独立性の検定・ピボットテーブル

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "独立性の検定・ピボットテーブル"

Transcription

1 II L04( Thu) : Time-stamp: Thu 12:48 JST hig 2, χ 2, V Excel ( ) L04 II(2016) 1 / 20

2 L03-Q1 Quiz : 1 { 0.95 (y = 10) P (Y = y X = 1) = 0.05 (y = 20) { (y = 10) P (Y = y X = 2) = (y = 20) ( ) L04 II(2016) 2 / 20

3 2 y\x P (Y = 10 X = 1)P (X = 1) P (X = 1 Y = 10) = x P (Y = 10 X = x)p (X = x) = = P (Y = 20 X = 2)P (X = 2) P (X = 2 Y = 20) = x P (Y = 20 X = x)p (X = x) = = ( ) L04 II(2016) 3 / 20

4 L03-Q2 Quiz : Y, X, P (X = Y = ) P (Y = X = )P (X = ) = P (Y = X = )P (X = ) + P (Y = X = )P (X = ) = = Y \X L03-Q3 Quiz : χ 2 ( ) L04 II(2016) 4 / 20

5 1 χ 2 = ( ) = k = C 1 = 4 1. α = 0.05, χ α (4 1) = > 16 3.,. L03-Q4 Quiz : χ 2 1. χ 2 = 1 ( ) = ( ) L04 II(2016) 5 / 20

6 2 α = 0.05, 1 6,., χ 2 k = C 1 = 6 1..,, χ 2 = , χ 0.05 (6 1) = > 4.2,. ( ) L04 II(2016) 6 / 20

7 2 : : V ( ) L04 II(2016) 7 / 20

8 2 : 2 Y \ X A A P( =A, = ) P( =A, = ) P( =A, = ) P( =A, = ) 1 A 2 A N = A, A A n 11 = 1 n 12 = 2 n 21 = 4 n 22 = 5 n ij, 1 i c, 1 j r. r, c. Excel ( ) L04 II(2016) 8 / 20

9 2 :?.., P ( =A, = ) =P ( =A ) P ( = ) f XY (x, y) =f X (x) f Y (y). ( ) L04 II(2016) 9 / 20

10 2 :, y \ x A A P ( = ) p 1 = 3 12 P ( =A ) q 1 = 5 12, (= ). A = N p 1 q 1 = = 1.25 ( ) L04 II(2016) 10 / 20

11 2 : : χ 2 A A Np 1 q 1 Np 1 q 2 Np 1 Np 2 q 1 Np 2 q 2 Np 2 Nq 1 Nq 2 N ( ) 2 = ( ) 2 : χ 2 ( ) p i (i = 1,..., r), q j (j = 1,..., c):. χ 2 = ( )2 = 1 i r,1 j c (n ij Np i q j ) 2 Np i q j ( ) L04 II(2016) 11 / 20

12 2 : χ 2 = (1 1.25) (2 1.75) (4 3.75) (5 5.25) = χ 2 ( ) 0 χ 2., (r 1)(c 1). Example Excel χ 2 Excel RaMMoodle >. ( ) L04 II(2016) 12 / 20

13 1 2 2 : V ( ) L04 II(2016) 13 / 20

14 1 α =..., 2 3, X,Y 4 χ 2 (c 1)(r 1). 5 χ 2 =... 6 χ 2 p,, α / / (X Y / ) ( ) L04 II(2016) 14 / 20

15 L04-Q1 Quiz( χ 2 ), 6,,, ( ) χ α = 0.05,. ( ), /. X Y /. ( ) L04 II(2016) 15 / 20

16 V : V ( ) L04 II(2016) 16 / 20

17 V V V χ 2 : χ 2, N:. V = = V = χ 2 N V χ 2, r 0 V 1 V = 0 V = 1 ( ) L04 II(2016) 17 / 20

18 B V : A = 1, A = 0. A B = 1, A B = 0.? A A A r.? 0 100? 0 1? 2 2 r V r = V ( ) L04 II(2016) 18 / 20

19 V I. RaMMoodle II(2016) /Math ryukoku.ac.jp manaba ( ) L04 II(2016) 19 / 20

20 V α, k, α = P (Y > χ 2 α(k)) χ 2 α(k). k\α ( ) L04 II(2016) 20 / 20

カテゴリ変数と独立性の検定

カテゴリ変数と独立性の検定 II L04(2015-05-01 Fri) : Time-stamp: 2015-05-01 Fri 22:28 JST hig 2, Excel 2, χ 2,. http://hig3.net () L04 II(2015) 1 / 20 : L03-S1 Quiz : 1 2 7 3 12 (x = 2) 12 (y = 3) P (X = x) = 5 12 (x = 3), P (Y =

More information

分散分析・2次元正規分布

分散分析・2次元正規分布 2 II L10(2016-06-30 Thu) : Time-stamp: 2016-06-30 Thu 13:55 JST hig F 2.. http://hig3.net ( ) L10 2 II(2016) 1 / 24 F 2 F L09-Q1 Quiz :F 1 α = 0.05, 2 F 3 H 0, : σ 2 1 /σ2 2 = 1., H 1, σ 2 1 /σ2 2 1. 4

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

データの分布と代表値

データの分布と代表値 I L01(2015-09-18 Fri) : Time-stamp: 2015-09-26 Sat 10:37 JST hig e, http://hig3.net ( ) L01 I(2015) 1 / 26 ? 1? 2? ( ) L01 I(2015) 2 / 26 ?,,.,., 1..,. (,, 1, 2 ),.,. ( ) L01 I(2015) 3 / 26 ? I. M (3 )

More information

時系列解析

時系列解析 B L12(2016-07-11 Mon) : Time-stamp: 2016-07-11 Mon 17:25 JST hig,, Excel,. http://hig3.net ( ) L12 B(2016) 1 / 24 L11-Q1 Quiz : 1 E[R] = 1 2, V[R] = 9 12 = 3 4. R(t), E[X(30)] = E[X(0)] + 30 1 2 = 115,

More information

2変量データの共分散・相関係数・回帰分析

2変量データの共分散・相関係数・回帰分析 2, 1, Excel 2, Excel http://hig3.net ( ) L04 2 I(2017) 1 / 24 2 I L04(2017-10-11 Wed) : Time-stamp: 2017-10-10 Tue 23:02 JST hig L03-Q1 L03-Q2 Quiz : 1.6m, 0.0025m 2, 0.05m. L03-Q3 Quiz : Sx 2 = 4, S x

More information

マルコフ連鎖の時間発展の数値計算

マルコフ連鎖の時間発展の数値計算 B L07(206-05-2 Mon : Time-stamp: 206-05-2 Mon 8:4 JST hig http://hig.net ( L07 B(206 / 20 L05-Q TA Prob and Sol:, {, 2}. M = ( 2 2 2.. 2 p(0 = ( 0 p(t. p(0 = 2 ( p(t. ( L07 B(206 2 / 20 M λ, λ 2, u, u

More information

データの分布

データの分布 I L01(2016-09-22 Thu) : Time-stamp: 2016-09-27 Tue 11:12 JST hig e LINE@, 4, http://hig3.net () L01 I(2016) 1 / 20 ? 1? 2? () L01 I(2016) 2 / 20 ?,,.,., 1..,. (,, 1, 2 ),.,. () L01 I(2016) 3 / 20 ? I.

More information

ユニセフ表紙_CS6_三.indd

ユニセフ表紙_CS6_三.indd 16 179 97 101 94 121 70 36 30,552 1,042 100 700 61 32 110 41 15 16 13 35 13 7 3,173 41 1 4,700 77 97 81 47 25 26 24 40 22 14 39,208 952 25 5,290 71 73 x 99 185 9 3 3 3 8 2 1 79 0 d 1 226 167 175 159 133

More information

広報なんと10月号

広報なんと10月号 2008.10 2 2008.10 3 2008.10 4 2008.10 5 2008.10 6 7 2008.10 8 2008.10 9 2008.10 2008.10 10 2008.10 11 12 2008.10 13 2008.10 2008.10 14 15 2008.10 16 2008.10 2008.10 17 18 2008.10 STAMP 2008.10 19 2008.10

More information

データの分布

データの分布 I L01(2014-09-19 Fri) /Excel /Excel http://hig3.net () L01 I(2014) 1 / 24 1 2 Quiz=? () L01 I(2014) 2 / 24 ,,.,., 1..,. (,, 1, 2 ),.,. () L01 I(2014) 3 / 24 I. M (3 ) II, II,! CPU,,, (AI), (machine learning)!!

More information

時系列解析と自己回帰モデル

時系列解析と自己回帰モデル B L11(2017-07-03 Mon) : Time-stamp: 2017-07-03 Mon 11:04 JST hig,,,.,. http://hig3.net ( ) L11 B(2017) 1 / 28 L10-Q1 Quiz : 1 6 6., x[]={1,1,3,3,3,8}; (. ) 2 x = 0, 1, 2,..., 9 10, 10. u[]={0,2,0,3,0,0,0,0,1,0};

More information

untitled

untitled 20 7 1 22 7 1 1 2 3 7 8 9 10 11 13 14 15 17 18 19 21 22 - 1 - - 2 - - 3 - - 4 - 50 200 50 200-5 - 50 200 50 200 50 200 - 6 - - 7 - () - 8 - (XY) - 9 - 112-10 - - 11 - - 12 - - 13 - - 14 - - 15 - - 16 -

More information

untitled

untitled 19 1 19 19 3 8 1 19 1 61 2 479 1965 64 1237 148 1272 58 183 X 1 X 2 12 2 15 A B 5 18 B 29 X 1 12 10 31 A 1 58 Y B 14 1 25 3 31 1 5 5 15 Y B 1 232 Y B 1 4235 14 11 8 5350 2409 X 1 15 10 10 B Y Y 2 X 1 X

More information

曲面のパラメタ表示と接線ベクトル

曲面のパラメタ表示と接線ベクトル L11(2011-07-06 Wed) :Time-stamp: 2011-07-06 Wed 13:08 JST hig 1,,. 2. http://hig3.net () (L11) 2011-07-06 Wed 1 / 18 ( ) 1 V = (xy2 ) x + (2y) y = y 2 + 2. 2 V = 4y., D V ds = 2 2 ( ) 4 x 2 4y dy dx =

More information

xyr x y r x y r u u

xyr x y r x y r u u xyr x y r x y r u u y a b u a b a b c d e f g u a b c d e g u u e e f yx a b a b a b c a b c a b a b c a b a b c a b c a b c a u xy a b u a b c d a b c d u ar ar a xy u a b c a b c a b p a b a b c a

More information

ユニセフ表紙_CS6_三.indd

ユニセフ表紙_CS6_三.indd 16 179 97 101 94 121 70 36 30,552 1,042 100 700 61 32 110 41 15 16 13 35 13 7 3,173 41 1 4,700 77 97 81 47 25 26 24 40 22 14 39,208 952 25 5,290 71 73 x 99 185 9 3 3 3 8 2 1 79 0 d 1 226 167 175 159 133

More information

a n a n ( ) (1) a m a n = a m+n (2) (a m ) n = a mn (3) (ab) n = a n b n (4) a m a n = a m n ( m > n ) m n 4 ( ) 552

a n a n ( ) (1) a m a n = a m+n (2) (a m ) n = a mn (3) (ab) n = a n b n (4) a m a n = a m n ( m > n ) m n 4 ( ) 552 3 3.0 a n a n ( ) () a m a n = a m+n () (a m ) n = a mn (3) (ab) n = a n b n (4) a m a n = a m n ( m > n ) m n 4 ( ) 55 3. (n ) a n n a n a n 3 4 = 8 8 3 ( 3) 4 = 8 3 8 ( ) ( ) 3 = 8 8 ( ) 3 n n 4 n n

More information

ランダムウォークの境界条件・偏微分方程式の数値計算

ランダムウォークの境界条件・偏微分方程式の数値計算 B L06(2018-05-22 Tue) : Time-stamp: 2018-05-22 Tue 21:53 JST hig,, 2, multiply transf http://hig3.net L06 B(2018) 1 / 38 L05-Q1 Quiz : 1 M λ 1 = 1 u 1 ( ). M u 1 = u 1, u 1 = ( 3 4 ) s (s 0)., u 1 = 1

More information

XX data 03.xls sheet(1) data 03.xls sheet(1) 2 1. n 1 2. m 1 3. O 11 O

XX data 03.xls sheet(1) data 03.xls sheet(1) 2 1. n 1 2. m 1 3. O 11 O 1 5 2017 5 8 4 : website: email: http://www3.u-toyama.ac.jp/kkarato/ kkarato@eco.u-toyama.ac.jp 1 2 1.1............................. 2 1.2 IF.................................... 3 1.3...............................

More information

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 32 ( ) ( ) 33 34 35 36 37 38 39 40 41 42 43 44 ) i ii i ii 45 46 47 2 48 49 50 51 52 53 54 55 56 57 58

More information

untitled

untitled i ii (1) (1) (2) (1) (3) (1) (1) (2) (1) (3) (1) (1) (2) (1) (3) (2) (3) (1) (2) (3) (1) (1) (1) (1) (2) (1) (3) (1) (2) (1) (3) (1) (1) (1) (2) (1) (3) (1) (1) (2) (1) (3)

More information

23 15961615 1659 1657 14 1701 1711 1715 11 15 22 15 35 18 22 35 23 17 17 106 1.25 21 27 12 17 420,845 23 32 58.7 32 17 11.4 71.3 17.3 32 13.3 66.4 20.3 17 10,657 k 23 20 12 17 23 17 490,708 420,845 23

More information

平成18年度「商品先物取引に関する実態調査」報告書

平成18年度「商品先物取引に関する実態調査」報告書 ... 1.... 5-1.... 6-2.... 9-3.... 10-4.... 12-5.... 13-6.... 15-7.... 16-8.... 17-9.... 20-10.... 22-11.... 24-12.... 27-13... 29-14.... 32-15... 37-16.... 39-17.... 41-18... 43-19... 45.... 49-1... 50-2...

More information

t χ 2 F Q t χ 2 F 1 2 µ, σ 2 N(µ, σ 2 ) f(x µ, σ 2 ) = 1 ( exp (x ) µ)2 2πσ 2 2σ 2 0, N(0, 1) (100 α) z(α) t χ 2 *1 2.1 t (i)x N(µ, σ 2 ) x µ σ N(0, 1

t χ 2 F Q t χ 2 F 1 2 µ, σ 2 N(µ, σ 2 ) f(x µ, σ 2 ) = 1 ( exp (x ) µ)2 2πσ 2 2σ 2 0, N(0, 1) (100 α) z(α) t χ 2 *1 2.1 t (i)x N(µ, σ 2 ) x µ σ N(0, 1 t χ F Q t χ F µ, σ N(µ, σ ) f(x µ, σ ) = ( exp (x ) µ) πσ σ 0, N(0, ) (00 α) z(α) t χ *. t (i)x N(µ, σ ) x µ σ N(0, ) (ii)x,, x N(µ, σ ) x = x+ +x N(µ, σ ) (iii) (i),(ii) z = x µ N(0, ) σ N(0, ) ( 9 97.

More information

stat2_slides-13.key

stat2_slides-13.key !2 !3 !4 !5 !6 !7 !8 !9 !10 !11 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

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

untitled

untitled 1 2 3 4 5 6 7 8 9 10 40% 2000 11 12 13 o o o o o o v v o v v v o v o v o o o o o o o o o o o o v o o o o o v o o v o o v v o o v o o v o o v o o o o o v o o o o o o o o o v 14 15 1980 16 17 18 19 20 8013%

More information

システムの概要

システムの概要 - i - - ii - 1 Excel BCS.CSV Excel BCS.CSV Excel A B C D Excel BCS.CSV - 1 - 2 Excel (V) (T) AB AB - 2 - 3 A B A B B C B C 1 B A - 3 - 1 C B 4 1 5 6 BCS - 4 - 4 1 Excel - 5 - 32 30 Excel Alt+Enter 1-6

More information

AHPを用いた大相撲の新しい番付編成

AHPを用いた大相撲の新しい番付編成 5304050 2008/2/15 1 2008/2/15 2 42 2008/2/15 3 2008/2/15 4 195 2008/2/15 5 2008/2/15 6 i j ij >1 ij ij1/>1 i j i 1 ji 1/ j ij 2008/2/15 7 1 =2.01/=0.5 =1.51/=0.67 2008/2/15 8 1 2008/2/15 9 () u ) i i i

More information

II Time-stamp: <05/09/30 17:14:06 waki> ii

II Time-stamp: <05/09/30 17:14:06 waki> ii II waki@cc.hirosaki-u.ac.jp 18 1 30 II Time-stamp: ii 1 1 1.1.................................................. 1 1.2................................................... 3 1.3..................................................

More information

untitled

untitled WINDOWS \ 1 2 3 4 5 6 (1) (2) Excel Type2.xls (3) Excel (4) Excel Excel Excel 7 8 A B C D E 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 \\ 26 27 28 29 30 31 Windows 2000 32 sienhi 33 34 Windows XP

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

「東京都子供・子育て支援総合計画」中間見直し版(案)第2章 子供と家庭をめぐる状況

「東京都子供・子育て支援総合計画」中間見直し版(案)第2章 子供と家庭をめぐる状況 1 (1) (2) (3) (4) (5) (6) (7) (8) 2 (1) (2) (3) (4) (5) (6) (7) 10,000 100% 8,000 78.7% 78.1% 79.3% 78.3% 74.1% 75.4% 73.2% 80% 6,000 46.6% 47.5% 50.9% 51.5% 49.9% 51.8% 52.2% 60% 4,000 2,000 3,713

More information

II

II II 16 16.0 2 1 15 x α 16 x n 1 17 (x α) 2 16.1 16.1.1 2 x P (x) P (x) = 3x 3 4x + 4 369 Q(x) = x 4 ax + b ( ) 1 P (x) x Q(x) x P (x) x P (x) x = a P (a) P (x) = x 3 7x + 4 P (2) = 2 3 7 2 + 4 = 8 14 +

More information

* n x 11,, x 1n N(µ 1, σ 2 ) x 21,, x 2n N(µ 2, σ 2 ) H 0 µ 1 = µ 2 (= µ ) H 1 µ 1 µ 2 H 0, H 1 *2 σ 2 σ 2 0, σ 2 1 *1 *2 H 0 H

* n x 11,, x 1n N(µ 1, σ 2 ) x 21,, x 2n N(µ 2, σ 2 ) H 0 µ 1 = µ 2 (= µ ) H 1 µ 1 µ 2 H 0, H 1 *2 σ 2 σ 2 0, σ 2 1 *1 *2 H 0 H 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

More information

2004/01/12 1 2004/01/23 2 I- - 10 2004/04/02 3-6 2004/04/03 4-1-5-1,-1-8-1,-2-2-1,-3-4-1,-3-5-1,-4-2-1, -5-4-2,-5-6-1,-6-2-1 4. _.doc 1

2004/01/12 1 2004/01/23 2 I- - 10 2004/04/02 3-6 2004/04/03 4-1-5-1,-1-8-1,-2-2-1,-3-4-1,-3-5-1,-4-2-1, -5-4-2,-5-6-1,-6-2-1 4. _.doc 1 4 2004 4 3 2004/01/12 1 2004/01/23 2 I- - 10 2004/04/02 3-6 2004/04/03 4-1-5-1,-1-8-1,-2-2-1,-3-4-1,-3-5-1,-4-2-1, -5-4-2,-5-6-1,-6-2-1 4. _.doc 1 - - I. 4 I- 4 I- 4 I- 6 I- 6 I- 7 II. 8 II- 8 II- 8 II-

More information

L P y P y + ɛ, ɛ y P y I P y,, y P y + I P y, 3 ŷ β 0 β y β 0 β y β β 0, β y x x, x,, x, y y, y,, y x x y y x x, y y, x x y y {}}{,,, / / L P / / y, P

L P y P y + ɛ, ɛ y P y I P y,, y P y + I P y, 3 ŷ β 0 β y β 0 β y β β 0, β y x x, x,, x, y y, y,, y x x y y x x, y y, x x y y {}}{,,, / / L P / / y, P 005 5 6 y β + ɛ {x, x,, x p } y, {x, x,, x p }, β, ɛ E ɛ 0 V ɛ σ I 3 rak p 4 ɛ i N 0, σ ɛ ɛ y β y β y y β y + β β, ɛ β y + β 0, β y β y ɛ ɛ β ɛ y β mi L y y ŷ β y β y β β L P y P y + ɛ, ɛ y P y I P y,,

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

1 911 9001030 9:00 A B C D E F G H I J K L M 1A0900 1B0900 1C0900 1D0900 1E0900 1F0900 1G0900 1H0900 1I0900 1J0900 1K0900 1L0900 1M0900 9:15 1A0915 1B0915 1C0915 1D0915 1E0915 1F0915 1G0915 1H0915 1I0915

More information

( 28 ) ( ) ( ) 0 This note is c 2016, 2017 by Setsuo Taniguchi. It may be used for personal or classroom purposes, but not for commercial purp

( 28 ) ( ) ( ) 0 This note is c 2016, 2017 by Setsuo Taniguchi. It may be used for personal or classroom purposes, but not for commercial purp ( 28) ( ) ( 28 9 22 ) 0 This ote is c 2016, 2017 by Setsuo Taiguchi. It may be used for persoal or classroom purposes, but ot for commercial purposes. i (http://www.stat.go.jp/teacher/c2epi1.htm ) = statistics

More information

1: n n e = φ(n) e 2 [2] 1 [1] 3 [3] 2 [2] 4 [2,2] 2 [2] 5 [5] 4 [2,2] 6 [2,3] 2 [2] 7 [7] 6 [2,3] 8 [2,2,2] 4 [2,2] 9 [3,3] 6 [2,3] 10 [2,5] 4 [2,2] 1

1: n n e = φ(n) e 2 [2] 1 [1] 3 [3] 2 [2] 4 [2,2] 2 [2] 5 [5] 4 [2,2] 6 [2,3] 2 [2] 7 [7] 6 [2,3] 8 [2,2,2] 4 [2,2] 9 [3,3] 6 [2,3] 10 [2,5] 4 [2,2] 1 (3) 25 7 17 1 φ n φ(n) n k n. k < n k n k φ(n). 10 1 10, 3 10, 7 10, 9 10 φ(10) = 4. φ(100) = 40, φ(1000) = 400. 100.,. φ.. 1. n p p p φ(p) = p 1. 2. n p 2 p p 2 φ(p 2 ) = p 2 p = p(p 1). 3. φ(p e ) =

More information

1 (1) (2)

1 (1) (2) 1 2 (1) (2) (3) 3-78 - 1 (1) (2) - 79 - i) ii) iii) (3) (4) (5) (6) - 80 - (7) (8) (9) (10) 2 (1) (2) (3) (4) i) - 81 - ii) (a) (b) 3 (1) (2) - 82 - - 83 - - 84 - - 85 - - 86 - (1) (2) (3) (4) (5) (6)

More information

- 2 -

- 2 - - 2 - - 3 - (1) (2) (3) (1) - 4 - ~ - 5 - (2) - 6 - (1) (1) - 7 - - 8 - (i) (ii) (iii) (ii) (iii) (ii) 10 - 9 - (3) - 10 - (3) - 11 - - 12 - (1) - 13 - - 14 - (2) - 15 - - 16 - (3) - 17 - - 18 - (4) -

More information

2 1980 8 4 4 4 4 4 3 4 2 4 4 2 4 6 0 0 6 4 2 4 1 2 2 1 4 4 4 2 3 3 3 4 3 4 4 4 4 2 5 5 2 4 4 4 0 3 3 0 9 10 10 9 1 1

2 1980 8 4 4 4 4 4 3 4 2 4 4 2 4 6 0 0 6 4 2 4 1 2 2 1 4 4 4 2 3 3 3 4 3 4 4 4 4 2 5 5 2 4 4 4 0 3 3 0 9 10 10 9 1 1 1 1979 6 24 3 4 4 4 4 3 4 4 2 3 4 4 6 0 0 6 2 4 4 4 3 0 0 3 3 3 4 3 2 4 3? 4 3 4 3 4 4 4 4 3 3 4 4 4 4 2 1 1 2 15 4 4 15 0 1 2 1980 8 4 4 4 4 4 3 4 2 4 4 2 4 6 0 0 6 4 2 4 1 2 2 1 4 4 4 2 3 3 3 4 3 4 4

More information

20 15 14.6 15.3 14.9 15.7 16.0 15.7 13.4 14.5 13.7 14.2 10 10 13 16 19 22 1 70,000 60,000 50,000 40,000 30,000 20,000 10,000 0 2,500 59,862 56,384 2,000 42,662 44,211 40,639 37,323 1,500 33,408 34,472

More information

I? 3 1 3 1.1?................................. 3 1.2?............................... 3 1.3!................................... 3 2 4 2.1........................................ 4 2.2.......................................

More information

(iii) x, x N(µ, ) z = x µ () N(0, ) () 0 (y,, y 0 ) (σ = 6) *3 0 y y 2 y 3 y 4 y 5 y 6 y 7 y 8 y 9 y ( ) *4 H 0 : µ

(iii) x, x N(µ, ) z = x µ () N(0, ) () 0 (y,, y 0 ) (σ = 6) *3 0 y y 2 y 3 y 4 y 5 y 6 y 7 y 8 y 9 y ( ) *4 H 0 : µ t 2 Armitage t t t χ 2 F χ 2 F 2 µ, N(µ, ) f(x µ, ) = ( ) exp (x µ)2 2πσ 2 2 0, N(0, ) (00 α) z(α) t * 2. t (i)x N(µ, ) x µ σ N(0, ) 2 (ii)x,, x N(µ, ) x = x + +x ( N µ, σ2 ) (iii) (i),(ii) x,, x N(µ,

More information

Copyright 2003 MapNet.Corp All rights reserved (1) V2.2 (2) (3) (4) (5) (6) 1 (7) (8) (9) 1/2500 1/250 1/10000 1/10000 20 5 1/2500 20 1/500 1/500 1/250 5 1 (1) (2) (3) OK One Point () and or

More information

ii 3.,. 4. F. ( ), ,,. 8.,. 1. (75% ) (25% ) =7 24, =7 25, =7 26 (. ). 1.,, ( ). 3.,...,.,.,.,.,. ( ) (1 2 )., ( ), 0., 1., 0,.

ii 3.,. 4. F. ( ), ,,. 8.,. 1. (75% ) (25% ) =7 24, =7 25, =7 26 (. ). 1.,, ( ). 3.,...,.,.,.,.,. ( ) (1 2 )., ( ), 0., 1., 0,. (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. ( ),..

More information

nakata/nakata.html p.1/20

nakata/nakata.html p.1/20 http://www.me.titech.ac.jp/ nakata/nakata.html p.1/20 1-(a). Faybusovich(1997) Linear systems in Jordan algebras and primal-dual interior-point algorithms,, Euclid Jordan p.2/20 Euclid Jordan V Euclid

More information

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

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

More information

A, B, C. (1) A = A. (2) A = B B = A. (3) A = B, B = C A = C. A = B. (3)., f : A B g : B C. g f : A C, A = C. 7.1, A, B,. A = B, A, A A., A, A

A, B, C. (1) A = A. (2) A = B B = A. (3) A = B, B = C A = C. A = B. (3)., f : A B g : B C. g f : A C, A = C. 7.1, A, B,. A = B, A, A A., A, A 91 7,.,, ( ).,,.,.,. 7.1 A B, A B, A = B. 1), 1,.,. 7.1 A, B, 3. (i) A B. (ii) f : A B. (iii) A B. (i) (ii)., 6.9, (ii) (iii).,,,. 1), Ā = B.. A, Ā, Ā,. 92 7 7.2 A, B, C. (1) A = A. (2) A = B B = A. (3)

More information

Super perfect numbers and Mersenne perefect numbers /2/22 1 m, , 31 8 P = , P =

Super perfect numbers and Mersenne perefect numbers /2/22 1 m, , 31 8 P = , P = Super perfect numbers and Mersenne perefect numbers 3 2019/2/22 1 m, 2 2 5 3 5 4 18 5 20 6 25 7, 31 8 P = 5 35 9, 38 10 P = 5 39 1 1 m, 1: m = 28 m = 28 m = 10 height48 2 4 3 A 40 2 3 5 A 2002 2 7 11 13

More information

x 3 a (mod p) ( ). a, b, m Z a b m a b (mod m) a b m 2.2 (Z/mZ). a = {x x a (mod m)} a Z m 0, 1... m 1 Z/mZ = {0, 1... m 1} a + b = a +

x 3 a (mod p) ( ). a, b, m Z a b m a b (mod m) a b m 2.2 (Z/mZ). a = {x x a (mod m)} a Z m 0, 1... m 1 Z/mZ = {0, 1... m 1} a + b = a + 1 1 22 1 x 3 (mod ) 2 2.1 ( )., b, m Z b m b (mod m) b m 2.2 (Z/mZ). = {x x (mod m)} Z m 0, 1... m 1 Z/mZ = {0, 1... m 1} + b = + b, b = b Z/mZ 1 1 Z Q R Z/Z 2.3 ( ). m {x 0, x 1,..., x m 1 } modm 2.4

More information

09 II 09/12/ (3D ) f(, y) = 2 + y 2 3D- 1 f(0, 0) = 2 f(1, 0) = 3 f(0, 1) = 4 f(1, 1) = 5 f( 1, 2) = 6 f(0, 1) = z y (3D ) f(, y) = 2 + y

09 II 09/12/ (3D ) f(, y) = 2 + y 2 3D- 1 f(0, 0) = 2 f(1, 0) = 3 f(0, 1) = 4 f(1, 1) = 5 f( 1, 2) = 6 f(0, 1) = z y (3D ) f(, y) = 2 + y 09 II 09/12/21 1 1 7 1.1 I 2D II 3D f() = 3 6 2 + 9 2 f(, y) = 2 2 + 2y + y 2 6 4y f(1) = 1 3 6 1 2 9 1 2 = 2 y = f() f(3, 2) = 2 3 2 + 2 3 2 + 2 2 6 3 4 2 = 8 z = f(, y) y 2 1 z 8 3 2 y 1 ( y ) 1 (0,

More information

ad bc A A A = ad bc ( d ) b c a n A n A n A A det A A ( ) a b A = c d det A = ad bc σ {,,,, n} {,,, } {,,, } {,,, } ( ) σ = σ() = σ() = n sign σ sign(

ad bc A A A = ad bc ( d ) b c a n A n A n A A det A A ( ) a b A = c d det A = ad bc σ {,,,, n} {,,, } {,,, } {,,, } ( ) σ = σ() = σ() = n sign σ sign( I n n A AX = I, YA = I () n XY A () X = IX = (YA)X = Y(AX) = YI = Y X Y () XY A A AB AB BA (AB)(B A ) = A(BB )A = AA = I (BA)(A B ) = B(AA )B = BB = I (AB) = B A (BA) = A B A B A = B = 5 5 A B AB BA A

More information

情報量・音声画像動画のA/D変換

情報量・音声画像動画のA/D変換 L06(2014-10-29 Wed), A/D..... http://hig3.net ( ) L06 A/D (2014) 1 / 24 : L05-S1 Quiz :int 16 2 15 x 2 15 1, 16 0 x 2 16 1. L05-S5 Quiz : 2 17 < 200000 2 18, 18. 2 10 = 1024, 2 16 = 65536. log 10 2, log

More information

ii 3.,. 4. F. (), ,,. 8.,. 1. (75% ) (25% ) =9 7, =9 8 (. ). 1.,, (). 3.,. 1. ( ).,.,.,.,.,. ( ) (1 2 )., ( ), 0. 2., 1., 0,.

ii 3.,. 4. F. (), ,,. 8.,. 1. (75% ) (25% ) =9 7, =9 8 (. ). 1.,, (). 3.,. 1. ( ).,.,.,.,.,. ( ) (1 2 )., ( ), 0. 2., 1., 0,. 23(2011) (1 C104) 5 11 (2 C206) 5 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

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

応用数学III-4.ppt

応用数学III-4.ppt III f x ( ) = 1 f x ( ) = P( X = x) = f ( x) = P( X = x) =! x ( ) b! a, X! U a,b f ( x) =! " e #!x, X! Ex (!) n! ( n! x)!x! " x 1! " x! e"!, X! Po! ( ) n! x, X! B( n;" ) ( ) ! xf ( x) = = n n!! ( n

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

10 10 10 1 2 3 4 2 2 4 5 6 7 8 1 9 1011 S 10 1 2 3 4 5 10 10 10 1 2 3 4 5 6 2 7 8 9 10 10 2 14 3 2 4 10 11 12 13 14 1521 15 16 15 16 ( ) 17 18 19 20 21 1 10 4 1 2 5 3 2 II 2 3 ( ) 5 21 5 22 2 3 23

More information