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2 1 1 ( ) ( % mm % A B A B A 1
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5 Excel 3 Excel 3.1 sum,mean,max,min,round,if... Σ [ ] 4
6 SUM =sum( =sum(a1:a5) AVERAGE sum MAX MIN 1 ROUND =round( ) =round( ,2) =round(b1,3) 1 5
7 IF =if( =if(a1>2,1,0) =if(a1=1,, ) 3.2 > > > > > > > 2 DATA01 2 = ( 2 /10000) 22 x ( 5 <= x <= 5 abs(x) <= 5) (x < 5 x > 5) 3 Word 2 3 6
8 3.3 DATA02 > > ( D)=Sheet1!A1:E7 =a1:e7) > > > 7
9 ( D)=Sheet1!A1:E7 =a1:e7) 8
10 専 攻 人 数 クラス1 クラス2 クラス3 クラス 中 国 語 韓 国 語 英 語 フランス 語 ドイツ 語 イタリア 語 言 語 3 4 DATA DATA01 9
11 階 級 度 数 累 積 度 数 相 対 度 数 累 積 相 対 度 数 % 18% % 38% % 66% % 90% % 94% % 100% 5 / 6 / 25 ヒストグラム 20 度 数 階 級 の 下 限 7 DATA
12 5 8 ( ) 9 () 10 () x = 1 n n x i = x 1 + x x n n n (x i x) = x 3 3. {x 1, x 2,, x t } t t 11
13 3 4 x t = x t 1 + x t + x t 3 x t = x t 1 + x t + x t+1 + x t+2 4 { x 2, x 3,, x n 1 } { x 3, x 4,, x n 2 } TOPIX 4 日 付 終 値 3 項 移 動 平 均 5 項 移 動 平 均 7 項 移 動 平 均 18/04/ /04/ /04/ /04/ /04/ /04/ /04/ /04/ /04/ /04/ /04/ /04/ /03/ /03/ /03/ TOPIX 4 Yahoo Japan Finance TOPIX 12
14 1, , , 終 値 3 項 移 動 平 均 半 月 移 動 平 均 一 ヶ 月 移 動 平 均 1, , , /2/7 06/2/22 06/3/9 06/3/24 06/4/8 TOPIX 4. x = n x 1 x 2 x n 1 x n 3 r 1 = 23%, r 2 = 27%, r 3 = 28% 10 R = (1 + r 1 )(1 + r 2 )(1 + r 3 ) 1 1 r = 3 (1 + r 1 )(1 + r 2 )(1 + r 3 ) 1 26% 13
15 % 10 r r = R 1 = % 25% r 3 (1 + r) 3 = (1 + 26%) (1 + 26%) (1 + 26%) DATA
16 Excel median 11 15
17 12 7 X = {4, 7, 2, 30, 9, 7, 1} X = {1, 2, 4, 7, 7, 9, 30} , 90, 10, 110, mode X = {2, 3, 2, 4, 6, 4, 6, 6, 7} range 16 R = max(x) min(x) 17 X = {2, 3, 2, 4, 6, 4, 6, 6, 7} X 2 7 = 7 2 = 5 16
18 7.4 mean deviation 18 X = {x 1, x 2,, x n } d = n x i x n = x 1 x + x 2 x + + x n x n x 19 X = {x 1, x 2,, x 10 } = {2, 3, 2, 4, 6, 4, 6, 6, 3, 4} x = = d = 10 = = x 7/5 7.5 Excel max( -min( 20 max(a1:a10)-min(a1:a10) MEDIAN( ) 21 =MEDIAN(a1:a10) MODE( ) 22 =MODE(a1:a10) 17
19 8 8.1 Variance) 23 S 2 = = n (x i x) 2 n n x2 i n x 2 = (x 1 x) 2 + (x 2 x) (x n x) 2 n x P n x2 i n x 2 s 2 = n (x i x) 2 n 1 n 8.2 Standard Deviation 24 S = S 2 = n (x i x) 2 n S = s 2 = n (x i x) 2 n 1 18
20 8.3 5 Yahoo! , ), (0.003, ) (-0.003, ) 株 価 の 収 益 率 の 例 収 益 率 収 益 率 トヨダ 自 動 車 収 益 率 日 産 自 動 車 収 益 率 新 日 本 製 鐵 T 8.4 C.V. = S x 5 19
21 z i = x i S x S x x T i = zi 8.6 Excel S 2 =VARP( ) s 2 =VAR( ) S =STDEVP( ) s: =STDEV( ) 26 Yahoo
22 X Y Z S 2 = = n (x i x) 2 n n x2 i n x 2 = (x 1 x) 2 + (x 2 x) (x n x) 2 n x = = 3 3 n Sx 2 = (x i x) 2 n = ( 2) S 2 x = n x2 i n S 2 y = S 2 z = = (1 3)2 + (5 3) 2 + (3 3) 2 3 = = 2.67 x 2 = = 2.67 ( ) = ( ) = S 2 z > S 2 y > S 2 x. Z X 1, 5, 3 Z 15, 50,
23 X Y Z 28 X = {x 1, x 2,, x 100 }, Y = {y 1, y 2,, y 100 }, Z{y 1, y 2,, y 100 } 0 1, 3, DATA
24 9.1 DATA01 体 重 図 1 身 長 と 体 重 の 散 布 図 身 長 GDP( GDP cm cm 175cm 50kg kg
25 図 2 一 人 当 たりGDPと 乳 児 死 亡 率 乳 児 死 亡 率 一 人 当 たりGDP 1: GDP SNA 9.3 X Y S xy X Y ρ xy X Y S xy = 1 n = 1 n n (x i x) (y i ȳ) n x i y i xȳ 30 24
26 ρ xy = S xy S x S y = 1 n n 1 n n (x i x) (y i ȳ) n (x i x) 2 1 n (y i ȳ) 2 n (x i x) (y i ȳ) (x i x) 2 n (y i ȳ) 2 n 1 ρ
27 ρ xy = ρ xy = ρ xy = ρ xy = ID a b c d e f g h i j (g) (kg)
28 (kg) (g) 5 ρ xy = x = {4, 3, 5, 1, 5}, y = {1, 3, 3, 0, 1} x y Excel 1. =COVAR( 1 2 ) 2. =CORREL( 1 2 ) DATA
29 10.1 x, y y = a + bx x y a = 3, b = 2 y = 3 + 2x x = 0 y = = 3, x = 2 y = = 7..., 4 x y y x
30 学 籍 番 号 身 長 (cm) 体 重 (kg) 体 重 身 長 29
31 Ìd W e i g h t H e i g h t g x y x i y i, i = 1, 2, 3,..., 12. ŷ = a + bx d i = y i ŷ = y i (a + bx i ) 30
32 S 12 S = (y i (a + bx i )) 2 S = n (y i (a + bx i )) 2. (1) 35 ( ) S b = n x iy i n xȳ n x2 i n x2 (2) a = ȳ b x (3) 11 a b S a b 1 ( n n ) ( n ) S = yi 2 + na 2 + b 2 + x i 2ab x 2 i ( n ) ( n ) y i 2a x i y i 2b (4) x i y i 3 a a ( ( n ) ( n ) S = na 2 y i 2a + x i 2ab ( nȳ 2 2nb xȳ + nb 2 x 2)) + ( nȳ 2 2nb xȳ + nb 2 x 2) + ( n n yi 2 + x 2 i ) ( n ) b 2 x i y i 2b S = n(a (ȳ b x))
33 S a = ȳ b x (5) 3 b b S = n x 2 i ( b S ( n x iy i a n x )) 2 i n + x2 i n x2 i b = ( n x iy i ) a n x i (6) 5 a 6 b = ( n x iy i ) n xȳ ( n x2 i ) n x2 (7) a = ȳ b x (8) 1 { S = 0 a S = 0 b
34 i x i y i x 2 i x i y i (x iy i ) 12 x i 12 y i 12 x2 i x ȳ b = ( n x iy i ) n xȳ ( n x2 i ) = = 0.80 n x a = ȳ b x = = ŷ = x 1cm 0.8kg 36 x = {2, 5, 6, 9}, y = {4, 6, 8, 9} 33
35 x, y y = a + bx x y a = 3, b = 2 y = 3 + 2x x = 0 y = = 3, x = 2 y = = 7..., 4 x y y x
36 12.2 学 籍 番 号 身 長 (cm) 体 重 (kg) 体 重 身 長 35
37 Ìd W e i g h t H e i g h t g x y x i y i, i = 1, 2, 3,..., 12. ŷ = a + bx d i = y i ŷ = y i (a + bx i ) 36
38 S 12 S = (y i (a + bx i )) 2 S = n (y i (a + bx i )) 2. (9) 37 ( ) S b = n x iy i n xȳ n x2 i n x2 (10) a = ȳ b x (11) 13 a b S a b 1 ( n n ) ( n ) S = yi 2 + na 2 + b 2 + x i 2ab x 2 i ( n ) ( n ) y i 2a x i y i 2b (12) x i y i 3 a a ( ( n ) ( n ) S = na 2 y i 2a + x i 2ab ( nȳ 2 2nb xȳ + nb 2 x 2)) + ( nȳ 2 2nb xȳ + nb 2 x 2) + ( n n yi 2 + x 2 i ) ( n ) b 2 x i y i 2b S = n(a (ȳ b x))
39 S a = ȳ b x (13) 3 b b S = n x 2 i ( b S ( n x iy i a n x )) 2 i n + x2 i n x2 i b = ( n x iy i ) a n x i (14) 5 a 6 b = ( n x iy i ) n xȳ ( n x2 i ) n x2 (15) a = ȳ b x (16) 1 { S = 0 a S = 0 b
40 i x i y i x 2 i x i y i (x iy i ) 12 x i 12 y i 12 x2 i x ȳ b = ( n x iy i ) n xȳ ( n x2 i ) = = 0.80 n x a = ȳ b x = = ŷ = x 1cm 0.8kg 38 x = {2, 5, 6, 9}, y = {4, 6, 8, 9} 39
41 x = {4, 6, 9}, y = {6, 6, 9} x y DATA03 Excel Excel Excel OK Y X OK 40
42 概 要 回 帰 統 計 重 相 関 R 重 決 定 R 補 正 R 標 準 誤 差 観 測 数 30 分 散 分 析 表 自 由 度 変 動 分 散 観 測 された 分 散 比 有 意 F 回 帰 残 差 合 計 係 数 標 準 誤 差 t P 値 下 限 95% 上 限 95% 下 限 95.0% 上 限 95.0% 切 片 X 値 y x 7 y = a + b x + u u a 94.4 b X y = x 17, 40 A 1 A = {... } 41 A = {2, 1, 5}, B = {3, 6, 5} A B A B A B = {2, 1, 5, 3, 6} A B A B A B = {5} 41
43 / / n n m n m/n p p p 42
44 A P (A) 0 P (A) 1 2. Ω P (Ω) = 1 3. A B A B P (A B) = P (A) + P (B) A B { { 4 1 1/4 47 A = { 2 }, B = { } P {A B}? 48 {A B} = { 2 } P {A B} = 1 49 C = { 2 } D = { 2 } 50 P (C D) C = { } D = { } C D C D = φ φ P {φ} = P {B} P {A B} = P {A} + P {B} ?{ 1 2 }?{ 1 2 }? 43
45 Excel Excel P = 0.5 OK X X = 0 0.5, X = x = 1 1 6, x = 2 1 6, X 0 1 0, 1, 2, 3, 4, 5, X x 1 x 2 x 3 x 4 P (x) P (x 1 ) P (x 2 ) P (x 3 ) P (x 4 ) x i P (x) P (x 1 ), P (x 2 )... x 44
46 F (x) {X x} F (x) = P ({X x}) F (x) = P ({X x}) = x i <x P (x i ) x x i F (x 3 ) = P (X x 3 ) = P (x 1 ) + P (x 2 ) + P (x 3 ) x i X E (X) E (X) = n x i P (x i ). X V (X) V (X) = E [ (X E (X)) 2] = n [ (xi E (X)) 2 P (x i ) ]. 53 () X P (x) X P (x) { P (x) = p P (x) = 1 p x = 1 x = 0 E (X) = 1 p + 0 (1 p) = p. V (X) = (0 p) 2 (1 p) + (1 p) 2 p = p 2 p 3 + p 2p 2 + p 3 = p p 2 45
47 F(x) p = F (x) = 1 p 1 x < 0 0 x < 1 x 1 46
48 F(x) p = 0.5 p = P (x) = C x np x (1 p) n x x = 0, 1, 2,..., n. Y i 0 p X = n ( n ) E (X) = E Y i = np V (X) = V ( n ) Y i = n(p p 2 ) C x n Cn x = P n x n!/ (n x)! = Px x x! n (n 1) (n 2) (n x + 1) = x (x 1) (x 2) 3 2 Y i 47
49 F (x) = x P (x i ) n x p = P (x) = e λ λ x p = λ/n. p = λ/n x! C x np x (1 p) n x = (n 1) (n 2) n n = n!/ (n x)! x! n n (n 1) n (n 2) n ( m ) x ( 1 m ) n x n n (n (x 1)) n x! ( m x 1 m ) n ( 1 m ) x n n ) x 1 ( 1 m n ) n e m n (n (x 1)) 1, ( 1 m n n n P (x) = e λ λ x x! E (X) = λ. V (X) = λ. F (x) = x P (x i ) n p 48
50 λ = 10 P (X 2) = F (2) = = e λ λ 0 0! 2 e λ i λ x i i=0 + e λ λ 1 1! x i! + e λ λ 2 2! = e ! = e ! + e ! p = 0.3, n = 10 P (6) Probability Density Funtion PDF x x x 1000 P (c) 6 = C c np c (1 p) n c x = 0, 1, 2,..., n. (17) 6 P (c), f (c), F (c) P (x), f (x), F (x) 49
51 f (c) = 1 σ (c µ) 2 2π e 2σ 2 (18) f (x) X c c
52 21 (Cumulative Distribution Function CDF) F (c) = P (X c) = c P (x i ) = c C x i n p x i (1 p) n x i x = 0, 1, 2,..., n. (19) 51
53 F (c) = P (X c) = c 1 σ (x µ) 2 2π e 2σ 2 dx (20) k k k F (x) c c c P (c) f (c) F (c) a b a < b X a b a b X 52
54 X c d c d X F (c) = P (X c) = c P (x i ) (21) F (c) = P (X c) = c f (x) dx (22) The area of this part F ( 1) 53
55 This value F ( 1) E (X) = n x i P (x i ). X V (X) V (X) = E [ (X E (X)) 2] = n [ (xi E (X)) 2 P (x i ) ]. E (X) = V (X) = E [ (X E (X)) 2] = xf (x) dx. (x E (X)) 2 f (x) dx. 54
56 22 1. P (x) = C x np x (1 p) n x x = 0, 1, 2,..., n. (23) x F (x) = P (x i ) (24) ( n ) E (X) = E Y i = np (25) ( n ) V (X) = V Y i = n(p p 2 ). (26) 2. f (c) = 1 σ (c µ) 2 2π e 2σ 2 (27) F (c) = 1 c σ e (x µ)2 2σ 2 dx (28) 2π 55
57 µ σ 2 E (X) = V (X) = = 1 σ 2π = µ = 1 σ 2π = σ 2 xf (x) dx c xe (x µ)2 2σ 2 dx (x µ) 2 f (x) dx c (x µ) 2 e (x µ)2 2σ 2 dx Excel Excel 10 p = x i, i = 1, 2,..., 100 x i 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, Excel p = 0.3, n = 6 P (3) C Excel 56
58 : 1 59 Excel p = 0.5, n = 10 x = 0, 1, 2,..., X 0 2 (1) 2 1 (2) 1 0 (3) P (X 1) P (X 0) f(0), f(1) 61 X 2 F ( 1) F (0) P (X 1) P (X 0) 57
59 : 58
60 Excel Cn x nc x = combin (n, x) C6 3 = combin (6, 3) P (x) = Cnp x x (1 p) n x x = 0, 1, 2,..., n. (29) x F (x) = P (x i ) (30) ( n ) E (X) = E Y i = np (31) ( n ) V (X) = V Y i = n(p p 2 ). (32) 2. µ σ 2 f (c) = 1 σ (c µ) 2 2π e 2σ 2 (33) F (c) = 1 c σ e (x µ)2 2σ 2 dx (34) 2π 59
61 3. P (x) = e λ λ x x! (35) p = λ/n E (X) = λ. (36) V (X) = λ. (37) 25 p n x P (x) = C x np x (1 p) n x x = 0, 1, 2,..., n. (38) P (x) = e λ λ x λ x x! (39) 26 1 X 0 2 (1) 2 1 (2) 1 0 (3) P (X 1) 60
62 P (X 0) f(0), f(1) X P (X c) P (X c) = F (c) = 1 σ 2π c e (x µ)2 2σ 2 dx (40) c P (X c) µ = 0 σ 2 =
63 27.1 E (X) = µ V (X) = σ 2 X X Z = X µ σ E (Z) = 0 V (Z) = 1 63 X µ = 0 σ 2 = 1 X 1.64 F (1.64) P (X 1.64) P (0 < X 1.64) X 1.64 P (X 1.64) = 0.95 P (0 < X 1.64) = P (X 1.64) P (X 0) P (X 0) P (X 1.64) P (0 < X 1.64) = P (X 1.64) P (X 0) = = X µ = 1 σ 2 = 4 X 4.28 µ = 1 σ 2 = 4 X µ = 0 σ 2 = 1 Z Z = X µ σ 2 = X 1 2 Z x = 4.38 x X z = x 1 = = P (X 4.28) = P (Z 1.64) =
64 28.1 X {X 1, X 2, X 3,..., X n } X E (X) µ V ar (X) = σ 2 65 (at random ) µ n X = 1 n X i σ 2 S 2 = 1 n 1 n (X i X) 2 X: 100 {X 1, X 2, X 3,..., X 100 } E (X) µ µ X = X i σ 2 S 2 = (X i X) 2 66 ( ) Γ γ E (γ) = Γ γ Γ 63
65 X S 2 µ σ 2 X E ( ( ) 1 n X) = E X i n = 1 n E (X i ) n X i X E (X i ) = µ = 1 n µ = µ. n ( ) X i µ σ 2 lim n X = µ. 68 ( ) Γ γ lim n γ = Γ γ Γ X µ
66 H 0 H H 0 H 1 : ( ) X i µ σ 2 n n X n( X µ) d N(0, 1). σ 31.3 σ σ σ
67 70 X 100 X = 175 σ = 10 µ 170 H 0 µ = 170 H 1 µ > 170 X µ = 170 σ = 10 N (170, 100) X 175 α 1% 5% µ = 170 n X Z 0 Z 0 1. H 0 µ = 170 H 1 µ > α α = 5% 3. α Z P (Z > Z ) = α Z P (Z > Z ) = 5% Z P (Z Z ) = 1 5% Z Z = Z 0 = n ( X µ ) /σ Z 0 = 100 ( ) /10 = 5 66
68 5. Z 0 Z Z 0 > Z Z 0 Z Z 0 = 5, Z = 1.65 Z 0 > Z µ = 170 µ > 170 Z 0 Z µ = % σ 2 = 9 25 X = 12 H 0 µ = 10 H 1 µ > 10 67
69 DATA01 µ 160 cm σ = σ 100 X = 3.2 cm σ 2 = 4 µ = 3 µ σ 2 X µ = 3 1. H 0 µ = 3 cm H 1 µ > 3 cm µ < 3 cm µ 3cm 2. α 1/2 α α = 5% 1/2 α = 2.5% 3. 1/2 α Z P (Z > Z ) = 1/2 α Z P (Z > Z ) = 2.5% Z P (Z Z ) = 1 2.5% z Z = Z 0 = n ( X µ ) /σ Z 0 = 100 (3.2 3) /2 = 1 68
70 5. Z 0 Z Z 0 > Z Z 0 < Z Z Z 0 Z Z 0 = 1, Z = 1.96 Z 0 < Z µ = 3 Z 0 Z Z µ = % 2.5%
71 , 9.3, 8, 7, 8.9, 9.8, 9.3, 9.2, 9, 8.9 α = 1% σ 2 = 4 37 σ n 37.1 σ σ n( X µ) σ d N(0, 1). σ σ σ s n( X µ) s n 1( X µ) s d N(0, 1) d t (n 1). 70
72 t (n 1) n 1 t n 1( X µ) t t n 1 t s 37.2 t t t t 3 t 37.3 σ σ t t σ 71 X 25 71
73 µ H 0 µ = 170 H 1 µ > α α = 5% 3. υ = n 1 υ = 25 1 = α n 1 t t t P (t > t ) = α t 24 t P (t > t ) = 5% t t = X s s 2 = n (x i x) 2 n 1 X = 175, s 2 = t 0 = n 1 ( X µ ) /s t 0 = 24 ( ) / t 0 4 t t 0 > t t 0 t t 0 = 2.5, t = 1.71 t 0 > t µ = 170 µ > cm 72
74 H 0 µ = µ 0 H 1 µ µ 0 2. α 1/2 α 3. υ = n 1 υ = 25 1 = /2 α n 1 t t t P (t > t ) = α t 5. X s s 2 = n (x i x) 2 n 1 6. t 0 = n 1 ( X µ ) /s 7. t 0 t t 0 > t t 0 < t t 0 t t 0 t 72 DATA01 9 µ > σ 73
75 100 X = 3.8 cm σ 2 = 4 µ = 3 µ σ 2 X µ = 3 1. H 0 µ = 3 cm H 1 µ > 3 cm µ < 3 cm µ 3cm 2. α 1/2 α α = 5% 1/2 α = 2.5% 3. 1/2 α Z P (Z > Z ) = 1/2 α Z P (Z > Z ) = 2.5% Z P (Z Z ) = 1 2.5% z Z = Z 0 = n ( X µ ) /σ Z 0 = 100 (3.2 3) /2 = 1 5. Z 0 z Z 0 > Z Z 0 < Z Z 0 Z Z 0 Z Z 0 = 1, Z = 1.96 Z 0 < Z µ = 3 Z 0 Z Z 74
76 µ = % 2.5% σ 4 10 cm, 12 cm, 15 cm, 9 cm 13 cm X = {3, 6, 9} Y = {2, 3, 8} 74 75
77 % [1] ˆb Y i = a + b X i (41) / n t ˆb (x i x) 2 t n 2 t Excel t 41.1 DATA03 Excel Excel 76
78 概 要 回 帰 統 計 重 相 関 R 重 決 定 R 補 正 R 標 準 誤 差 観 測 数 30 分 散 分 析 表 自 由 度 変 動 分 散 観 測 された 分 散 比 有 意 F 回 帰 残 差 合 計 係 数 標 準 誤 差 t P 値 下 限 95% 上 限 95% 下 限 95.0% 上 限 95.0% 切 片 X 値 y x 7 y = a + b x + u u a 94.4 b X y = x x t t p t p p [1] 2002) 77
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1 ( S/E) 006 7 30 0 (1 ) 01 Excel 0 7 3 1 (-4 ) 5 11 5 1 6 13 7 (5-7 ) 9 1 1 9 11 3 Simplex 1 4 (shadow price) 14 5 (reduced cost) 14 3 (8-10 ) 17 31 17 3 18 33 19 34 35 36 Excel 3 4 (11-13 ) 5 41 5 4
2 Excel =sum( ) =average( ) B15:D20 : $E$26 E26 $ =A26*$E$26 $ $E26 E$26 E$26 $G34 $ E26 F4
1234567 0.1234567 = 2 3 =2+3 =2-3 =2*3 =2/3 =2^3 1:^, 2:*/, 3:+- () =2+3*4 =(2+3)*4 =3*2^2 =(3*2)^2 =(3+6)^0.5 A12 =A12+B12 ( ) ( )0.4 ( 100)0.9 % 1 2 Excel =sum( ) =average( ) B15:D20 : $E$26 E26 $ =A26*$E$26
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確率論と統計学の資料
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【補足資料】確率・統計の基礎知識
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