 P (X = 4) = 7 36 (2), {X 5} = {X 5, Y 2} = (1, 5) (1, 6) (2, 5) (2, 6) (3, 5) (3, 6) (4, 5) (4, 6) (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) (5, 1) (5, 2) (6, 1) (6, 2) (1, 5) (1, 6) (2, 5) (2, 6),
, (3) X. P (Y 2 X 5) = P (Y 2, X 5) P (X 5) = 8/36 20/36 = 2 5. E(X) = k 1 2 3 4 5 6 P (X = k) 1/36 3/36 5/36 7/36 9/36 11/36 6 kp (X = k) k=1 = 1 1 36 + 2 3 36 + 3 5 36 + 4 7 36 + 5 9 36 + 6 11 36 = 161 36 (4) σ XY = E[XY ] E[X]E[Y ]., XY = ij i, j, E[XY ] = E[ij] = E[i]E[j] = 7 2 7 2 = 49 4 (3),, E(Y ) = 91 36 σ XY = 49 4 161 36 91 49 9 36 161 91 15876 14651 15876 14651 = 36 36 2 = 36 2 = 36 2 = 1225 36 2 [8] H 0 : p = 1 2, H 1 : p 1, α = 0.05. 400 2 X, X B(400, 1/2) N(200, 10 2 ). H 1 W : x 200 10 1.96 x = 215 W., 5% H 0,.,, 2., p, 2., p = 0.525 ( ) p = 0.6 ( ). p = 0.525, p = 0.5,, β 0.5.,,, ( 2 ). p = 0.6, 2, β 0.025., β = 0.05 p 0.6., 10 2, 219.6 + 1.64 10 = 236 p = 0.59., (p = 0.5), p 0.59,, 2 0.05.
p p p p β β
( 2, ) (2013.07.25) [1] [7]. [8] [9] 1. (100 ).. (,, ).,. [1] ( ) L 1 2., 1.5. (10 ) [2] ( ) 2 X. (, X.) X. (10 ) [3] ( ) 2 E, F, P (E) = 1 3, P (F ) = 1 2, P (E F ) = 2 3., E c E. (10 ). (1) P (E c F ). (2) P (E F c ). [4] ( ). (10 ) (1) X N(2, 3 2 ) P (X a) = 0.123 a. (2) (6 ) 720, 1 135. [5] ( ) 100 A. 600 22.1%,, 95% 22.1 ± 3.3%.. (20 ) (1),. (2) 99%,,. (3) 95% 1/10 22.1% ± 0.33% 10. (4) 10 1000, 95% 10. [6] ( ) 400 224.? 5%. 1%? 2. (10 )
[7] ( ), A 100 3. B, 90%, 5%.. (%, 1 ). (10 ) [8] ( ) 400, 215. 5% 2., 2 5%. (20 ) [9] ( ) O 1 1, O X. X F (x) = P (X x).,,,,. (20 ) P = 1 z e x2 /2 dx 2π z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359 0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753 0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141 0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517 0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879 0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224 0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549 0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852 0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133 0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389 1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621 1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830 1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015 1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177 1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319 1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441 1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545 1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633 1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706 1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767 2.0 0.4773 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817 2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857 2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890 2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916 2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936 2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952 2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964 2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974 2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981 2.9 0.4981 0.4982 0.4983 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986 3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990 0
(2013.07.25 ) [1] 2/5 2/5, 4/5. [2] 1 i, 2 j,, Ω = {ω = (i, j) ; i, j {1, 2, 3, 4, 5, 6}}, P ({ω}) = P ({(i, j)}) = 1 36. X = min{i, j},... i\j 1 2 3 4 5 6 1 1 1 1 1 1 1 2 1 2 2 2 2 2 3 1 2 3 3 3 3 4 1 2 3 4 4 4 5 1 2 3 4 5 5 6 1 2 3 4 5 6 k 1 2 3 4 5 6 P (X = k) 11/36 9/36 7/36 5/36 3/36 1/36, E[X] = 6 kp (X = k) k=1 = 1 11 36 + 2 9 36 + 3 7 36 + 4 5 36 + 5 3 36 + 6 1 36 = 91 36. V[X] = E[X 2 ] E[X] 2 = 301 36 ( ) 91 2 = 2555 36 36 2. [3] E F E F (1) P (E F ) = P (E) + P (F ) P (E F ),,, 2 3 = 1 3 + 1 P (E F ). 2 P (E F ) = 1 6.
, (2), P (E c F ) = P (F ) P (E F ) = 1 2 1 6 = 1 3. P (E F c ) = P (E) P (E F ) = 1 3 1 6 = 1 6., P (E F c ) = P (E F c ) P (F c ) = 1/6 1/2 = 1 3 [4] (1), Z N(0, 1), P (Z 1.16) = 0.123 = 0.5 0.377. N(0,1),, a = 1.48. X 2 3 = Z 1.16 X 1.48 (2) 720 1 B(720, 1/6) N(120, 10 2 ).,, ( ) ( ) X 120 134.5 120 X 120 P (X 135) = P (X 134.5) = P = P 1.45 10 10 10,, 0.5 0.4265 = 0.0735. [5] X ± z σ n,., 95% z = 1.96, 99% z = 2.58. (1) 95%,. (2) 99%, 99%,,. (3) 2z σ n., 1/10 100. (4),.
[6] H 0 : p = 1 2, H 1 : p 1, α = 0.05. 400 2 X, X B(400, 1/2) N(200, 10 2 ). H 1 W : x 200 10 1.96 x = 224 W., 5% H 0,. α = 0.01, W : x 200 10 2.58, x = 224 W., 1% H 0,. [7] A P (A) = 3 100, P (Ac ) = 97 100. B,,,, 36%. P (B A) = 0.9 P (B A c ) = 0.05 P (A)P (B A) P (A B) = P (A)P (B A) + P (A c )P (B A c ) = 3 100 3 100 0.9 97 0.9 + 100 0.05 = 2.7 2.7 + 4.85 = 2.7 7.55 = 0.3576... [8] H 0 : p = 1 2, H 1 : p 1, α = 0.05. 400 2 X, X B(400, 1/2) N(200, 10 2 ). H 1 W : x 200 10 1.96 x = 215 W., 5% H 0,.,, 2., p, 2., p = 0.525 ( ) p = 0.6 ( ). p = 0.525, p = 0.5,, β 0.5.,,, ( 2 ). p = 0.6, 2, β 0.025., β = 0.05 p 0.6., 10 2, 219.6 + 1.64 10 = 236 p = 0.59., (p = 0.5), p 0.59,, 2 0.05.
p p p p β β [9] x X 0 X 1, x < 0 F (x) = 0, x 1 F (x) = 1., 0 x 1. X x 1 O x, O x., F (x) = P (X x) = πx2 π = x2., : 0, x 0, F (x) = x 2, 0 x 1, 1, x 1. f(x) = { 2x, 0 x 1, 0,., σ 2 = 1 0 m = 1 x 2 f(x)dx m 2 = 0 xf(x)dx = 2 3. 1 0 2x 3 dx m 2 = 1 2 ( 2 3 ) 2 = 1 18
( 2, ) (2013.07.26) [1] [6]. [7] [8] 1. (100 ).. (,, ).,. [1] ( ) A,B 2. A 2/5, B 3/5. 5, 10000. A 4, B 2.? (10 ) [2] ( ) 2 X. (, X.) X. (10 ) [3] ( ). (20 ) (1) (6 ) 720, 1 105. (2), 5%., 68, 8.,. [4] ( ), A 100 2. B, 90%, 5%... (10 ) [5] ( ) 100 A. 600 22.1%,, 95% 22.1 ± 3.3%. (20 ) (1)?. (2) 95%, 22.1% ± 0.33%,?.
[6] ( ) 400 224.? 5%. 1%? (10 ) [7] ( ) 400, 215. 5% 2., 2 5%. (20 ) [8] ( ) O 1 1, O X. X F (x) = P (X x).,,,,. (20 ) P = 1 z e x2 /2 dx 2π z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359 0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753 0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141 0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517 0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879 0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224 0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549 0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852 0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133 0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389 1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621 1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830 1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015 1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177 1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319 1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441 1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545 1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633 1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706 1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767 2.0 0.4773 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817 2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857 2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890 2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916 2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936 2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952 2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964 2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974 2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981 2.9 0.4981 0.4982 0.4983 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986 3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990 0
(2013.07.26 ) [1] A 1, B 3,, 98 : 27. P (A) = 2 5 + 3 5 2 5 + 3 5 3 5 2 50 + 30 + 18 = = 98 5 125 125 P (B) = 3 5 3 5 3 5 = 27 125 A : 10000 98 27 = 7840, B : 10000 125 125 = 2160 [2] 1 i, 2 j,, Ω = {ω = (i, j) ; i, j {1, 2, 3, 4, 5, 6}}, P ({ω}) = P ({(i, j)}) = 1 36. X = min{i, j},... i\j 1 2 3 4 5 6 1 1 1 1 1 1 1 2 1 2 2 2 2 2 3 1 2 3 3 3 3 4 1 2 3 4 4 4 5 1 2 3 4 5 5 6 1 2 3 4 5 6 k 1 2 3 4 5 6 P (X = k) 11/36 9/36 7/36 5/36 3/36 1/36,, E[X] = E[X 2 ] = 6 kp (X = k) k=1 = 1 11 36 + 2 9 36 + 3 7 36 + 4 5 36 + 5 3 36 + 6 1 36 = 91 36. 6 k 2 P (X = k) k=1 = 1 2 11 36 + 22 9 36 + 32 7 36 + 42 5 36 + 52 3 36 + 62 1 36 = 301 36. V[X] = E[X 2 ] E[X] 2 = 301 36 ( ) 91 2 = 2555 36 36 2.
[3] (1) 720 1 X, X B(720, 1/6) N(120, 10 2 )., Z = X 120 N(0, 1)., 10 ( ) X 120 105.5 120 P (X 105) = P (X 105.5) = P = P (Z 1.45). 10 10,, 0.5 0.4265 = 0.0735. (2) X N(68, 8 2 ), P (X a) = 0.05 a., Z N(0, 1), P (Z 1.64) = 0.05., Z = X 68 8 1.64 X 81.12, 82. [4] A P (A) = 2 100, P (Ac ) = 98 100. B,,, P (B A) = 0.9 P (B A c ) = 0.05 P (A)P (B A) P (A B) = P (A)P (B A) + P (A c )P (B A c ) = 2 100 2 100 0.9 98 0.9 + 100 0.05 = 1.8 1.8 + 4.9 = 2.7 6.7 = 0.402... = 40% [5] (1). (2) 95% X ± 1.96 σ n, 1/10 100. [6] H 0 : p = 1 2, H 1 : p 1, α = 0.05. 400 2 X, X B(400, 1/2) N(200, 10 2 ). H 1 W : x 200 10 1.96 x = 224 W., 5% H 0,. α = 0.01, W : x 200 10 2.58
, x = 224 W., 1% H 0,. [7] H 0 : p = 1 2, H 1 : p 1, α = 0.05. 400 2 X, X B(400, 1/2) N(200, 10 2 ). H 1 W : x 200 10 1.96 x = 215 W., 5% H 0,.,, 2., p, 2., p = 0.525 ( ) p = 0.6 ( ). p = 0.525, p = 0.5,, β 0.5.,,, ( 2 ). p p p p β β p = 0.6, 2, β 0.025., β = 0.05 p 0.6., 10 2, 219.6 + 1.64 10 = 236 p = 0.59., (p = 0.5), p 0.59,, 2 0.05. [8] x X 0 X 1, x < 0 F (x) = 0, x 1 F (x) = 1., 0 x 1. X x 1 O x, O x., F (x) = P (X x) = πx2 π = x2.
, 0, x 0, F (x) = x 2, 0 x 1, 1, x 1. : f(x) = { 2x, 0 x 1, 0,., σ 2 = 1 0 m = 1 x 2 f(x)dx m 2 = 0 xf(x)dx = 2 3. 1 0 2x 3 dx m 2 = 1 2 ( 2 3 ) 2 = 1 18