I [ ] N(µ, σ 2 ) σ 2 (X 1,..., X n ) X := 1 n (X X n ): µ X N(µ, σ 2 /n) Z = X µ σ/ n N(, 1) < α < 1/2 Φ(z) =.5 α z α

Size: px
Start display at page:

Download "3 3.3. I 3.3.2. [ ] N(µ, σ 2 ) σ 2 (X 1,..., X n ) X := 1 n (X 1 + + X n ): µ X N(µ, σ 2 /n) 1.8.4 Z = X µ σ/ n N(, 1) 1.8.2 < α < 1/2 Φ(z) =.5 α z α"

Transcription

1 : : 2 : ( ): : ( ): : : : ( ) ( ) ( ) : ( pp ) : 2.2. ( ). i X i (i = 1, 2,..., n) X 1, X 2,..., X n X i (X 1, X 2,..., X n ) ( ) n (x 1, x 2,..., x n ) (X 1, X 2,..., X n ) : X 1, X 2,..., X n 1

2 I [ ] N(µ, σ 2 ) σ 2 (X 1,..., X n ) X := 1 n (X X n ): µ X N(µ, σ 2 /n) Z = X µ σ/ n N(, 1) < α < 1/2 Φ(z) =.5 α z α z z(α) α α z(α) II : z(.5) = z(.25) = 1.96 z(.5) = Z = X µ σ/ n N(, 1) II z(α/2) P ( Z < z(α/2)) = 1 α P ( X µ σ/ n ) < z(α/2) = 1 α 2

3 X µ X σ n z(α/2) < µ < X + σ n z(α/2) 1 α 1 α ( ) 1 (1 α)% µ µ 1 (1 α)% [ : ( )] σ 2 N(µ, σ 2 ) n (X 1,..., X n ) X µ 1(1 α)% X σ z(α/2) < µ < X + σ z(α/2) n n 1. (p.91) N(µ, (5.54) 2 ) cm µ 95% ( ) n = 5 σ = 5.54 X = α =.5 II z(α/2) = z(.25) = 1.96 : < µ < < µ < % : % µ 5 µ 3.4. II [ ] (n 5) ( 1.8.5) σ 2 S 2 ( ) (X 1,..., X n ) σ 2 σ : S 2 := 1 n 1 {(X 1 X) (X n X) 2 } : S := S ( ) S 2 = 1 n 1 {(X X 2 n) n X 2 } 3

4 ( ) (X 1 X) (X n X) 2 = X X 2 n 2(X X n ) X + n X 2 = X X 2 n 2n X 2 + n X 2 = X X 2 n n X ( ) S 2 E(S 2 ) = σ 2 σ 2 n ( ) (X 1,..., X n ) n µ E(X i ) = µ (i = 1, 2,..., n) V (X i ) = σ 2 (i = 1, 2,..., n) E(X 2 i ) = V (X i ) + {E(X i )} 2 = σ 2 + µ 2 (1) X = (X X n )/n X 1,..., X n E( X) = 1 n {E(X 1) + + E(X n )} = 1 n (nµ) = µ (1), (2) (3) E(S 2 ) = 1 n 1 V ( X) = 1 n 2 {V (X 1) + + V (X n )} = 1 n 2 (nσ2 ) = σ2 n E( X 2 ) = V ( X) + {E( X)} 2 = σ2 n + µ2 (2) E(S 2 ) = 1 n 1 {E(X2 1) + + E(X 2 n) ne( X 2 )} (3) { ( σ n(σ 2 + µ 2 2 ) n n + µ2 )} = 1 n 1 (n 1)σ2 = σ n n 1 [ : ( )] ( ) n ( 5) X S 2 µ 1 (1 α)% X S n z(α/2) < µ < X + S n z(α/2) 2. (p.94) g 23g 4

5 99% ( ) (n = 12) X = 348 S = 23 α =.1 II z(α/2) = z(.5) = : < µ < < µ < [ : χ 2 ] χ 2 X 1, X 2,..., X n N(, 1) χ 2 := X X X 2 n n χ 2 1 p χ 2 n (x) = 2 n/2 Γ(n/2) xn/2 1 e x/2 (x > ) (x ) Γ(s) := e x x s 1 dx (s > ) χ 2 n (p ) < α < 1 P (χ 2 x) = α x χ 2 n(α) χ 2 α α χ 2 ( 4, p.194 ) 5

6 1. (χ 2 α ) χ 2 a, b (1) χ 2 1 χ 2 P (χ 2 a) =.1 (2) χ 2 15 χ 2 P (χ 2 < 25.) = b ( ) (1) χ 2 a = χ 2 1 (.1) = (2) χ2 15 (α) = 25. α χ2 α =.5 b = 1 α = 1.5 = (χ 2 ) E(χ 2 ) = n V (χ 2 ) = 2n n χ 2 χ 2 ( ) E(χ 2 ) = 1 2 n/2 Γ(n/2) x x n/2 1 e x/2 dx x x n/2 1 e x/2 dx = E(χ 2 ) = E((χ 2 ) 2 ) = (2y) n/2 e y (2dy) (y = x/2 ) ( ) n + 2 = 2 n/2+1 y (n+2)/2 1 e y dy = 2 n/2+1 Γ 2 = 2 n/2 2 n ( n ) 2 Γ n/2 Γ(n/2) 2n/2 nγ(n/2) = n 1 2 n/2 Γ(n/2) x 2 x n/2 1 e x/2 dx 6

7 x 2 x n/2 1 e x/2 dx = E((χ 2 ) 2 ) = (2y) n/2+1 e y (2dy) (y = x/2 ) = 2 n/2+2 y n/2+1 e y dy = 2 n/2+2 y (n+4)/2 1 e y dy ( ) n + 4 = 2 n/2 4Γ = n ( n ) = 2 n/2 n(n + 2)Γ 2 ( n + 2 Γ 2 ) = n ( n ) 2 n/2 Γ(n/2) 2n/2 n(n + 2)Γ = n(n + 2) 2 V (χ 2 ) = E((χ 2 ) 2 ) {E(χ 2 )} 2 = n 2 + 2n n 2 = 2n n ( n ) 2 Γ ( n χ 2 ) n n χ 2 (X 1, X 2,..., X n ) N(µ, σ 2 ) χ 2 := 1 n σ 2 (X i µ) 2 i= ( ) n χ 2 := (X 1, X 2,..., X n ) N(µ, σ 2 ) X := 1 n S 2 := 1 n 1 n X i, i=1 n (X i X) 2 i=1 (n 1)S2 σ 2 = 1 σ 2 n (X i X) 2 n 1 χ 2 µ [ ] (X 1, X 2,..., X n ) N(µ, σ 2 ) (µ ) n χ 2 = 1 n σ 2 (X i µ) 2 n χ i=1 i=1 χ 2 χ 2 n(α/2) χ 2 n(1 α/2) P ( χ 2 n(1 α/2) < χ 2 < χ 2 n(α/2) ) = 1 α 7

8 P ( ) χ 2 n(1 α/2) < 1 n σ 2 (X i µ) 2 < χ 2 n(α/2) = 1 α i=1 n i=1 (X i µ) 2 σ 2 n i=1 (X i µ) 2 n χ 2 < σ 2 i=1 < (X i µ) 2 n(α/2) χ 2 n(1 α/2) 1 α [ : ( )] N(µ, σ 2 ) (µ ) n (X 1,..., X n ) σ 2 1 (1 α)% n i=1 (X i µ) 2 χ 2 n(α/2) < σ 2 < n i=1 (X i µ) 2 χ 2 n(1 α/2) 1. (p.97) 1 2.5mg 1 98% 2.46, 2.51, 2.52, 2.48, 2.49, 2.5, 2.54, 2.53, 2.49, 2.52 (mg) ( ) n = 1 α =.2 χ 2 χ 2 1 (α/2) = χ2 1 (.1) = 23.2 χ2 1 (1 α/2) = χ 2 1 (.99) = 2.56 µ = i=1 (X i µ) 2 =.56 : < σ 2 < < σ 2 < [ ] µ µ X χ 2 = 1 n σ 2 (X i X) 2 = n 1 σ 2 S 2 n 1 χ i=1 [ : ( )] N(µ, σ 2 ) (µ ) n (X 1,..., X n ) X S 2 σ 2 1 (1 α)% n i=1 (X i X) 2 χ 2 n 1 (α/2) < σ 2 < (n 1)S 2 χ 2 n 1 (α/2) < σ2 < n i=1 (X i X) 2 χ 2 n 1 (1 α/2) (n 1)S 2 χ 2 n 1 (1 α/2) 2. (p.98) 1 2.5mg 98% 8

9 ( ) 1 1 = 9 α =.2 χ 2 χ 2 9 (α/2) = χ2 9 (.1) = 21.7 χ 2 9 (1 α/2) = χ2 9 (.99) = 2.9 X = i=1 (X i X) 2 =.544 : < σ 2 < < σ 2 < [ : ] C C 2 C C p q N = n C P = N/n p p n (1) N B(n, p) : P (N = k) = n C k p k (1 p) n k (k =, 1,..., n) (2) n np 5 nq 5 Z = N np npq = P p (pq)/n N(, 1) ( ) ( ) (1) (2) (p.74) A 38% K 78 A 39 ( ) 78 A N N 9

10 B(78,.38) n = 78 p =.38 np = nq = N N(np, npq) Z = (N np)/ npq N(, 1) npq = P (N 39) B(78,.38) = P (N 39.5) N(29.6,18.4) ( ) N = P = P (Z 2.7) =.5 Φ(2.7) = = [ ] C C C ( ) p n N: i.e. C P = N/n: n np 5 nq 5 Z = n (P p) p (1 p) N(, 1) II z(α/2) P ( Z < z(α/2)) = 1 α P ( ) n (P p) < z(α/2) = 1 α p (1 p) p 1 (1 α)% P p (1 p) n z(α/2) < p < P + 1 p (1 p) n z(α/2)

11 p p P [ : ( )] C p n P n np 5 nq 5 p 1 (1 α)% : P P (1 P ) n z(α/2) < p < P + P (1 P ) n z(α/2) 1. (p.11) M 1 37 M p 95% ( ) n = 1 P = 37/1 =.37 np np = 1.37 = 37 5 nq = 1 37 = 63 5 α =.5 z(α/2) = z(.25) = 1.96 :.37 (1.37).37 (1.37) < p < < p < < p <

12 [ ] N(µ, σ 2 ) (σ 2 ) µ µ [ ] (1) H : µ = µ (2) H (a) µ > µ (b) µ < µ (c) µ µ (a) (b) (c) (3) : n X H X N(µ, σ 2 /n). Z = X µ σ/ n N(, 1) ( ) (4) : (= ) 1 α % H 1 (a) µ > µ P (Z > z(α)) = α Z > z(α) (b) µ < µ P (Z < z(α)) = α Z < z(α) (c) µ µ P ( Z > z(α/2)) = α Z > z(α/2) ( ) ( ) ( ) Z Z Z > z(α) Z < z(α) Z > z(α/2) 2 : 12

13 1 α 2 H : µ = µ 2 (5) : X ( ) Z (6) : Z (4) H 1 α% H 1 Z H 1 α% H ( ) (n 5) (i) ( ) (ii) σ 2 S 2 ( ) 1 (p.121) 5mm mm.9mm 5% ( ) (n = 1) σ 2 S 2 [ ( )] µ µ = 5 n = 1 ( ) σ = S =.9 X = (1) H : µ = 5 (2) H 1 : µ < 5 ( ) (3) : X H Z = X µ σ/ n N(, 1) (4) : 5% 1α = 5 α =.5 z(.5) = : Z < (5) : σ S Z = / (6) : Z H 5% ( 5%) 13

14 [ ] N(µ, σ 2 ) σ 2 σ 2 [ ] (1) H : σ 2 = σ 2 (2) H (a) σ 2 > σ 2 (b) σ 2 < σ 2 (c) σ 2 σ 2 (3) : N(µ, σ 2 ) n S 2 H χ 2 = (n 1)S2 σ 2 n 1 χ 2 ( ) (4) : 1α% H 1 (a) σ 2 > σ 2 P (χ2 > χ 2 n 1 (α)) = α χ2 > χ 2 n 1 (α) ( ) (b) σ 2 < σ 2 P (χ2 < χ 2 n 1 (1 α)) = α χ2 < χ 2 n 1 (1 α) ( ) (c) σ 2 σ 2 P (χ2 > χ 2 n 1 (α/2) χ2 < χ 2 n 1 (1 α/2)) = α χ 2 < χ 2 n 1 (1 α/2) χ2 > χ 2 n 1 (α/2) ( ) (5) : S 2 ( ) χ 2 (6) : χ 2 (4) H 1 α% 14

15 1 (p.134) 2 S 2 = % H : σ 2 = 14.2 H 1 : σ 2 < 14.2 ( ) [ ] σ 2 σ 2 = 14.2 n = 2 S 2 = 13.6 (1) H : σ 2 = 14.2 (2) H 1 : σ 2 < 14.2 ( ) (3) : S 2 H χ 2 = (n 1)S2 σ 2 n 1 χ 2 (4) : 5% 1α = 5 α =.5 χ 2 19 (.95) = 1.12 : χ 2 < 1.12 (5) : χ 2 = (2 1) = 18.2 (6) : χ 2 H 5% 14.2 ( 5%) [ ] p p [ ] (1) H : p = p (2) H (a) p > p (b) p < p (c) p p (3) : n P n np 5 nq 5 H P N(p, p (1 p )/n) Z = P p p (1 p )/n N(, 1) ( ) (4) : 1α% H 1 15

16 (a) p > p P (Z > z(α)) = α Z > z(α) ( ) (b) p < p P (Z < z(α)) = α Z < z(α) ( ) (c) p p P ( Z > z(α/2)) = α Z > z(α/2) ( ) (5) : P ( ) Z (6) : Z H 1 α% 1 (p.137) % ( ) ( ) [ ( )] p p = 1/6 n = 12 P = 28/ (1) H : p = 1/6 (2) H 1 : p > 1/6 ( ) (3) : P np = 12/6 = 2 > 5 nq = 12 2 = 1 > 5 H Z = P p p (1 p )/n N(, 1) (4) : 5% 1α = 5 α =.5 z(.5) = : Z > (5) : Z = ( ) /12 (6) : Z H 5% 1 ( 5%) 16

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

統計的仮説検定とExcelによるt検定

統計的仮説検定とExcelによるt検定 I L14(016-01-15 Fri) : Time-stamp: 016-01-15 Fri 14:03 JST hig 1,,,, p, Excel p, t. http://hig3.net ( ) L14 Excel t I(015) 1 / 0 L13-Q1 Quiz : n = 9. σ 0.95, S n 1 (n 1)

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

1 1 3 1.1 (Frequecy Tabulatios)................................ 3 1........................................ 8 1.3.....................................

1 1 3 1.1 (Frequecy Tabulatios)................................ 3 1........................................ 8 1.3..................................... 1 1 3 1.1 (Frequecy Tabulatios)................................ 3 1........................................ 8 1.3........................................... 1 17.1................................................

More information

2

2 1 12123456789012345678901234 12123456789012345678901234 12123456789012345678901234 12123456789012345678901234 12123456789012345678901234 12123456789012345678901234 12123456789012345678901234 12123456789012345678901234

More information

受賞講演要旨2012cs3

受賞講演要旨2012cs3 アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート α β α α α α α

More information

Part. 4. () 4.. () 4.. 3 5. 5 5.. 5 5.. 6 5.3. 7 Part 3. 8 6. 8 6.. 8 6.. 8 7. 8 7.. 8 7.. 3 8. 3 9., 34 9.. 34 9.. 37 9.3. 39. 4.. 4.. 43. 46.. 46..

Part. 4. () 4.. () 4.. 3 5. 5 5.. 5 5.. 6 5.3. 7 Part 3. 8 6. 8 6.. 8 6.. 8 7. 8 7.. 8 7.. 3 8. 3 9., 34 9.. 34 9.. 37 9.3. 39. 4.. 4.. 43. 46.. 46.. Cotets 6 6 : 6 6 6 6 6 6 7 7 7 Part. 8. 8.. 8.. 9..... 3. 3 3.. 3 3.. 7 3.3. 8 Part. 4. () 4.. () 4.. 3 5. 5 5.. 5 5.. 6 5.3. 7 Part 3. 8 6. 8 6.. 8 6.. 8 7. 8 7.. 8 7.. 3 8. 3 9., 34 9.. 34 9.. 37 9.3.

More information

第86回日本感染症学会総会学術集会後抄録(II)

第86回日本感染症学会総会学術集会後抄録(II) χ μ μ μ μ β β μ μ μ μ β μ μ μ β β β α β β β λ Ι β μ μ β Δ Δ Δ Δ Δ μ μ α φ φ φ α γ φ φ γ φ φ γ γδ φ γδ γ φ φ φ φ φ φ φ φ φ φ φ φ φ α γ γ γ α α α α α γ γ γ γ γ γ γ α γ α γ γ μ μ κ κ α α α β α

More information

0.45m1.00m 1.00m 1.00m 0.33m 0.33m 0.33m 0.45m 1.00m 2

0.45m1.00m 1.00m 1.00m 0.33m 0.33m 0.33m 0.45m 1.00m 2 24 11 10 24 12 10 30 1 0.45m1.00m 1.00m 1.00m 0.33m 0.33m 0.33m 0.45m 1.00m 2 23% 29% 71% 67% 6% 4% n=1525 n=1137 6% +6% -4% -2% 21% 30% 5% 35% 6% 6% 11% 40% 37% 36 172 166 371 213 226 177 54 382 704 216

More information

10 117 5 1 121841 4 15 12 7 27 12 6 31856 8 21 1983-2 - 321899 12 21656 2 45 9 2 131816 4 91812 11 20 1887 461971 11 3 2 161703 11 13 98 3 16201700-3 - 2 35 6 7 8 9 12 13 12 481973 12 2 571982 161703 11

More information

24.15章.微分方程式

24.15章.微分方程式 m d y dt = F m d y = mg dt V y = dy dt d y dt = d dy dt dt = dv y dt dv y dt = g dv y dt = g dt dt dv y = g dt V y ( t) = gt + C V y ( ) = V y ( ) = C = V y t ( ) = gt V y ( t) = dy dt = gt dy = g t dt

More information

aisatu.pdf

aisatu.pdf 1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71

More information

61“ƒ/61G2 P97

61“ƒ/61G2 P97 σ σ φσ φ φ φ φ φ φ φ φ σ σ σ φσ φ σ φ σ σ σ φ α α α φα α α φ α φ α α α φ α α α σ α α α α α α Σα Σ α α α α α σ σ α α α α α α α α α α α α σ α σ φ σ φ σ α α Σα Σα α σ σ σ σ σ σ σ σ σ σ σ σ Σ σ σ σ σ

More information

3 3.1 *2 1 2 3 4 5 6 *2 2

3 3.1 *2 1 2 3 4 5 6 *2 2 Armitage 1 2 11 10 3.32 *1 9 5 5.757 3.3667 7.5 1 9 6 5.757 7 7.5 7.5 9 7 7 9 7.5 10 9 8 7 9 9 10 9 9 9 10 9 11 9 10 10 10 9 11 9 11 11 10 9 11 9 12 13 11 10 11 9 13 13 11 10 12.5 9 14 14.243 13 12.5 12.5

More information

1 + 1 + 1 + 1 + 1 + 1 + 1 = 0? 1 2003 10 8 1 10 8, 2004 1, 2003 10 2003 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 ( )?, 1, 8, 15, 22, 29?, 1 7, 1, 8, 15, 22,

More information

第85 回日本感染症学会総会学術集会後抄録(III)

第85 回日本感染症学会総会学術集会後抄録(III) β β α α α µ µ µ µ α α α α γ αβ α γ α α γ α γ µ µ β β β β β β β β β µ β α µ µ µ β β µ µ µ µ µ µ γ γ γ γ γ γ µ α β γ β β µ µ µ µ µ β β µ β β µ α β β µ µµ β µ µ µ µ µ µ λ µ µ β µ µ µ µ µ µ µ µ

More information

2 CHAPTER 2. ) ( ) 2 () () 2.1.1 10 Octave rand() octave:27> A=rand(10,1) A = 0.225704 0.018580 0.818762 0.634118 0.026280 0.980303 0.014780 0.477392

2 CHAPTER 2. ) ( ) 2 () () 2.1.1 10 Octave rand() octave:27> A=rand(10,1) A = 0.225704 0.018580 0.818762 0.634118 0.026280 0.980303 0.014780 0.477392 Chapter 2 2.1 (cf. ) (= ) 76, 86, 77, 88, 78, 83, 86, 77, 74, 79, 82, 79, 80, 81, 78, 78, 73, 78, 81, 86, 71, 80, 81, 88, 82, 80, 80, 70, 77, 81 10? () ( 1 2 CHAPTER 2. ) ( ) 2 () () 2.1.1 10 Octave rand()

More information

1,000 700-1 -

1,000 700-1 - 23 9 () - 0 - 1,000 700-1 - 2 3 ( 16:0017:00 ( 8:15 8:30 10:3010:50 8:00 8:10 8:10 9:30 11:0011:20 11:3015:30 16:0016:40 16:0016:10 16:50 21:00 4:00 4:006:00 6:00 6:1511:00 11:3012:00 12:3014:30 (1) ()

More information

(interval estimation) 3 (confidence coefficient) µ σ/sqrt(n) 4 P ( (X - µ) / (σ sqrt N < a) = α a α X α µ a σ sqrt N X µ a σ sqrt N 2

(interval estimation) 3 (confidence coefficient) µ σ/sqrt(n) 4 P ( (X - µ) / (σ sqrt N < a) = α a α X α µ a σ sqrt N X µ a σ sqrt N 2 7 2 1 (interval estimation) 3 (confidence coefficient) µ σ/sqrt(n) 4 P ( (X - µ) / (σ sqrt N < a) = α a α X α µ a σ sqrt N X µ a σ sqrt N 2 (confidence interval) 5 X a σ sqrt N µ X a σ sqrt N - 6 P ( X

More information

07.報文_及川ら-二校目.indd

07.報文_及川ら-二校目.indd 8 01 01 4 4 1 5 16 18 6 006 H 18 4 011 H 6 4 1 5 1 5 007 H 19 5 009 1 5 006 007 009 011 9 10 4 000 H 1 4 5 004 H 16 4 004 009 H 1 5 4 4 5 1 4 006 011 1 1 4m 5m 10m 007 1 7 009 009 1 5 10 1 000kg 10a 006

More information

... 3... 3... 3... 3... 4... 7... 10... 10... 11... 12... 12... 13... 14... 15... 18... 19... 20... 22... 22... 23 2

... 3... 3... 3... 3... 4... 7... 10... 10... 11... 12... 12... 13... 14... 15... 18... 19... 20... 22... 22... 23 2 1 ... 3... 3... 3... 3... 4... 7... 10... 10... 11... 12... 12... 13... 14... 15... 18... 19... 20... 22... 22... 23 2 3 4 5 6 7 8 9 Excel2007 10 Excel2007 11 12 13 - 14 15 16 17 18 19 20 21 22 Excel2007

More information

一般演題(ポスター)

一般演題(ポスター) 6 5 13 : 00 14 : 00 A μ 13 : 00 14 : 00 A β β β 13 : 00 14 : 00 A 13 : 00 14 : 00 A 13 : 00 14 : 00 A β 13 : 00 14 : 00 A β 13 : 00 14 : 00 A 13 : 00 14 : 00 A β 13 : 00 14 : 00 A 13 : 00 14 : 00 A

More information

P-12 P-13 3 4 28 16 00 17 30 P-14 P-15 P-16 4 14 29 17 00 18 30 P-17 P-18 P-19 P-20 P-21 P-22

P-12 P-13 3 4 28 16 00 17 30 P-14 P-15 P-16 4 14 29 17 00 18 30 P-17 P-18 P-19 P-20 P-21 P-22 1 14 28 16 00 17 30 P-1 P-2 P-3 P-4 P-5 2 24 29 17 00 18 30 P-6 P-7 P-8 P-9 P-10 P-11 P-12 P-13 3 4 28 16 00 17 30 P-14 P-15 P-16 4 14 29 17 00 18 30 P-17 P-18 P-19 P-20 P-21 P-22 5 24 28 16 00 17 30 P-23

More information

cm H.11.3 P.13 2 3-106-

cm H.11.3 P.13 2 3-106- H11.3 H.11.3 P.4-105- cm H.11.3 P.13 2 3-106- 2 H.11.3 P.47 H.11.3 P.27 i vl1 vl2-107- 3 h vl l1 l2 1 2 0 ii H.11.3 P.49 2 iii i 2 vl1 vl2-108- H.11.3 P.50 ii 2 H.11.3 P.52 cm -109- H.11.3 P.44 S S H.11.3

More information

z z x = y = /x lim y = + x + lim y = x (x a ) a (x a+) lim z z f(z) = A, lim z z g(z) = B () lim z z {f(z) ± g(z)} = A ± B (2) lim {f(z) g(z)} = AB z

z z x = y = /x lim y = + x + lim y = x (x a ) a (x a+) lim z z f(z) = A, lim z z g(z) = B () lim z z {f(z) ± g(z)} = A ± B (2) lim {f(z) g(z)} = AB z Tips KENZOU 28 6 29 sin 2 x + cos 2 x = cos 2 z + sin 2 z = OK... z < z z < R w = f(z) z z w w f(z) w lim z z f(z) = w x x 2 2 f(x) x = a lim f(x) = lim f(x) x a+ x a z z x = y = /x lim y = + x + lim y

More information

卒論 提出用ファイル.doc

卒論 提出用ファイル.doc 11 13 1LT99097W (i) (ii) 0. 0....1 1....3 1.1....3 1.2....4 2....7 2.1....7 2.2....8 2.2.1....8 2.2.2....9 2.2.3.... 10 2.3.... 12 3.... 15 Appendix... 17 1.... 17 2.... 19 3.... 20... 22 (1) a. b. c.

More information

Micro-D 小型高密度角型コネクタ

Micro-D 小型高密度角型コネクタ Micro- 1 2 0.64 1.27 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 1.09 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 3 4 J J

More information

136 pp p µl µl µl

136 pp p µl µl µl 135 2006 PCB C 12 H 10-n Cl n n 1 10 CAS No. 42 PCB: 53469-21-9, 54 PCB: 11097-69-1 0.01 mg/m 3 PCB PCB 25 µg/l 136 pp p µl µl µl 137 1 γ 138 1 γ γ γ µl µl µl µl µl µl µl l µl µl µl µl µl l 139 µl µl µl

More information

14 12 10 8 6 4 2 0 1 2 3 4 5 6 7 8 9 10 11 12 20 0 0 0 0 0 0 0 0 0 0 0 0 21 0 0 0 0 0 0 0 0 0 0 0 0 22 0 0 0 0 0 0 0 0 0 0 0 0 23 0 0 0 0 0 0 0 0 0 0 0 0 24 0 0 0 0 0 0 0 0 0 0 0 0 25 0 0 0 0 0 0 0 0 0

More information

arakidonsan.rtf

arakidonsan.rtf DHA (ARA) ARA DHA 50 DHA 20 DHA( ) γ 300 mg 170 mg 150 mg 78 mg 57 mg 3,200 mg 2,300 mg 1,700 mg 1,700 mg 1,700 mg β- E FAO DHA( ARA) DHA 2ARA 1 DHA 70mg 100g 35mg 100g 6 6 63 EPA DHA 63 6 EHA DHA 50 WEB

More information

N N 1,, N 2 N N N N N 1,, N 2 N N N N N 1,, N 2 N N N 8 1 6 3 5 7 4 9 2 1 12 13 8 15 6 3 10 4 9 16 5 14 7 2 11 7 11 23 5 19 3 20 9 12 21 14 22 1 18 10 16 8 15 24 2 25 4 17 6 13 8 1 6 3 5 7 4 9 2 1 12 13

More information

http://banso.cocolog-nifty.com/ 100 100 250 5 1 1 http://www.banso.com/ 2009 5 2 10 http://www.banso.com/ 2009 5 2 http://www.banso.com/ 2009 5 2 http://www.banso.com/ < /> < /> / http://www.banso.com/

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

1 913 10301200 A B C D E F G H J K L M 1A1030 10 : 45 1A1045 11 : 00 1A1100 11 : 15 1A1115 11 : 30 1A1130 11 : 45 1A1145 12 : 00 1B1030 1B1045 1C1030

1 913 10301200 A B C D E F G H J K L M 1A1030 10 : 45 1A1045 11 : 00 1A1100 11 : 15 1A1115 11 : 30 1A1130 11 : 45 1A1145 12 : 00 1B1030 1B1045 1C1030 1 913 9001030 A B C D E F G H J K L M 9:00 1A0900 9:15 1A0915 9:30 1A0930 9:45 1A0945 10 : 00 1A1000 10 : 15 1B0900 1B0915 1B0930 1B0945 1B1000 1C0900 1C0915 1D0915 1C0930 1C0945 1C1000 1D0930 1D0945 1D1000

More information

12-7 12-7 12-7 12-7 12-8 12-10 12-10 12-10 12-11 12-12 12-12 12-14 12-15 12-17 12-18 10 12-19 12-20 12-20 12-21 12-22 12-22 12-23 12-25 12-26 12-26 12-29 12-30 12-30 12-31 12-33 12-34 12-3 12-35 12-36

More information

n=360 28.6% 34.4% 36.9% n=360 2.5% 17.8% 19.2% n=64 0.8% 0.3% n=69 1.7% 3.6% 0.6% 1.4% 1.9% < > n=218 1.4% 5.6% 3.1% 60.6% 0.6% 6.9% 10.8% 6.4% 10.3% 33.1% 1.4% 3.6% 1.1% 0.0% 3.1% n=360 0% 50%

More information

4 4. A p X A 1 X X A 1 A 4.3 X p X p X S(X) = E ((X p) ) X = X E(X) = E(X) p p 4.3p < p < 1 X X p f(i) = P (X = i) = p(1 p) i 1, i = 1,,... 1 + r + r

4 4. A p X A 1 X X A 1 A 4.3 X p X p X S(X) = E ((X p) ) X = X E(X) = E(X) p p 4.3p < p < 1 X X p f(i) = P (X = i) = p(1 p) i 1, i = 1,,... 1 + r + r 4 1 4 4.1 X P (X = 1) =.4, P (X = ) =.3, P (X = 1) =., P (X = ) =.1 E(X) = 1.4 +.3 + 1. +.1 = 4. X Y = X P (X = ) = P (X = 1) = P (X = ) = P (X = 1) = P (X = ) =. Y P (Y = ) = P (X = ) =., P (Y = 1) =

More information

土壌環境行政の最新動向(環境省 水・大気環境局土壌環境課)

土壌環境行政の最新動向(環境省 水・大気環境局土壌環境課) 201022 1 18801970 19101970 19201960 1970-2 1975 1980 1986 1991 1994 3 1999 20022009 4 5 () () () () ( ( ) () 6 7 Ex Ex Ex 8 25 9 10 11 16619 123 12 13 14 5 18() 15 187 1811 16 17 3,000 2241 18 19 ( 50

More information

syuryoku

syuryoku 248 24622 24 P.5 EX P.212 2 P271 5. P.534 P.690 P.690 P.690 P.690 P.691 P.691 P.691 P.702 P.702 P.702 P.702 1S 30% 3 1S 3% 1S 30% 3 1S 3% P.702 P.702 P.702 P.702 45 60 P.702 P.702 P.704 H17.12.22 H22.4.1

More information

,..,,.,,.,.,..,,.,,..,,,. 2

,..,,.,,.,.,..,,.,,..,,,. 2 A.A. (1906) (1907). 2008.7.4 1.,.,.,,.,,,.,..,,,.,,.,, R.J.,.,.,,,..,.,. 1 ,..,,.,,.,.,..,,.,,..,,,. 2 1, 2, 2., 1,,,.,, 2, n, n 2 (, n 2 0 ).,,.,, n ( 2, ), 2 n.,,,,.,,,,..,,. 3 x 1, x 2,..., x n,...,,

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

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

38

38 3 37 38 3.1. 3.1.1. 3.1-1 2005 12 5 7 2006 5 31 6 2 2006 8 10 11 14 2006 10 18 20 3.1-1 9 00 17 3 3.1.2. 3.1-2 3.1-1 9 9 3.1-2 M- M-2 M-3 N- N-2 N-3 S- S-2 S-3 3.1.2.1. 25 26 3.1.2.2. 3.1-3 25 26 39 3.1-1

More information

O1-1 O1-2 O1-3 O1-4 O3-1 O3-2 O3-3 O3-4 ES1-1 ES1-2 ES1-3 ES2-1 ES2-2 ES2-3 ES2-4 O2-1 O2-2 O2-3 O2-4 O2-5 O4-1 O4-2 O4-3 O4-4 O5-1 O5-2 O5-3 O5-4 O7-1 O7-2 O7-3 O7-4 O9-1 O9-2 O9-3 O9-4 O12-1 O12-2

More information