I L01(2017-09-20 Wed) : Time-stamp: 2017-09-20 Wed 07:38 JST hig e, http://hig3.net ( ) L01 I(2017) 1 / 19
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? I. M (3 ) II, B I( )=, A( ), B( ),!,, (AI), (machine learning)!! 2,3 ( ) L01 I(2017) 4 / 19
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? I 100 25 : quiz,e,, 30 : (11 ) http://www.toukei-kentei.jp3. 45 : ( )..,. 15, ( ),..,,, 5 (. )... ( ) L01 I(2017) 7 / 19
? : hig-probstat : 1-502 3.5(1-539) 4(1-502), Math - (1-614) Web : http://hig3.net ( QR ),. ( ) L01 I(2017) 8 / 19
?. https://www.amazon.co.jp/product/dp/4781913555 Microsoft Excel. Office365. http://portal.office.com ( ).. ( ) L01 I(2017) 9 / 19
? Learn Math Moodle http://hig3.net Learn Math Moodle ( ) I http://hig3.net I. https://g.mail.ryukoku.ac.jp Gmail. ( ) L01 I(2017) 10 / 19 1 1 09:00 (=Trial ) Learn Math Moodle Trial. 2 1 Trial(= ) ( 1, 1 Trial ) 3 1... 4 1 Trial 5 3.5(1-539) 4(1-502), Math - (1-614)
? 1? 2? ( ) L01 I(2017) 11 / 19
? 1 + (1 ). 148cm 148.5cm 149cm. 185cm http://girlsgroupguide.com/ ( ),?,?. ( ) L01 I(2017) 12 / 19
1? 2? ( ) L01 I(2017) 13 / 19
I 5.1.1 145 150 7 0.09 150 155 17 0.22 155 160 29 0.37 160 165 19 0.24 165 170 4 0.05 170 175 1 0.01 175 180 0 0.00 180 185 1 0.01 185 190 0 0.00 78 1.00 ( ) 1., 1,.,, (= ), ( ) L01 I(2017) 14 / 19
I 5.1.1 =,? bin.?,. = = = = = n = / (% ) ( ) L01 I(2017) 15 / 19
1? 2? ( ) L01 I(2017) 16 / 19
I 5.1.1,. = 1 =??, ( ) L01 I(2017) 17 / 19
L01-Q1 Quiz( ).. 14.1 15.2 16.4 16.2 17.9 17.1 17.9 17.6 17.1 17.0 17.6 18.1 18.4 18.4 18.1 18.4 19.0 19.0 19.2 20.7 20.8 21.4 21.7 21.5 21.2 5. 10-15,15-20,..., 4. 12-16,16-20,...,,,,,. 22.0 22.0 22.1 23.7 23.2 23.2 23.0 24.2 26.9 ( ) L01 I(2017) 18 / 19
7-002 1-503, http://hig3.net. 3.5(1-539) 4(1-502), Math - (1-614), ( )... Trial 5.1.2 5.1.3.. 2017-11-26, ). ( ) L01 I(2017) 19 / 19
I L02(2017-09-27 Wed) : Time-stamp: 2017-10-03 Tue 09:48 JST hig,,,,, 5.1.2, 5.1.6 I,,,, 5.1.3, 5.1.6 I http://hig3.net ( ) L02 I(2017) 1 / 27
: L01-Q1 Quiz : ( ) ( ) 10 15 1 15 20 18 20 25 14 25 30 1 34,. 12 16 2 16 20 17 20 24 13 24 28 2 34,. ( ) L02 I(2017) 2 / 27
3 : 4 (,range) ( ) L02 I(2017) 3 / 27
: 1! 5.1.2 = 150cm? 170cm? 148 152... 170 (cm) f i 145 150 7 150 155 17 155 160 29 160 165 19 165 170 4 170 175 1 77 ( ) L02 I(2017) 4 / 27
(+ ) (, ),, IQR,, ( ) < < ( ) ( ) L02 I(2017) 5 / 27
(median) / / (quartile) x, x (1) x (2) x (N). n = 11 i 1 2 3 4 5 6 7 8 9 10 11 x i 9 17 9 12 13 10 15 13 13 12 11 i 1 2 3 4 5 6 7 8 9 10 11 x (i) 9 9 10 11 12 12 13 13 13 15 17 ( ) L02 I(2017) 6 / 27
p.96 Q 0 = x (1) x ( 0 4 N) 1 Q 1 = x ( 1 4 N) 2 Q 2 = x ( 2 4 N)= 3 Q 3 = x ( 3 4 N) Q 4 = x ( 4 4 N) ( ) L02 I(2017) 7 / 27
I p.96 5 Q 0, Q 4. x N+1 = (N ) ( Q 2 = 2 ) 1 2 (x ( N + x 2 ) ( N = (N ) 2 +1)) Q 1, Q 2 (Q 2 ) Q 3, Q 2 (Q 2 ) Q 2 1 : y 10 11 12 12 13 13 13 15 17 ( ) L02 I(2017) 8 / 27
5.6 = ( + )/2 (cm) m i f i 145 150 147.5 7 150 155 17 155 160 29 160 165 19 165 170 4 N 77 ( ) L02 I(2017) 9 / 27
( ) L02 I(2017) 10 / 27
3 : 4 (,range) ( ) L02 I(2017) 11 / 27
=mode : : /, 1(30 50 55 55 60 70 70 70 75 100) (cm) f i 145 150 7 150 155 17 155 160 29 160 165 19 165 170 4 170 175 1 77 ( ) L02 I(2017) 12 / 27
3 : 4 (,range) ( ) L02 I(2017) 13 / 27
=mean 5.1.2 n x 1, x 2,..., x N, x = 1 N N i=1 x i x m, m x. : 30 50 55 55 60 70 70 70 75 100 ( ) L02 I(2017) 14 / 27
L02-Q1 Quiz( ). 14cm, 14cm, 15cm, 16cm, 18cm, 18cm, 18cm, 25cm 1 Q 1, Q 2, Q 3. 2 ( ) 3 ( ) L02 I(2017) 15 / 27
5.1.6 x 1 n k k i=1 m i f i = m if i k i=1 f i i=1 i m i, f i. x G = i x im i i M i i x i, M i ( ) L02 I(2017) 16 / 27
L02-Q2 Quiz( ). 1 2 ( ) 3 ( ) L02 I(2017) 17 / 27
(,range) 3 : 4 (,range) ( ) L02 I(2017) 18 / 27
(,range) 1,3? ( ) L02 I(2017) 19 / 27 Frequency 0 1 2 3 4 Frequency 0 1 2 3 4 16 18 20 22 24 26 28 16 18 20 22 24 26 28 Frequency 0 1 2 3 4 Frequency 0 1 2 3 4 16 18 20 22 24 26 28 16 18 20 22 24 26 28
(,range) ( ) L02 I(2017) 20 / 27
(,range) I I p.97 ( ) = (interquartile range) IQR= (quartile deviation) = L02-Q3 Quiz( ),,,. 14 14 15 16 18 18 18 25 ( ) L02 I(2017) 21 / 27
(,range) ( ) L02 I(2017) 22 / 27
3 : 4 (,range) ( ) L02 I(2017) 23 / 27
I p.98 : x 1, x 2,..., x N. (variance): ( ) 2 S 2 = 1 N N (x i x) 2 i=1 (standard deviation)= (mean deviation): d = 1 N N x i x i=1 ( ) L02 I(2017) 24 / 27
( ) (77 ) I n 1 = 77 1. 77 OK. x = 148+148.5+ +172.3 77 = 158(cm) S 2 = (148 158)2 +(148.5 158) 2 + +(172.3 158) 2 77 = 26.0 (cm 2 ) S = 26.0 = 5.1 (cm) ( ) L02 I(2017) 25 / 27
( ) (77 ) II L02-Q4 Quiz( ) 87kg, 93kg, 89kg, 91kg, 90kg. ( ) I 5.1(p.100) I p.104 ( ) L02 I(2017) 26 / 27
1-503 http://hig3.net. ( )... Learn Math Moodle trial. 3.5(1-539) 4(1-502), Math - (1-614) 5.1.4, 5.1.5 ( 2017-10-09 ). 3. 2 : 3 ( ) 2 3 2 ( ) ( ) L02 I(2017) 27 / 27
, 1 2, http://hig3.net ( ) L03 2 I(2017) 1 / 28 2 I L03(2017-10-04 Wed) : Time-stamp: 2017-10-04 Wed 13:22 JST hig
: L02-Q1 Quiz : 1 Q 2 = 17cm, Q 1 = 14.5cm, Q 3 = 18cm. 2 18cm. 3 (14 + + 25)/8 = 17.25cm. L02-Q2 Quiz : N = 9. 1 Q 2 = x (5). 21 23. x (5) 21 + 2 1.5 2 = 22.5. 2, 10 3 1 9 (10 3 + 22 2 + 24 2 + 26 2) = 19.3 ( ) L03 2 I(2017) 2 / 28
: L02-Q3 Quiz : Q 4 Q 0 = 25 14 = 11, Q 3 Q 1 = 18 14.5 = 3.5, 1 2 (Q 3 Q 1 ) = 1.75. L02-Q4 Quiz : = 90kg, = 4kg 2, = 2kg. ( ) L03 2 I(2017) 3 / 28
2 2 : 3 2 4 2 2 2 ( ) L03 2 I(2017) 4 / 28
2 (Box Plot, Box and Whisker diagram) p.97 Q 0, Q 4, Q 1, Q 2, Q 3 I Q 0, Q 4 Q 1,Q 2,Q 3 x Q 2 Q 1,Q 3 Q 0,Q 4, + 1., : (cm), : ( ) L03 2 I(2017) 5 / 28
2, (,tail) = / = / ( ) L03 2 I(2017) 6 / 28
2 2 : 3 2 4 2 2 2 ( ) L03 2 I(2017) 7 / 28
2 I L03-Q1 Quiz( ), 60, 10. 60, 20. 70 70,? 1. 1 2 3 4 5 60 ( ) L03 2 I(2017) 8 / 28
2 5.1.4 x y x 1, x 2,..., x n, x x, S 2 x, S x. y i = ax i + b (a, b ). y 1, y 2,..., y n, y y, S 2 y, S y? : y = 1.8(m) x = 80(cm) y = ax + b, ( ) L03 2 I(2017) 9 / 28
2,, 5.2(p.101) y = ax + b 1 y = ax + b 2 S 2 y = a 2 S 2 x 3 S y = a S x L03-Q2 Quiz( ) ( 100cm ), cm 2 x cm, 60cm, 25cm 2. m y m. ( ) L03 2 I(2017) 10 / 28
2 2 : 3 2 4 2 2 2 ( ) L03 2 I(2017) 11 / 28
2! 5.1.5 Berryz, 20cm, 3cm. =,. (coefficient of variation) ( x ) = S x x 100.... ( ) L03 2 I(2017) 12 / 28
2 (standard score, z-score, z ) ( x i ) z i = x i x S x,,. n = 5 i 1 2 3 4 5 x i 15 13 12 11 9 12 2 z i 1.50 0.5 0 0.5 1.50 0 1 L03-Q3 Quiz( ) x 87, 93, 89, 91, 90. 87. ( ) L03 2 I(2017) 13 / 28
2 z z = S 2 z =, S z = z.,. 180cm, 80cm, 1.8m. z =ax + b = 1 x x = 0. S x S x S z = a S z = 1 S Sx x = 1. ( ) L03 2 I(2017) 14 / 28
2 (?). 1 1,,. a =, b = ( x i ) w =10z i + 50 = x i x S x 10 + 50.,., (1000 100 ) ( ) L03 2 I(2017) 15 / 28
2 L03-Q4 Quiz( ) ( ), ( )? 1 0 2 75 3 ( ) 50 4 100, 5 50 1/2 6 60 15%. ( ) L03 2 I(2017) 16 / 28
2 2 2 : 3 2 4 2 2 2 ( ) L03 2 I(2017) 17 / 28
2 2 2 1. 2. (x, y). x, y. x y ( ) z J Div1. 2014 34. n = 18( ). ( ) x y z 9 347 50 18 512 39. http://www.j-league.or.jp/data/... (x, y) =( (cm), (kg)), ( ( ), (m 2 ), (, ), (, ).... ( ) L03 2 I(2017) 18 / 28
2 2 = 5.2.2? ( ) L03 2 I(2017) 19 / 28
2 2 x:, y ( ) 2 18. y \x 0 5 10 15 20 200 250 1 1 250 300 1 1 300 350 2 3 1 6 350 400 1 4 3 8 400 450 1 1 450 500 0 0 500 550 1 1 1 4 7 6 18 ( ) L03 2 I(2017) 20 / 28
2 2 2 : 3 2 4 2 2 2 ( ) L03 2 I(2017) 21 / 28
Y 0 2 4 6 8 10 X Y 0 2 4 6 8 10 X 2 2 Y 0 2 4 6 8 10 X Y 0 2 4 6 8 10 X Y 0 2 4 6 8 10 X 5.2.3 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 r = 0.99 r = 0.55 r = 0 r = 0.55 r = 0.99 : x y : x y / : / r:. ( ) L03 2 I(2017) 22 / 28
2 2 I 5.2.3 x x = 1 N x Sx 2 = 1 N y, Sy 2. N i=1 x i N (x i x) 2 = 1 N i=1 N (x i x)(x i x) i=1 (covariance) x, y C xy = 1 N N (x i x) (y i y) i=1 : C xy = S xy, x S 2 x = S xx, y S 2 y = S yy. ( ) L03 2 I(2017) 23 / 28
2 2 L03-Q5 Quiz( ) x 1 x, y 2 x, y., y =. y 1 5 3 15 4 14 5 11 7 20 122 5 = 4.94 ( ) L03 2 I(2017) 24 / 28
2 2 Y (,+) (+,+) p.110 Y の平均値 (, ) (+, ) X Xの平均値 (+, ) = (x i x, y i y ). / / (?) ( ) ( ) L03 2 I(2017) 25 / 28
2 2 I p.111 x, y 1 5.4(p.112),,. (correlation coefficient) x, y r = C xy S x S y ( ) L03 2 I(2017) 26 / 28
2 2 1 r +1 5.5(p.114) r = 0 ( ) r = ±1 / y x 1. 5.7(p.115) r x, y 1 5.6(p.114) ( ) L03 2 I(2017) 27 / 28
2 2 1-609... Excel. Excel 4 2. https://moodle.media.ryukoku.ac.jp 1-503 http://hig3.net. ( )... Learn Math Moodle trial. 3.5(1-539) 4(1-502), Math - (1-614) 5.2.4, 5.2.5, 5.2.6 ( 2017-10-09 ). 3. ( ) L03 2 I(2017) 28 / 28
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 = 2. z = (87 90)/2 = 1.5. w = ( 1.5) 10 + 50 = 35. x = 90, ( ) L04 2 I(2017) 2 / 24
2 2 1 2 2 2 2 Excel ( ) L04 2 I(2017) 3 / 24
2 2 2 1. 2. (x, y). x, y. x y ( ) z J Div1. 2014 34. n = 18( ). ( ) x y z 9 347 50 18 512 39. http://www.j-league.or.jp/data/... (x, y) =( (cm), (kg)), ( ( ), (m 2 ), (, ), (, ).... ( ) L04 2 I(2017) 4 / 24
2 2 = 5.2.2? ( ) L04 2 I(2017) 5 / 24
2 2 x:, y ( ) 2 18. y \x 0 5 10 15 20 200 250 1 1 250 300 1 1 300 350 2 3 1 6 350 400 1 4 3 8 400 450 1 1 450 500 0 0 500 550 1 1 1 4 7 6 18 ( ) L04 2 I(2017) 6 / 24
2 2 1 2 2 2 2 Excel ( ) L04 2 I(2017) 7 / 24
Y 0 2 4 6 8 10 X 2 2 Y 0 2 4 6 8 10 X Y 0 2 4 6 8 10 X Y 0 2 4 6 8 10 X Y 0 2 4 6 8 10 X 5.2.3 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 r = 0.99 r = 0.55 r = 0 r = 0.55 r = 0.99 : x y : x y / : / r: r xy.. ( ) L04 2 I(2017) 8 / 24
2 2 I 5.2.3 x x = 1 N x Sx 2 = 1 N y, Sy 2. N i=1 x i N (x i x) 2 = 1 N i=1 N (x i x)(x i x) i=1 (covariance) x, y C xy = 1 N N (x i x) (y i y) i=1 : C xy = S xy, x S 2 x = S xx, y S 2 y = S yy. ( ) L04 2 I(2017) 9 / 24
2 2 p.110 Y (,+) (+,+) Y の平均値 (, ) (+, ) X Xの平均値 (+, ) = (x i x, y i y ). / / (?) ( ) ( ) L04 2 I(2017) 10 / 24
2 2 I p.111 x, y 1 5.4(p.112),,. (correlation coefficient) x, y r = C xy S x S y ( ) L04 2 I(2017) 11 / 24
2 2 1 r +1 5.5(p.114) r = 0 ( ) r = ±1 / y x 1. 5.7(p.115) r x, y 1 5.6(p.114) ( ) L04 2 I(2017) 12 / 24
2 2 L04-Q1 Quiz( ( )) (xg, ycm) 1 x, y 2 x, y., y 122 = 5 = 4.94(cm). x(g) y(cm) 1 5 3 15 4 14 5 11 7 20 ( ) L04 2 I(2017) 13 / 24
2 1 2 2 2 2 Excel ( ) L04 2 I(2017) 14 / 24
2 5.2.4 (regression), = =1 2 (x, y) r = ±1 (x, y) ( ) y = ax + b! a, b. shoot.received 250 300 350 400 450 400 420 440 460 480 500 520 FK y: ( ) x: ( )? x y ( ) L04 2 I(2017) 15 / 24
2 1 2 2. 2 2 d 2 n n L(a, b) = d 2 i = (y i (ax i + b)) 2 L Y a = L b i=1 i=1 = 0 a, b. I X ( ) L04 2 I(2017) 16 / 24
2 L(a, b) = N(1 r 2 )Sy. 2 ( ) L04 2 I(2017) 17 / 24 5.8, (5.11) x i, y i (i = 1,..., n) x, y, S x, S y, r., y = r S y S x (x x) + y = ax + b. a = r Sy S x = Cxy, b = ( (x, y) ) Sx 2 a: (x 1 y ) r 2 : ( ) 5.2.4
2 I, 2S x, 2S y Sy S x?,, 0, r. (x, y) (m,kg). r.. y (kg). r Sy(kg) S x(m) x(m) + b(kg), S x /S y. ( ) L04 2 I(2017) 18 / 24
2 L04-Q2 Quiz( ) 2 (x, y). x x 9 y y 4 x s 2 x 49 y s 2 y 36 x, y s xy 25 (x, y) n 16,, x, y.. ( ) L04 2 I(2017) 19 / 24
2 Excel 1 2 2 2 2 Excel ( ) L04 2 I(2017) 20 / 24
2 Excel R... SPSS. Excel,. Office365. Excel. >Excel 2016 > > > Excel > > OK. ( ) L04 2 I(2017) 21 / 24
2 Excel (Excel) I,,, > > average, > > var.p, stdev.p, mode > > > median, quartile, > > frequency + > > = > >, > > >, covar=covariance.p, correl > > linest > > >, =, n 1 n, var.p. ( ) L04 2 I(2017) 22 / 24
2 Excel Excel, 1 ( ).,,. 2 (n ), S xy r xy S xx S yx S xy S yy, r xx r xy r yx r yy. S yy r yy, y = x S xy, r., S yy = Sy, 2 r yy = 1. n 3 n n. R = r R2 = r 2 = b X 1 = a n 3 (x1, x 2,..., x n 1, y), X 2,. ( ) L04 2 I(2017) 23 / 24
2 Excel 2017-10-18 2017-10-25, trial. Excel. 2017-10-25 23:59 2017-10-25,,, ( ) trial. 1-503, http://hig3.net. ( )... 3.5(1-539) 4(1-502), Math - (1-614) 1.4, 2.1, 2.2, 2.3. ( ) L04 2 I(2017) 24 / 24
I L05(2017-10-25 Wed) : Time-stamp: 2017-10-24 Tue 08:38 JST hig 1.4, 2 A B,,, http://hig3.net ( ) L05 I(2017) 1 / 20
2 L04-Q1 Quiz : ( ) x = 4(g), Sx 2 = 4(g 2 ), S x = 2(g). y = 13(cm), Sx 2 = 122/5 = 24.4(cm 2 ), S y = 122/5 = 4.94(cm). S xy = 1 5 [(1 4)(5 13) + (3 4)(15 13) + (4 4)(14 13) + (5 4)(11 13) + (7 4)(20 13)] = 41/5 = 8.2(g cm). r = 2 122/5 41/5 = 0.83. L04-Q2 Quiz : y + 4 = 25 36 49 36 49 (x 9). ( ) L05 I(2017) 2 / 20
1 2 2 ( ) L05 I(2017) 3 / 20
1 1 2 1 2 1 ( ). {, } = {,, }..,, CPU (= ) 77, 77 2,.. ( ) L05 I(2017) 4 / 20
A ( 1 ) ( 1 ) 1. Ω = { 1,..., K}. A = { 1, 2,...} = { x a(x)} ω Ω Ω. Ω A c = Ω \ A. A. A B, A B, A, B A B =. ( ) L05 I(2017) 5 / 20
A =P (A) = a(x) =P (a(x)) Ω =( ), P ({ 1,..., K}) = P (X ) = ( ) P ({ 1}) = P (X 1) = ( 1 ) P ({ 1,..., K, 1,..., K}) = P (X ) = (X ) 1.3 1.1 1.1(p.15) ( ) L05 I(2017) 6 / 20
1 2 2 ( ) L05 I(2017) 7 / 20
2 2, 3. 3, x X. X, Ω = {0, 1, 2, 3}. X.,,. : ( ) L05 I(2017) 8 / 20
x. 0 1 0 f(x) 0 1 10 = 1/ 5C 3 1 6 10 = 2 3/ 5C 3 2 3 10 = 1 3/ 5C 3 3 0. 0 1 ( ) 2.1.1 2.1 1 10 (x = 0) 6 f(x) = 10 (x = 1) 3 10 (x = 2) 0 ( ) 0 f(x) 1. f(x) = 1. x ( ) L05 I(2017) 9 / 20
1 2 2 ( ) L05 I(2017) 10 / 20
ϕ(x) 2.2.1 2.7 AB ϕ(x) E[ϕ(X)] X f(x) =, E[ϕ(X)] = x f(x) ϕ(x) ϕ. : ϕ(x) = x 2, e x, ( ),... E[1] = 1. (ϕ(x) = 1 x f(x) = 1 ) m = E[X]. (ϕ(x) = x ). (x ) = V [X] = E[(X m) 2 ]. (ϕ(x) = (x m) 2 ) = V [X] ( ) L05 I(2017) 11 / 20
A a(x), 1 [a(x)] (x) = { 1 (a(x) ) 0 (a(x) ) P (A) = P (a(x)) = E[1 [a(x)] (X)] 1 [X 2 4](x) = { 1 ( 2 x 2) 0 ( ) ( ) L05 I(2017) 12 / 20
L05-Q1 Quiz( ) X. 4 12 (x = 1) 5 f(x) = 12 (x = 0) 3 12 (x = 2) 0 ( ) 1 E[e X ]. 2 X. 3 X. 4 X. 5 X 1. ( ) L05 I(2017) 13 / 20
( ) L05 I(2017) 14 / 20
, 2.7(p.48) B X:, a, b R:, E[aX + b] = f(x) (ax + b) x ( = a ) f(x)x + b x x f(x) = ae[x] + b. E[ϕ 1 (X) + ϕ 2 (X)] = x f(x) (ϕ 1 (X) + ϕ 2 (X)) =E[ϕ 1 (X)] + E[ϕ 2 (X)]. E[ϕ(X)] ϕ(e[x]), E[X 2 ] (E[X]) 2., sin(x 2 ) (sin(x)) 2 +. ( ) L05 I(2017) 15 / 20
B X:, a, b R:, V[aX + b] = a 2 V[X]. 2.12(p.54) B V[X] = E[X 2 ] (E[X]) 2 ( ) L05 I(2017) 16 / 20
L05-Q2 Quiz( ) X,. V[X] = 9, E[X] = 2. 1 E[ X 2 + 2X 3]. 2 Y = 2X 3 V[ 2X 3]. ( ) L05 I(2017) 17 / 20
L05-Q3 Quiz( ) X. { x f(x) = 55 (0 x 10) 0 ( ) 1 P (X 5). 2 E[X]. 3 V[X]. ( ) L05 I(2017) 18 / 20
L05-Q4 2.3(p.44) L05-Q5 2.1(p.59) L05-Q6 2.6(p.59) ( ) L05 I(2017) 19 / 20
Excel Learn Math Moodle 2017-10-27. 2017-11-01 1 2017-11-22 1 1-503, http://hig3.net. ( )... 3.5(1-539) 4(1-502), Math - (1-614) 1.5. ( ) L05 I(2017) 20 / 20
I L06(2017-11-01 Wed) : Time-stamp: 2017-10-31 Tue 07:53 JST hig,,, 2.3 1.5.4 http://hig3.net ( ) L06 I(2017) 1 / 20
L05-Q1 Quiz : 1 E[e X ] = 4 12 e 1 + 5 12 e0 + 3 12 e2. 2 E[X] = 4 12 ( 1) + 5 12 0 + 3 12 2 = 1 6 (= µ). 3 V[X] = E[(X µ) 2 ] = 4 12 ( 1 1 6 )2 + 5 12 (0 1 6 )2 + 3 12 (2 1 6 )2 = 47 4 V[X] = 47 36. 5 E[1 [a(x)] (X)] = 4 12 1 + 5 12 1 + 3 12 0 = 9 12 = 3 4. L05-Q2 Quiz : E[X 2 ] = V[X] + E[X] 2 = 13. 1 E[ X 2 + 2X 3] = E[X 2 ] + 2E[X] 3E[1] = 13 + 2 2 3 1 = 12. 2 V[ 2X 3] = V[ 2X] = ( 2) 2 V[X] = 36. 36. ( ) L06 I(2017) 2 / 20
2 1 2 2 ( ) L06 I(2017) 3 / 20
2 2 B 6 1. 7 8 9 8 9 9 2 X =, Y = 0( ), 1( ) (x, y) 2.,, joint distribution. 1 3 ((x, y) = (8, 0)) 1 6 ((x, y) = (9, 0)) f XY (x, y) = 1 3 ((x, y) = (9, 1)) 1 6 ((x, y) = (7, 0)) 0 ( ).,. y\x 7 8 9 1 0 6 1 3 1 6 1 0 0 1 3 ( ) L06 I(2017) 4 / 20
2 f XY (x, y), X f X (x), Y f Y (y), f X (x) = y f XY (x, y), f Y (y) = x f XY (x, y), ( ) L06 I(2017) 5 / 20
2 B E[ϕ(X, Y )] = + + x= y= f XY (x, y) ϕ(x, y) ( ) L06 I(2017) 6 / 20
2 L06-Q1 Quiz( ) 2 X, Y. f XY (x, y). y\x 1 2 3 0 0 2/12 1/12 2 4/12 0 5/12 1 E[X + 2Y ]. 2 E[1 [Y 1] (X, Y )]. 3 f X (x), f Y (y). ( ) L06 I(2017) 7 / 20
2 2 B 2.7(p.48) E[ϕ 1 (X, Y ) + ϕ 2 (X, Y )] =E[ϕ 1 (X, Y )] + E[ϕ 2 (X, Y )] E[X + Y ] =E[X] + E[Y ], E[X + Y ] = f XY (x, y) (x + y) x y = f XY (x, y) x + f XY (x, y) y x y x y =E[X] + E[Y ]. ( ) L06 I(2017) 8 / 20
2 X, Y x, y, = =. E[ϕ(X)] = x E[ϕ(Y )] = y f XY (x, y) ϕ(x) = y x f XY (x, y) ϕ(y) = x y ϕ(x) y ϕ(y) x f XY (x, y) = x f XY (x, y) = y ϕ(x) f X (x) ϕ(y) f Y (y) ( ) L06 I(2017) 9 / 20
1 2 2 ( ) L06 I(2017) 10 / 20
B 2.9(p.57) covariance X, Y, µ X = E[X], µ Y = E[Y ], Cov[X, Y ] ( C(X, Y )). Cov[X, Y ] =E[(X µ X )(Y µ Y )] = 2.10(p.58) = E[XY ] E[X] E[Y ]. correlation X, Y,. ρ[x, Y ] = Cov[X, Y ] V[X] V[Y ] 1 ρ[x, Y ] 1. ( ) L06 I(2017) 11 / 20
L06-Q2? ( ) L06 I(2017) 12 / 20
L06-Q3 Quiz( ) X, Y. E[X] = 2, E[Y ] = 3, V[X] = 5, V[Y ] = 11, Cov[X, Y ] = 7. 1 E[ 2X + 3Y ]. 2 V[ 2X + 3Y ]. ( ) L06 I(2017) 13 / 20
1 2 2 ( ) L06 I(2017) 14 / 20
B 1.5.4 X, Y f XY (x, y). X, Y, f XY (x, y) = f X (x) f Y (y) (, ). X, Y, X, Y A, B P (A B) = P (A) P (B) 1.5.3. 2.10(p.57) X, Y, Cov[X, Y ] = 0.. Cov[X, Y ] = 0, X, Y. ( ) L06 I(2017) 15 / 20
L06-Q4 Quiz(2 ) X, Y.. y\x 1 3 2 1/7 2/7 4 0 4/7 1 X, Y. 2 V[X]. 3 Cov[X, Y ]. ( ) L06 I(2017) 16 / 20
L06-Q5 Quiz( ) 2 (X, Y ). f XY (x, y) ( X, Y zero ). y\x 2 3 3 2/12 1/12 7 A B X, Y, A, B. ( ) L06 I(2017) 17 / 20
X, Y 2.13(p.57) E[ϕ 1 (X) ϕ 2 (Y )] =E[ϕ 1 (X)] E[ϕ 2 (Y )] E[XY ] =E[X] E[Y ] 2.9(p.49) Cov[X, Y ] =(E[XY ] E[X] E[Y ] =)0 Cov=0 V[X + Y ] =V[X] + V[Y ] 4.2(p.85) E[XY ] = x = x = x f XY (x, y) x y y f X (x) f Y (y) x y y f X (x) x f Y (y) y = E[X] E[Y ] y V[X + Y ] =E[(X + Y ) 2 ] E[X + Y ] 2 =E[X 2 ] + 2E[XY ] + E[Y 2 ] (E[X] 2 + 2E[X]E[Y ] + E[Y ] 2 ) =V[X] + 2Cov[X, Y ] + V[Y ] ( ) L06 I(2017) 18 / 20
L06-Q6 Quiz( ) X, Y. E[X] = 2, E[Y ] = 3, V[X] = 5, V[Y ] = 11. 1 E[( 2X + 3Y )(X + 5Y )]. 2 V[ 2X + 3Y ]. ( ) L06 I(2017) 19 / 20
L06-Q7 2.3(p.50) L06-Q8 2.4(p.59) L06-Q9 2.5(p.59) 2017-11-22 1. 1-609. ( ( ) PC. Excel ). 2017-11-08 5 3 ( ) =. 2017-11-08 1. 1-503, http://hig3.net. ( )... 3.5(1-539) 4(1-502), Math - ( ) L06 I(2017) 20 / 20
I L07(2017-11-08 Wed) : Time-stamp: 2017-11-08 Wed 07:15 JST hig 2.2.1 2.3 4.1 http://hig3.net ( ) L07 I(2017) 1 / 24
2 L06-Q1 Quiz : 1 E[X + 2Y ] = 0 (1 + 2 0) + 2 1 12 (2 + 2 0) + 2) + 0(2 + 2 2) + 5 62 (3 + 2 2) = 12 12. 12 (3 + 2 0) + 4 12 (1 + 2 2 E[1 [Y 1] (X, Y )] = 0 0 + 2 12 0 + 1 12 0 + 4 12 1 + 0 1 + 5 12 1 = 9 3 4/12 (x = 1) 2/12 (x = 2) 3/12 (y = 0) f X (x) = f Y (y) = 9/12 (y = 2) 6/12 (x = 3) 0 ( ) 0 ( ) 4 (1 ) 12. ( ) L07 I(2017) 2 / 24
2 L06-Q4 Quiz : 1 E[X + 2Y ] = 0 (1 + 2 0) + 2 1 12 (2 + 2 0) + 2) + 0(2 + 2 2) + 5 62 (3 + 2 2) = 12 12. 12 (3 + 2 0) + 4 12 (1 + 2 2 E[1 [Y 1] (X, Y )] = 0 0 + 2 12 0 + 1 12 0 + 4 12 1 + 0 1 + 5 12 1 = 9 3 4/12 (x = 1) 2/12 (x = 2) 3/12 (y = 0) f X (x) = f Y (y) = 9/12 (y = 2) 6/12 (x = 3) 0 ( ) 0 ( ) 4 (1 ) L06-Q6 Quiz : 12. ( ) L07 I(2017) 3 / 24
2 1 E[ 2X + 3Y ] = 2E[X] + 3E[Y ] = 5. 2 V[ 2X + 3Y ] = E[( 2X + 3Y ) 2 ] E[ 2X + 3Y ] 2 = ( 2) 2 V[X] + 2( 2)(3)Cov[X, Y ] + 3 2 V[X] = 20 84 + 99 = 35. L06-Q7 Quiz :2 E[X] = 1 7 1 + 6 7 3 = 19 7, E[Y ] = 3 7 2 + 4 7 4 = 22 7, E[X 2 ] = 1 7 12 + 6 7 32 = 55 7, E[XY ] = 1 7 2 1 + 2 7 3 2 + 4 7 3 4 = 62 7. 1. 2 V[X] = E[X 2 ] E[X] 2 = 24 49. 3 Cov[X, Y ] = E[XY ] E[X]E[Y ] = 16 49. ( ) L07 I(2017) 4 / 24
2 L06-Q8 Quiz : 2 1, 12 + 1 12 + A + B = 1., { 3 f Y (y) = 12 (y = 3) 9 12 (y = 7), f XY (2, 3) =f X (2) 3 12 = 2 12, f XY (3, 3) =f X (3) 3 12 = 1 12, f XY (2, 7) =f X (2) 9 12 = A, f XY (3, 7) =f X (3) 9 12 = B. A, B, f X (2), f X (3), A = 6 12, B = 3 L06-Q9 Quiz : 12. ( ) L07 I(2017) 5 / 24
2 1 X, Y, E[XY ] = E[X]E[Y ], E[( 2X + 3Y )(X + 5Y )] = E[ 2X 2 ] + E[ 7XY ] + E[15Y 2 ] = 2(V[X] + E[X] 2 ) 7E[X]E[Y ] + 15(V[Y ] + E[Y 2 ]) 2, V[XY ] = V[X] + V[Y ], V[ 2X + 3Y ] = V[ 2X] + V[3Y ] = 4V[X] + 9V[Y ]. ( ) L07 I(2017) 6 / 24
1 2 2 ( ) L07 I(2017) 7 / 24
+ I L07-Q1 Quiz( ) X. { x f 10 (x) = 55 (0 x 10) 0 ( ) 1 P (X 5). 2 E[X]. 3 V[X]. ( ) L07 I(2017) 8 / 24
B 2.4(p.36) X, X n, p B(n, p). { nc x p x (1 p) n x (x = 0, 1, 2, 3,..., n) P (X = x) = f(x) = 0 ( ) : p n, x. B(40, 0.1), B(40, 0.5), B(40, 0.7), B(4, 0.8), B(20, 0.8), B(40, 0.8) ( ) L07 I(2017) 9 / 24
2.2(p.44). E[X] =, V[X] = A 2.1(p.36) E[1] = n (a + b) n = nc x a x b n x x=0 ( ) L07 I(2017) 10 / 24
L07-Q2 Quiz( ) p = 2 3 100. 1 40... 2,. ( ) L07 I(2017) 11 / 24
2.3(p.36) n = 1 B(1, p) 1 p (x = 0) P (X = x) = f(x) = p (x = 1) 0 ( ) : =( ). p. x = 1. 2.2(p.44) E[X] =, V[X] = ( ) L07 I(2017) 12 / 24
2.2(p.37) X 1, X 2,..., X n X i B(1, p), U n = X 1 + + X n U n B(n, p). ( ) L07 I(2017) 13 / 24
( ) X 1, X 2, U 2 = X 1 + X 2. E[U 2 ] = E[X 1 + X 2 ] = E[X 1 ] + E[X 2 ]. X 1, X 2 V[U 2 ] = V[X 1 + X 2 ] = V[X 1 ] + V[X 2 ]. ( ) L07 I(2017) 14 / 24
L07-Q3 Quiz( ), 2. 0.05. 1000, 0. Y ( ). 1 Y, B(1, p) X. 2 Y.. ( ) L07 I(2017) 15 / 24
1 2 2 ( ) L07 I(2017) 16 / 24
Chebyshev s inequality 2.11(p.52) X: ( ) µ = E[X]: σ 2 = V[X]: a > 0:. P ( X µ aσ) 1 a 2 X. ( ) L07 I(2017) 17 / 24
( ) ( ) L07 I(2017) 18 / 24 P ( X µ aσ) P ( X µ aσ) = x 1 [ X µ aσ] (x) f(x) (x µ)2 1 [ X µ aσ] (x) (aσ) 2 f(x) x (x µ) 2 (aσ) 2 f(x) x = 1 (aσ) 2 (x µ) 2 f(x) x = 1 (aσ) 2 V[X] = 1 a 2.
L07-Q4 2.2(p.59) ( ) L07 I(2017) 19 / 24
4.1 (i.i.d.) 4.1 / X 1, X 2,..., X n,, ( f(x)). X 1,..., X n (i.i.d.=independent and identically-distributed). : U n = X 1 + + X n E[X i ] = µ, V[X i ] = σ 2, U n n? E[U n ] = E[X i ] = n µ. V[U n ] = i=1 n V[X i ] = n σ 2. 4.2(p.85) i=1 ( ) L07 I(2017) 20 / 24
W n? : W n = 1 n U n = 1 n (X 1 + + X n ) E[W n ] =E [ 1 n U ] 1 n = n n µ. V[W n ] =V [ 1 n U n] = ( 1 n) 2 n σ 2. ( ) L07 I(2017) 21 / 24
L07-Q5 Quiz( ) X 1,..., X 100 E[X i ] = 3, V[X i ] = 7.. 1 A = 1 100 (X 1 + X 2 + X 3 + + X 100 ) 2 B = 1 10 (X 1 + X 2 + X 3 + + X 100 100 3) 3 C = 1 10 (X 7 1 + X 2 + X 3 + + X 100 100 3) ( ) L07 I(2017) 22 / 24
( ) 4.1(p.84) X 1,..., X n, E[X i ] = µ, V[X i ] = σ 2,W = 1 n n i=1 X i, ϵ > 0 lim P ( W n µ ϵ) = 0. n + n W n E[W n ] = E[X i ] ( ). E[W n ] = µ, V[W n ] = σ 2 /n, W n, P ( W n µ a σ n ) 1 a 2 a = ϵ σ n, n + P ( W n µ ϵ) σ2 ϵ 2 n 0., ( ) L07 I(2017) 23 / 24
2017-11-22. 1-609. 30.. 2017-11-16,. ( ) PC(Moodle). Excel.,,,,, (L02),, ( ) (L03),, (L04, ) 1,,,,, n (L05) 1 2,,,,, (L05,L06) 2,,,,,,,, (L06),,,,, (L07) X i W, X, W, (L05,L07),,,,, n (L08) :, ( )., trial. Learn Math Moodle,,. ( ) L07 I(2017) 24 / 24
I L08(2017-11-15 Wed) : Time-stamp: 2017-11-14 Tue 18:39 JST hig 3 B,,,, http://hig3.net ( ) L08 I(2017) 1 / 22
L07-Q1 L07-Q2 Quiz : 1 B(100, 2 3 ) X. P (X = 40), 100 C 40 p 40 (1 p) 100 40 = 100! 40!60! ( 2 3 )40 (1 2 3 )60. 2 E[X] = n p = 200 200 3. V[X] = n p(1 p) = 9. L07-Q3 Quiz : 1 B(1, 0.05) X, Y = 1000X. 2 E[Y ] = E[1000X] = 1000E[X] = 1000p = 50.( ) V[Y ] = V[1000X] = 1000 2 V[X] = 1000 2 p(1 p) = 47500. ( 2 ) ( ) L08 I(2017) 2 / 22
L08-Q4 Quiz :, E[aX + by + c] = ae[x] + be[y ] + c, V[aX + by + c] = a 2 V[X] + b 2 V[Y ]. 1 A 3, 7 100. 2 B 0, 7. 3 C 0, 1. ( ) L08 I(2017) 3 / 22
1 2 ( ) L08 I(2017) 4 / 22
0 : 4, 3, 2, 1, 0 0k 4 2 3 1 s f(s) 4 0.1 3 0.3 2 0.3333 1 0.2 0 0.0667 Score 4 3 2 1 0 0.05 0.1 0.15 Probability 0.2 0.25 0.3 0.35 0.4 Probability 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 4 3 2 1 0 Score ( ) L08 I(2017) 5 / 22
x cm x cm, s = 4 x ( ). 0.4 x 0.35 0.3 0 1 4 Probability 0.25 0.2 0.15 0.1 0.05 0 0 1 2 3 4 5 6 Distance from center r = 0.5cm 0.9 cm. r = 1.0cm.. x f(x)! f(x) (x ) f(x) (x ) ( ) L08 I(2017) 6 / 22
3.1 X,, f(x). x f(x) 0 0.1 1 0.3. x f(x) 0 f(x). f(x) 1. p(x) f(x) II ( ) L08 I(2017) 7 / 22
p p p 2.0 2.0 2.0 0.4 0.35 1.5 1.5 1.5 0.3 0.25 1.0 1.0 1.0 Probability 0.2 0.15 0.5 0.5 0.5 0.1 0.05 0.5 1.0 1.5 2.0 y 0.5 1.0 1.5 2.0 s 0.5 1.0 1.5 2.0 s 0 0 1 2 3 4 5 6 Distance from center P (a X < b) = ( ) = b a f(x) dx ( ) ( ) L08 I(2017) 8 / 22
E[ϕ(X)] = x f(x) ϕ(x) lim E[ϕ(X)] = f(x i ) x = i + f(x) dx. f(x) ϕ(x) dx : µ = E[X], V[X] = E[(X µ) 2 ] E[aX + b] = ae[x] + b V[aX + b] = a 2 V[X] V[X] = E[X 2 ] E[X] 2 2.12(p.54) ( ) L08 I(2017) 9 / 22
( ) L08 I(2017) 10 / 22
L08-Q1 Quiz( ) X. { 8x (0 x < 1 f(x) = 2 ) 0 ( ) 1 X + 1 4. 2 E[X]. 3 V[X]. 4 1 E[ X ]. 3.2 ( ) L08 I(2017) 11 / 22
( ) L08 I(2017) 12 / 22
P ( ) = P ( ) = E[1 [ ] (X)] P (a X < b) = E[1 [a X<b] (X)] = = 1 = + + f(x)1 [a X<b] (x) dx = f(x) dx = E[1] b, x = a cm?. 0.4 a f(x) dx 0.35 0.3 0.25 Probability 0.2 0.15 0.1 0.05 0 0 1 2 3 4 5 6 Distance from center 1 [X ] (x) = { 1 (x ) 0 ( ) ( ) L08 I(2017) 13 / 22
L08-Q2 Quiz( ) X. 1 4x (0 x < 2) f(x) = 1 2 (2 x < 3) 0 ( ) 1 E[X]. 2 P (X 1). 3 E[ 1 X ]. ( ) L08 I(2017) 14 / 22
( ) L08 I(2017) 15 / 22
1 2 ( ) L08 I(2017) 16 / 22
3.4(p.66) U(a, b) X, X [a, b) U(a, b). { C( ) (a x < b) f(x) = 0 ( ) L08-Q3 Quiz( ) X U(a, b). 1 C. 2 E[X]. 3 V[X]. ( ) L08 I(2017) 17 / 22
( ) L08 I(2017) 18 / 22
f(x) 0.4 0.35 0.3 0.25 Probability 0.2 0.15 0.1 0.05 0 0 1 2 3 4 5 6 Distance from center ( ) L08 I(2017) 19 / 22
Y = ax + b X U(r, s), Y = ax + b U(ar + b, as + b). E[aX + b] = V[aX + b] = f(x) 2.0 1.5 1.0 0.5 2 4 6 8 10 12 x -0.5 X U(3, 5), Z = 1 4 X + 21 4, Y = 2X + 1. ( ) L08 I(2017) 20 / 22
1-503, http://hig3.net. ( )... 3.5(1-539) 4(1-502), Math - (1-614) ( ) L08 I(2017) 21 / 22
2017-11-22. 1-609. 30.. 2017-11-16,. ( ) PC(Moodle). Excel.,,,,, (L02),, ( ) (L03),, (L04, ) 1,,,,, n (L05) 1 2,,,,, (L05,L06) 2,,,,,,,, (L06),,,,, (L07) X i W, X, W, (L05,L07),,,,, n (L08), ( )., trial. Learn Math Moodle,,. ( ) L08 I(2017) 22 / 22
I L09(2017-11-29 Wed) : Time-stamp: 2017-11-30 Thu 07:24 JST hig. p.68-69,. 4.2 http://hig3.net ( ) L09 I(2017) 1 / 20
L08-Q1 Quiz : 1 2 3 4 + f(x)1 1 dx = [X 4 ](x) E[X] = 1/2 0 1/2 1/4 f(x) x dx = 1/3. 8x dx = 3 4. V[X] = E[X 2 ] (E[X]) 2 = 1 8 ( 1 3 )2 = 1 72. E[ 1 X ] = 2 5/2 /3. L08-Q2 Quiz : ( ) L09 I(2017) 2 / 20
1 23 12 2 7 8 3 1 2 + 1 2 (log 3 log 2). L08-Q3 Quiz : 1 E[1] = 1, C = 1 b a. 2 E[X] = a+b 2. V[X] = b a 3 12 b a 3.5. ( ) L09 I(2017) 3 / 20
1 2 ( ) L09 I(2017) 4 / 20
B 3.5(p.68) =normal ( ) N(b, a 2 ) b, a 2 : f(x; b, a 2 ) = 1 (x b)2 e 2a 2. 2πa 2 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 N(0,1) N(3,2 2 ) -2 0 2 4 6 8 x, b = 0, a = 1. N(0, 1 2 ) z = x b a x = az + b y = f(z; 0, 1), a, b, 1/ a 2 y = f(x; b, a 2 ) ( ) L09 I(2017) 5 / 20
N(0, 1 2 ) N(0, 1 2 ) f(z; 0, 1 2 ) = 1 2π e z2 2. E[1] =1, (3.7)(p.68) II E[Z] =0,. 3.2(p.68) V[Z] =1 3.2(p.68) II ( ) L09 I(2017) 6 / 20
, f(x) X, F (a) = a f(x)dx = P (X < a). N(0, 1 2 ) Φ(z) = Q(z) = 1 Φ(z) = z z z. z f(z ; 0, 1 2 )dz. f(z ; 0, 1 2 ) dz = P (Z > z) ( ) L09 I(2017) 7 / 20
. Q( ) = 1, Q(+ ) = 0. f(z; 0, 1 2 ) Q( z) = 1 Q(z), Q(0) = 1 2. P (c < z < d) Q(z) ( 0 < z < + ). Q(z) B(p.188) 0.4 σ σ 0.4 σ σ 0.3 0.3 0.2 0.2 0.1 0.1 0.0 μ 1σ 2σ3σ 0.6827 0.9545 0.9973-4 -2 0 2 4 0.0 μ 1.96σ 2.58σ -4-2 0 2 4 0.9500 0.9900 ( ) L09 I(2017) 8 / 20
L09-Q1 Quiz( ) Z N(0, 1 2 ). Z < 2, Q(z) ( 0 < z < + ).. L09-Q2 Quiz( ) Z N(0, 1 2 ). 1 E[Z 2 ]. 2 P ( 0.56 < Z < +1.23) Q(z) ( 0 < z < + ),. ( ) L09 I(2017) 9 / 20
N(b; a 2 ) Z = X b a X = az + b, Z N(0, 1 2 ). E[1] =1, (3.7)(p.68) II µ X = E[X] =E[aZ + b] = b, 3.2(p.68) σ 2 X = V[X] =V[aZ + b] = a 2,, b = µ X = m, a 2 = σ 2 X = v. 3.2(p.68) II ( ) N(µ, σ 2 ) 3.5(p.68) µ, σ 2 N(µ, σ 2 ), f(x; µ, σ 2 ) = 1 (x µ)2 e 2σ 2. 2πσ 2 ( ) L09 I(2017) 10 / 20
L09-Q3 Quiz( ) X, f(x) = (x 4)2 1 e 2 3 2 2π 3 2. 1 f(x). 2 E[X]. 3 V[X]. ( ) L09 I(2017) 11 / 20
N(µ, σ 2 ) I Prob x Μ 1.5Σ p x 1.0 0.8 0.6 0.4 0.2 6 4 2 2 4 6 x Z = X µ X σ X N(0, 1 2 ),. X N(µ X, σ 2 X ), Z N(0, 12 ) P (c < X < d) = P ( c µ X σ < X µ X σ < d µ X σ ) = P ( c µ X σ < Z < d µ X σ ) ( ) L09 I(2017) 12 / 20
L09-Q4 Quiz( ) X 3, 4., Q(z) ( 0 < z < + ),,. 1 X 5 2 +1 X 7 ( ) L09 I(2017) 13 / 20
1 2 ( ) L09 I(2017) 14 / 20
4.2 ( ) X 1,..., X n µ, σ 2, n + U n = X 1 + + X n,, W n = 1 n (X 1 + + X n ), Z n = Wn µ σ n, ( ) L09 I(2017) 15 / 20
( ) 4.3(p.87) X 1, X 2,..., X n, µ, σ 2.. 1 n (X 1+ +X n) µ Z n = σ n, Z n, n +, N(0, 1 2 ). b lim P (a Z 1 n < b) = e 1 2 x2 dx n + a 2π Z n N(0, 1 2 ) Z,. E[Z n ] = 0, V[Z n ] = 1 II()L ( ) L09 I(2017) 16 / 20
8.4 I L09-Q5 Quiz( ) X 1,..., X 100 E[X i ] = 1, V[X i ] = 1 4.., n = 100,. 1 U = X 1 + X 2 + X 3 + + X 100. P (U > 110). 2 W = 1 100 (X 1 + X 2 + X 3 + + X 100 ). P (W < 1.01). ( ) L09 I(2017) 17 / 20
( ) L09 I(2017) 18 / 20
B 8.4 L09-Q6 Quiz( ) 1 10, 9 U. 10, 400, 1 U. B(?,?). 2 U. N(?,?). 3 31, Q(z),. ( ) L09 I(2017) 19 / 20
7-002. trial. 1-503, http://hig3.net. ( )... 3.5(1-539) 4(1-502), Math - (1-614) 6.1-6.3 ( ) L09 I(2017) 20 / 20
I L10(2017-12-06 Wed) : Time-stamp: 2017-12-06 Wed 12:08 JST hig,. 4.2 http://hig3.net ( ) L10 I(2017) 1 / 26
L09-Q1 Quiz : (z = 0 ), P (Z < 2) = 2 f(z) dz = + L09-Q2 Quiz :. +2 f(z) dz = Q(2) = 0.0228. 1 E[Z 2 ] = V[Z] + (E[Z]) 2 = 1 0 2. 2 P ( 0.56 < Z < +1.23) = 1.23 0.56 f(z) dz = f(z) dz 0.56 f(z) dz 1.23 f(z) dz = 1 Q(1.23) Q(0.56) = 1 0.1093 0.2877 = 0.6030. ( ) L10 I(2017) 2 / 26
L09-Q3 Quiz :,, X N(4, 3 2 ), 1 E[X] = 4. 2 V[X] = 3 2. L09-Q4 Quiz : N(0, 1 2 ). Z = X 3 2, Z 1 P (X 5) = P (Z 5 3 2 ) = 1 f(z) dz. 1 f(z) dz = Q(1.00) = 0.1587. 1 f(z) dz = 0 1 f(z) dz + 0 f(z) dz = I(1.00) + 1 2. ( ) L10 I(2017) 3 / 26
2 Z = X 3 2, Z. P (1 X 7) = P ( 1 Z 2) = 2 f(z) dz. 2 1 f(z) dz = 1 Q(2.00) Q(1.00) = 0.8186. 2 1 f(z) dz = 1 0 f(z) dz + 2 0 f(z) dz = I(1) + I(2) = 0.8186. 1 ( ) L10 I(2017) 4 / 26
1 2 ( ) ( ) ( ) L10 I(2017) 5 / 26
4.1(p.84) I(2017)L07 X 1,..., X n. E[X i ] = µ, V[X i ] = σ 2. : U n = X 1 + + X n E[U n ] = V[U n ] = n E[X i ] = n µ. i=1 n V[X i ] = n σ 2. i=1 : W n = 1 n U n = 1 n (X 1+ +X n ) E[W n ] =E [ 1 n U ] n = 1 n n µ. V[W n ] =V [ 1 n U n] = ( 1 n) 2 n σ 2. ( ) L10 I(2017) 6 / 26
4.2 ( ) X 1,..., X n µ, σ 2, n + U n = X 1 + + X n,, W n = 1 n (X 1 + + X n ), Z n = Wn µ σ/ n, ( ) L10 I(2017) 7 / 26
( ) 4.3(p.87) X 1, X 2,..., X n, µ, σ 2.. 1 n (X 1+ +X n) µ Z n = σ n, Z n, n +, N(0, 1 2 ). b lim P (a Z 1 n < b) = e 1 2 x2 dx n + a 2π Z n N(0, 1 2 ) Z,. E[Z n ] = 0, V[Z n ] = 1 II()L ( ) L10 I(2017) 8 / 26
B 8.4 I L10-Q1 Quiz( ) 1 10, 9 U. 10, 400, 1 U. B(?,?). 2 U. N(?,?). 3 31, Q(z),. ( ) L10 I(2017) 9 / 26
( ) L10 I(2017) 10 / 26
( U 1, U 4, U 9 ) X n B(1, 2 3 ) f(x) = { 2 3 (x = 1) 3 4 5 6 1 3 (x = 0) 1 2 U n = X 1 + + X n. n 1 2 3 4 5 6 7 8 9 (1 6) X n (0 1) U n (0 9) * * * https://manaba.ryukoku.ac.jp. ( ) L10 I(2017) 11 / 26
1 2 ( ) ( ) ( ) L10 I(2017) 12 / 26
(1) 6.1 AKB48 AKB48 ( ) x i x = 1 N N i=1 x i! 1,, X E[X]. 1? 74. 5 (= ). 5 (= ) =, =, =. ( ) L10 I(2017) 13 / 26
(2) or, ( ). X ( ). E[X] = x f(x) x. (f(x) ),. +,. 5 (= ). 5 (= ). =, =, =. ( ) L10 I(2017) 14 / 26
6.1,6.2 population =.,,,,. sample ( ) = sample( )= sampling ( ) estimate( ) = estimation ( ).. ( ) L10 I(2017) 15 / 26
( ) 1 2 ( ) ( ) ( ) L10 I(2017) 16 / 26
( ) ( ) B X 1, X 2,..., X n n. X i (i = 1,..., n) µ = E[X i ], σ 2 = V[X i ]. µ, σ 2. p.132 X (n) = 1 n (X 1 + + X n ) = W n, µ. µ, X (n), = (X (n) ) ( ) L10 I(2017) 17 / 26
( ) L10-Q2 Quiz(,, ) (= ), 6,. 117g, 109g, 109g, 119g, 100g, 112g. 1. 2. 3 110g. ( ) L10 I(2017) 18 / 26
( )? 6.1(p.132) X (n) X (n) =X i E[W n ] = µ n E[X (n) ] = µ X (n) n, X (n) µ 0. ( ) ϵ > 0 lim P ( X (n) µ > ϵ ) 0 n + ( ) L10 I(2017) 19 / 26
( ) ( ) B p.134 (p.134) (p.134) s 2 = 1 n 1 [(X 1 X) 2 + + (X n X) 2 ] [ = n 1 Xi 2 ( X ) ] 2 n 1 n,., X, (X (n) ). n 1 : E[s 2 ] = σ 2. X X i, (X i X) 2 (X i µ) 2 ( n 1 n ). ( ) n = 1, ( ) L10 I(2017) 20 / 26 i
( ) E[s 2 ] = σ 2 n = 2 ( 6.2,6.3 ) = 1 2 1 E[(X 1 X) 2 + (X 2 X) 2 ] =E[X 2 1 + X 2 2 2(X 1 + X 2 )X + 2X 2 ] =E[X 2 1 + X 2 2 2X 2 ] =E[X 2 1] + E[X 2 2] 2E[X 2 ], σ 2 = V[X 1 ] = E[X 2 1 ] (E[X 1]) 2 = E[X 2 1 ] µ2, σ 2 n = V[X] = E[X2 ] (E[X]) 2 = E[X 2 ] µ 2,, =(µ 2 + σ 2 ) + (µ 2 + σ 2 ) 2(µ 2 + σ2 2 ) =σ 2 = ( ) L10 I(2017) 21 / 26
( ) 1 2 ( ) ( ) ( ) L10 I(2017) 22 / 26
( ) { 165cm ( ), L10 165cm }. I(2017) 23 / 26 =ratio 7.5.1,8.4 Y B(1, p),. Y,, 2. X, Y = 1 [ ] (X), X > 10 Y = 1. =, x A Y = 1. B(1, p) p. f(x) B(1, p),,., x p = = E[Y ]
( ) : p ( )!, A? n. A %? n., %? n.? n. ( ) L10 I(2017) 24 / 26
( ) ( ) n k, ˆp = k n p. p = E[Y ] n k, E[Y ] = Y = 1 n [1 } + {{ + 1 } + 0 } + {{ + 0 } ] k n k = k n = ˆp. ( ) L10 I(2017) 25 / 26
( ) 7-002. trial. 1-503, http://hig3.net. ( )... 3.5(1-539) 4(1-502), Math - (1-614) 8.1-8.4. ( ) L10 I(2017) 26 / 26
I L11(2017-12-13 Wed) : Time-stamp: 2017-12-25 Mon 17:33 JST hig ( 8.2, 8.3 ) 8.4 t 6.4.4 http://hig3.net ( ) L11 I(2017) 1 / 24
L10-Q1 Quiz : 1 U, B(400, 1 10 )., E[U] = 40, V[U] = 36. 2 i, { 1 ( ) X i = 0 ( ), U = X 1 + + X 400. X i (i = 1,..., 400), n = 400,, U N(40, 36). 3 Z = U 40 36 N(0, 1 2 ).,, P (U > 31) = P (Z > 9 6 ) = Q( 3 2 ) Q( ) = (1 Q( 3 2 )) 0 = 0.9332. ( ) L11 I(2017) 2 / 24
L10-Q2 Quiz :,, 1, 1 6 (117 + + 112) = 111g, 111g. 2 1, 6 1 [(117 111)2 + + (112 111) 2 ] = 46g 2, 46g 2. 3, 1 6 [1 + 0 + 0 + 1 + 0 + 1] = 0.5, 0.5. ( ) L11 I(2017) 3 / 24
(, ) 10 11 (, ) (, ) ( p) ( ) L11 I(2017) 4 / 24
(, ). B, C. ( ) L11 I(2017) 5 / 24,, A? 8.1, B C 1 α = 0.95. 1 α = 0.95 B C
(, ) (, ) B 8.2 N(µ, σ 2 ) ( ), n,, X (n)., W = X (n) N(µ, σ 2 /n). X (n) µ σ 2 /n N(0, 12 ). II()L, n +., n (30 ),, X (n) µ σ 2 /n N(0, 12 ). ( 1 α = 1 0.05) Q(1.96) = 0.05/25, P ( 1.96 < X (n) µ σ 2 /n < +1.96) = 1 0.05. P (µ 1.96 σ 2 /n < X (n) < µ + 1.96 σ 2 /n) = 1 0.05. µ, P (X (n) 1.96 σ 2 /n < µ < X (n) + 1.96 σ 2 /n) =1 0.05. ( ) L11 I(2017) 6 / 24
(, ) ( ) 0.4 σ σ 0.4 σ σ 0.3 0.3 0.2 0.2 0.1 0.1 0.0-4 -2 0.0 μ0 2 4 1σ 2σ3σ -4-2 μ0 2 4 1.96σ 2.58σ 0.6827 0.9545 0.9500 0.9973 0.9900 0.05/2, 0.01/2 Q(z), P ( 1.96 < Z < 1.96) =0.95, P ( 2.58 < Z < 2.58) =0.99 ( ) L11 I(2017) 7 / 24
(, ) (, ) B 8.2 N(µ, σ 2 ), σ 2, n, µ 1 α = 0.95 (95% ),1 α = 0.99 (99% ), X (n) 1.96 σ 2 /n <µ < X (n) + 1.96 σ 2 /n, X (n) 2.58 σ 2 /n <µ < X (n) + 2.58 σ 2 /n, µ 0.95 or 0.99. B, 1 α = 0.95 1.96. a < µ < b, [a, b]. σ 2, ( 1 1 n i (Xi X)2 n 1 ) S2 = ( ) L11 I(2017) 8 / 24
(, ) L11-Q1 Quiz( ( )) Xg,., X σ 2 = 9g 2. 4,. 51g, 52g, 47g, 50g. 1 µ = E[X], 1 α = 0.95. 2 µ = E[X], 1 α = 0.99. L11-Q2 8.1 ( ) L11 I(2017) 9 / 24
(, ) ( ) L11 I(2017) 10 / 24
(, ). Quiz( ),? 1, 2, 3, 4, ( ) L11 I(2017) 11 / 24
(, ) 10 11 (, ) (, ) ( p) ( ) L11 I(2017) 12 / 24
(, ) (, ) 8.3 µ σ 2.. σ 2 s 2 ( )., T = X (n) µ s 2 /n, N(0, 1 2 ) n 1 Student t.. t 6.4.4 k + N(0, 1 2 ). k, N(0, 1 2 ). f k (x) = A k (1 + 1k ) k+1 2 x2. ( ) L11 I(2017) 13 / 24
(, ) t B2 α = 0.025, 0.005, k, α = P (T > tk(α)) tα(k). t0.025(k) 1.960, t0.005(k) 2.576 (k + ). t-distribution, k=5 0.4 t-distribution(k = 2, 5, 10), N(0, 1) 0.4 0.3 0.2 0.2 α 0.05 α 1-α 0.95 2 0.025 α 2 0.025 -t 0.025 (5) 0 t 0.025 (5) t-distribution, k=5-5 -3-1 1 3 5 0.4 0.1-4 -2 2 4 t 0.2 α 0.01 α 1-α 0.99 2 0.005 α 2 0.005 -t -5 0.005 (5) -3-1 0 t 0.005 (5) 5 1 3 ( ) L11 I(2017) 14 / 24
(, ) ( ) 8.3 ( ) N(µ,? 2 ), n, µ 1 α X (n) t α/2 (n 1) s 2 /n < µ < X (n) + t α/2 (n 1) s 2 /n., s 2 :, n:, t α/2 (n 1): n 1 t α/2 ( ). ( ) L11 I(2017) 15 / 24
(, ) L11-Q3 Quiz( ( )) Xg, 4,. 51g, 52g, 47g, 50g. 1 µ = E[X], 1 α = 0.95. 2 µ = E[X], 1 α = 0.99. L11-Q4 8.2, 8.2(p.166), 8.2(p.171) ( ) L11 I(2017) 16 / 24
(, ) ( ) L11 I(2017) 17 / 24
(, ) (, ) B n 1, t N(0, 1 2 ).,,,. L11-Q5 Quiz( (, )) Xg. 400,. 51g, 52g, 47g,..., 50g.,, m = 51g, s 2 = 4g 2. 1 µ = E[X i ], 1 α = 0.95. 2 µ = E[X i ], 1 α = 0.99. ( ) L11 I(2017) 18 / 24
( p) 10 11 (, ) (, ) ( p) ( ) L11 I(2017) 19 / 24
( p) B 8.4 A %? n. %? n.? n.,, x = 1 x = 0., X B(1, p) ( ) : E[X] = p, V[X] = p(1 p)., p.. n, X = = 1 X i = Y n n Y X = 1. ˆp = y n., Y B(n, p). ( ) L11 I(2017) 20 / 24 i
( p) X B(n, p), ˆp(1 ˆp) X, (, ), ( ) 8.4 X n, ˆp = y/n, 1 α = 0.95, 1 α = 0.99 1 ˆp 1.96 n ˆp(1 ˆp) <p < ˆp + 1.96 1 n ˆp(1 ˆp), 1 ˆp 2.58 n ˆp(1 ˆp) <p < ˆp + 2.58 1 n ˆp(1 ˆp). ( ) L11 I(2017) 21 / 24
( p) L11-Q6 Quiz( ), 50 35 A.. A ( ). 1 A, ( ) 2 A, 1 α = 0.95. 3 A, 1 α = 0.99. :, 0,1, 0,1. L11-Q7 8.4,8.5, 8.4,8.5, 8.4(p.170) ( ) L11 I(2017) 22 / 24
( p), ( ) 1.96,2.58 1.96,2.58, ( ) t t n + 1.96,2.58 ( ). 1.96,2.58,,,. ( ) L11 I(2017) 23 / 24
( p) 1-503, http://hig3.net. ( )... 3.5(1-539) 4(1-502), Math - (1-614) 7 ( ) L11 I(2017) 24 / 24
I L12(2017-12-20 Wed) : Time-stamp: 2017-12-25 Mon 17:31 JST hig 7.1, 7.2, 7.3 t 7.4.2 ( ) 7.5.1 http://hig3.net ( ) L12 I(2017) 1 / 25
L11-Q1 Quiz : ( ) 1 m = 50g., 0.95 9 50 1.96 4 < µ < 50 + 1.96 9 4., 47.06 < µ < 52.94. 2, 9 50 2.58 4 < µ < 50 + 2.58, 46.13 < µ < 53.87. L11-Q2 L11-Q3 Quiz : ( ) 9 4. ( ) L12 I(2017) 2 / 25
1 m = 50g. s 2 = 1 4 1 14g2. k = n 1 = 3 t, 0.95 50 3.182 1 4 14 3 < µ < 50 + 3.182 1 4 14 3. 2, 50 5.841 1 4 14 3 < µ < 50 + 5.841 1 4 14 3. L11-Q4 L11-Q5 Quiz : (, ) 1, t,. 0.95 4 51 1.96 400 < µ < 51 + 1.96 4 400. ( ) L12 I(2017) 3 / 25
2, 4 51 2.58 400 < µ < 51 + 2.58 4 400. L11-Q6 Quiz : A X = 1, X = 0. 1 ˆp = 35 50 = 0.7. p 0.7. 2 X 0.7 (1 0.7) = 0.21. p 1 α = 0.95, 1 0.7 1.96 50 0.21 <p < 0.7 + 1.96 1 50 0.21 0.7 0.13 <p < 0.7 + 0.13 0.57 <p < 0.83 0.95 (, 0.05 ). ( ) L12 I(2017) 4 / 25
3 p 0.99, L11-Q7 0.7 2.58 0.0042 <p < 0.7 + 2.58 0.0042 0.7 0.17 <p < 0.7 + 0.17 0.53 <p < 0.87 0.99,. ( ) L12 I(2017) 5 / 25
i Y i X(cm) s 2 (cm 2 ) 1 4 2 169.5 97.7 2 5 1 165.8 5.7 3 2 2 175.0 50 4 7 4 169.7 24.9 5 4 1 167.5 21.7 6 3 2 167.7 30.3 7 4 1 161.0 62 7.5 5 2 185.0 250 8 3 1 170.0 1.0 9 5 1 175.8 35.2 10 8 3 168.8 19.6 11 7 3 165.0 39.7 12 4 1 169.5 51.0 13 7 1 168.9 171.8 ( ) L12 I(2017) 6 / 25
Team Number 14 12 10 8 6 4 0.95 0.99 sample size 7 4 7 8 5 4 5 3 3 4 7 2 5 4 0 0 50 100 150 200 250 Height(cm) :,.. ( ) L12 I(2017) 7 / 25
Team Number 14 7 0.95 0.99 12 sample size 4 7 10 8 5 8 3 5 4 6 3 4 4 7 2 2 5 4 0 0 0.2 0.4 0.6 0.8 1 Ratio : 0.5, 2 1.96 0.5/ n. ( ) L12 I(2017) 8 / 25
11 12 t ( ) ( ) L12 I(2017) 9 / 25
7.1 µ xx µ yy zz ( 1 α ) µ xx α ) or ( =, X( ) 55g., 5,., 55 g?( 55 g )., 1 63. 5.. ( ) L12 I(2017) 10 / 25
( ) 7.1 = 0.01 ( ), A, ( = )., 10 X = 2. A, ( ), P (X 2) =P (X = 2) + P (X = 3) +... + P (X = 10) =1 P (X = 0) P (X = 1) =1 10! 10!0! 0.010 0.99 10 10! 9!1! 0.011 0.99 9 =1 0.904 0.091 = 0.0042662., 0.0042662. ( ) α = 0.05,., X = 1. X = 2, 3, 4,.... p = 0.01, > 0.01. 7.5.1 ( ) L12 I(2017) 11 / 25
?,,. Yes/No, (test)= (statistical hypothesis test) 55g,, 54g,,.. 55g.,,,.,.. ( ) L12 I(2017) 12 / 25
H 0 : (null hypothesis) = = µ 55g H 1 : (alternative hypothesis) = = µ 55g. H 1 : µ > 55 ( ). significance level α ( 1 ). α. ( α ). ( ) L12 I(2017) 13 / 25
( α ) Y, ( α ) / Y, ( α ), H 0 (reject) H 1 (accept) (significant) H 1., H 0 H 0 (not significant) H 0 ( ) L12 I(2017) 14 / 25
t 11 12 t ( ) ( ) L12 I(2017) 15 / 25
t t I L12-Q1 Quiz( ( )=t ) X i g,. 57g, 5,. 52g, 52g, 53g, 48g, 50g. X i g 57g..,,. t,. ( ) L12 I(2017) 16 / 25
t ( ) L12 I(2017) 17 / 25
t 7.3(p.155), 7.4(p.156), 7.3(p.157), 7.1(p.162) ( ) L12 I(2017) 18 / 25
t... 1 α = (2,3 ) 2 3 Y 4 y = 5 (y ) / ( ) Y Y,...,,,.. ( ) L12 I(2017) 19 / 25
t p =p = (y 1 ). p. α > p α < p y y 1 y y 1 y t t α/2 (n 1) < t t α/2 (n 1) > t ( ) L12 I(2017) 20 / 25
t L12-Q2 Quiz( t ),, 9:00 10:00 196. 4,. 204, 208, 188, 200.? L12-Q3 160cm ( ).,. ( ),,. ( ) L12 I(2017) 21 / 25
( ) 11 12 t ( ) ( ) L12 I(2017) 22 / 25
( ) ( ) 7.5.1 1 α ( ) 2 p = ˆp p 3 Z = p(1 p)/n N(0, 1 2 ) (n ) 4 5 ( ) L12-Q4 Quiz( ), p = 0.5,, 68, 25. p = 0.5,? (,, p > 0.5, ). ( ) L12 I(2017) 23 / 25
( ) 2018-01-10 9:20. t. Learn Math Moodle, 1 5. 2018-01-10, 10, 11, 15,16 Math. 6.4.3, 7.4.3, 8.3 1-503, http://hig3.net. ( )... 3.5(1-539) 4(1-502), Math - (1-614) ( ) L12 I(2017) 24 / 25
( ),,, -, -, ( ). ( ) L12 I(2017) 25 / 25
I L13(2018-01-10 Wed) : Time-stamp: 2018-01-13 Sat 09:45 JST hig 6.4.3 S8.3 / 7.4.3 http://hig3.net ( ) L13 I(2017) 1 / 24
L12-Q1 Quiz : ( )=t 1 0.05, t. 2 µ 57g. 3 n = 5 X, s 2, T = X µ 0 s 2 /n, 5 1 t. 4, t = x µ 0 s 2 n = 51 57 1 16 5 5 1 = 3 5 = 6.708. ( ) L13 I(2017) 2 / 24
5 t, t 0.05/2 (4) = 2.776 < t,. 57g,. ( :, significant., µ 57g, µ 55,, ) L12-Q2 Quiz : t 1 0.05, t. 2, µ 196. 3 n = 4 X, s 2,, 4 1 t. T = X 196 s 2 /n ( ) L13 I(2017) 3 / 24
4, X = 200, s 2 = 224 t = 200 196 = 0.92582. 1 224 4 3 4 1 = 74.7., 5 t, t 0.05/2 (3) = 3.182 > t,.. ( :, µ 196g, ).,,,. t,. L12-Q3 Quiz : 1 0.05,. 2, p = 0.5. ( ) L13 I(2017) 4 / 24
3 n = 68 ˆp, Z = ˆp 0.5 0.5(1 0.5)/68,. 4, ˆp = 35/68 = 0.5147, z = 0.2425. 5 ( t t 0.05/2 ( ) ), 1.960 > z., z.,., p = 0.5,.. p 0.5 p > 0.5.,, ( ). ( ) L13 I(2017) 5 / 24
12 13 ( ) L13 I(2017) 6 / 24
( ) S?? µ X = 1 n [X 1 + ] X < µ < X + σ 2 s 2 = 1 n 1 [(X 1 X) 2 + ] < σ 2 < ( ), ( ) L13 I(2017) 7 / 24
Z N(0, 1 2 ) ( ) X 1 = 2Z X 2 = Z + 3 1.0 p 0.8 X 3 = 2Z + 3 W k = Z 1 + Z 2 + + Z k 0.6 0.4 0.2 p 1.0 3 2 1 1 2 3 x 0.8 Y 1 = Z 2 ( :V[Z] = E[Z 2 ] 0 2 ) 0.6 0.4 0.2 x 3 2 1 1 2 3 Y 2 = Z 2 1 + Z2 2. Y k = Z 2 1 + Z2 2 + + Z2 k ( ) L13 I(2017) 8 / 24
6.4(p.142) Z 1,..., Z k, N(0, 1 2 ), Y = Z 2 1 + + Z2 k. Y, k χ 2 (k). x X χ X χ 2 (k) 6.2(p.141) f k (y) = { C k y k 2 1 e 1 2 y (y 0) 0 ( ) E[Y ] = E[Z1 2 + + Z2 k ] = k, V[Y ] = = 2k. ( ) L13 I(2017) 9 / 24
12 13 ( ) L13 I(2017) 10 / 24
6.5(p.142) X N(µ, σ 2 ). n s 2 = 1 n 1 [(X 1 X) 2 + + (X n X) 2 ], Y = (n 1) s2 σ 2 k = n 1 χ 2 (n 1). 1,, Y n 1. (Y χ2 (n 1)) ( ) L13 I(2017) 11 / 24
A4.2(p.182) X i N(µ, σ 2 ) (i = 1,..., n), [ (X1 ) µ 2 ( ) ] Xn µ 2 + + n 1 n σ σ n χ 2 (n). s 2, [ (X1 ) 2 ( ) 2 ] Y = (n 1) s2 1 X Xn X = (n 1) + + σ2 n 1 σ σ n 1 χ 2 (n 1). µ X n 1. ( ) L13 I(2017) 12 / 24
χ 2 distribution,k=3 0.2 0.1 α 2 0.025 1-α 0.95 α 2 0.025 χ 2 α(k) 6.3(p.143) α = P (Y > χ 2 α(k)). χ2 1-α/2 (k) 5 χα/2 2 (k) 10 15 P (χ 2 s2 1 α/2 (n 1) < (n 1) σ 2 < χ2 α/2 (n 1)) = 1 α σ 2, 8.3 s 2, σ 2 1 α n 1 χ 2 α/2 (n 1) s2 < σ 2 n 1 < s2 (n 1) χ 2 1 α/2 s 2, (n 1)/χ 2 1. ( ) L13 I(2017) 13 / 24
L13-Q1 Quiz( ) S. 9 S, 9. 80g, 72g 2. 1 α = 0.95. 8.3(p.167), 8.3(p.167), 8.1,8.2(p.171) ( ) L13 I(2017) 14 / 24
L13-Q2 i Y i X(cm) s 2 (cm 2 ) 1 4 2 169.5 97.7 2 5 1 165.8 5.7 3 2 2 175.0 50 4 7 4 169.7 24.9 5 4 1 167.5 21.7 6 3 2 167.7 30.3 7 4 1 161.0 62 7.5 5 2 185.0 250 8 3 1 170.0 1.0 9 5 1 175.8 35.2 10 8 3 168.8 19.6 11 7 3 165.0 39.7 12 4 1 169.5 51.0 13 7 1 168.9 171.8 Excel, > >,.. 1 : 2 1 [(170.0 169.5)2 (169.0 169.5) 2 ] = 0.5. 1 : 2 1 [(170 170)2 (169.0 170) 2 ] = 1.0. 4 0. ( ) L13 I(2017) 15 / 24
t t 6.4,6.5(p143,144) Z N(0, 1 2 ), Y k χ 2 (k), Z Y, T = Z k (, Y/k )t., X T = X µ s 2. X µ = σ2 s 2 = Z t Y/k σ 2 k, k + Y Z. ( ) L13 I(2017) 16 / 24
12 13 ( ) L13 I(2017) 17 / 24
( ) 7.4.3 I, σ 2 0!(σ 2 < σ 2 0, σ2 > σ 2 0 ) H 1 σ < σ 0 ( σ > σ 0 ) H 0 σ = σ 0. P ((n 1) s2 σ 2 0 ) < χ 2 1 α(n 1) = α. P (χ 2α(n ) 1) < (n 1) s2 = α. σ 2 < σ 2 0 ( σ2 > σ 2 0 ) σ 2 σ 2 0 σ 2 0 ( ) L13 I(2017) 18 / 24
( ) 7.4.3 II α, (n 1) s2 σ 2 0 < χ 2 1 α(n 1), χ 2 α(n 1) < (n 1) s2 σ 2 0, α/2.. t ( ). t µ < µ 0 ( µ > µ 0 ) t µ µ 0 ( ) L13 I(2017) 19 / 24
L13-Q3 TA Prob and Sol: S, 4g 2., S 9, ( g). 76, 76, 76, 76, 80, 84, 84, 84, 84. S σ 2, 2 2?, α = 0.05,. 1 α = 0.05,. ( ) L13 I(2017) 20 / 24
2, σ 2, 2 2 2 2. 3 n s 2, Y = (n 1) s2 2 2, n 1.. 4 Y = (n 1) s2 = (9 1) 16 = 32. 2 2 2 2 5, Y > χ 2 α(n 1) = 15.5073,. 2 2.,. 2 2. 7.5(p.158), 7.4(p.158), 7.2(p.162) ( ) L13 I(2017) 21 / 24
L13-Q4 Quiz( ) S, 4g 2., S 9, ( g). 76, 76, 76, 76, 80, 84, 84, 84, 84. S σ 2, 2 2?, α = 0.05,. ( ) L13 I(2017) 22 / 24
t. Learn Math Moodle, 1 5. 2018-01-10, 10, 11, 15,16 Math.. Learn Math Moodle, 1 5. 2018-01-17, 15 22 Math. Excel p 1-503, http://hig3.net. ( )... 3.5(1-539) 4(1-502), Math - (1-614) ( ) L13 I(2017) 23 / 24
... Excel.,., Trial. :. 2018-01-18. (L05, ) (L06, ) N(µ, σ 2 ), (L09) (L10) (L10,L11) (L10,L13) (L10,L11) t (L12) (L13) (p,, ) (L10,L12, ) (,, t, ) ( ) ( ) L13 I(2017) 24 / 24
, p, I L14(2018-01-17 Wed) : Time-stamp: 2018-01-17 Wed 11:52 JST hig / 7.4.3 p Excel t, http://hig3.net ( ) L14, p, I(2017) 1 / 20
L13-Q1 Quiz : n = 9, 9 1, σ 2 0.95, n 1 χ 2 α/2 (n 1) s2 <σ 2 < χ 2 1 α/2 8 17.53 72 <σ2 < 8 2.180 72 32.85 <σ 2 < 264.2 n 1 s2 (n 1) ( ) L14, p, I(2017) 2 / 20
( = ) Team Number 14 12 10 8 6 4 2 0 7 4 7 8 5 3 5 4 3 4 7 2 5 4 0.95 0.99 sample size 0 500 1000 1500 2000 Variance of Height(cm 2 ) ( ) L14, p, I(2017) 3 / 20
, p, 13 14, p,,,, p p, Excel ( ) L14, p, I(2017) 4 / 20
, p, ( ) 7.4.3 I, σ 2 0!(σ 2 < σ 2 0, σ2 > σ 2 0 ) ( ) H 1 σ 2 < σ 2 0 ( σ > σ 0) H 0 σ 2 = σ 2 0 P ((n 1) s2 σ 2 0 ) < χ 2 1 α(n 1) = α. P (χ 2α(n ) 1) < (n 1) s2 = α. σ 2 0 ( ) L14, p, I(2017) 5 / 20
, p, ( ) 7.4.3 II α, (n 1) s2 σ 2 0 < χ 2 1 α(n 1), χ 2 α(n 1) < (n 1) s2 σ 2 0 χ 2 distribution,k=3 χ 2 distribution,k=3 0.2 0.1 α 1-α 0.95 α 2 χ 1-α (k) 5 10 15 5 χ 2 α (k) 10 15 ( ) L14, p, I(2017) 6 / 20
, p, ( ) σ 2 = σ 2 0 σ 2 < σ 2 0 ( σ2 > σ 2 0 ) σ 2 σ 2 0, α/2.. t t ( ). ( ) µ = µ 0 t µ < µ 0 ( µ > µ 0 ) t µ µ 0 ( ) L14, p, I(2017) 7 / 20
, p, L14-Q1 TA Prob and Sol: S, 4g 2., S 9, ( g). 76, 76, 76, 76, 80, 84, 84, 84, 84. S σ 2, 2 2?, α = 0.05,. 1 α = 0.05,. ( ) L14, p, I(2017) 8 / 20
, p, 2, σ 2, 2 2 2 2. 3 n s 2, Y = (n 1) s2 2 2, n 1.. 4 Y = (n 1) s2 = (9 1) 16 = 32. 2 2 2 2 5, Y > χ 2 α(n 1) = 15.5073,. 2 2.,. 2 2. 7.5(p.158), 7.4(p.158), 7.2(p.162) ( ) L14, p, I(2017) 9 / 20
, p, L14-Q2 Quiz( ) S, 4g 2.,., S 7, ( g). 79, 79, 79, 80, 81, 81, 81 S σ 2, 2 2?, α = 0.05,. ( ) L14, p, I(2017) 10 / 20
, p,,,, p 13 14, p,,,, p p, Excel ( ) L14, p, I(2017) 11 / 20
, p,,,, p,, 7.3 H 0 H 0 H 0 2 ( β ) H 0 1 ( α ) α: 1 α: 1 β: or α, β,., α, β. - ( ) L14, p, I(2017) 12 / 20
, p,,,, p L14-Q3 Quiz( ) 1. 1, 2,, 3, 4,, ( ) L14, p, I(2017) 13 / 20
, p, p, Excel 13 14, p,,,, p p, Excel ( ) L14, p, I(2017) 14 / 20
, p, p, Excel p Step5, s 2 ( X) α. 2 χ 2 1 α (n 1) > < (Step5) χ 2 α(n 1) < > y 1 = (n 1) s2 σ 2 or chisq.inv.rt chisq.dist.rt α > < p p (p-value), 0.2 χ 2 distribution,k=3 0.2 χ 2 distribution,k=3. α > p 0.1 α 5 χα 2 (k) 10 15 0.1 p 5 y1 10 15 ( ) L14, p, I(2017) 15 / 20
, p, p, Excel Excel 2016 Excel average var.s s for sample ( ), p for population ( ) stdev.s : average, var.p, stdev.p. Excel Excel,. R II, II ( ) L14, p, I(2017) 16 / 20
, p, p, Excel Excel 2016 χ 2 distribution,k=3 χ 2 distribution,k=3 0.2 0.2 0.1 0.1 α p 5 χα 2 (k) 10 15 5 y1 10 15 k: t Excel p =chisq.dist.rt(y 1, k) 1 p =chisq.dist.rt(y 0, k) chisq.inv.rt(α, k)= χ 2 α(k) chisq.inv.rt(1 α, k)= χ 2 1 α (k) ( ) L14, p, I(2017) 17 / 20
, p, p, Excel Excel 2016 t k: t Excel p 2 =t.dist.2t(t, k) t.inv.2t( α 2, k)= t α/2(k), t, 2 t II ( ) L14, p, I(2017) 18 / 20
, p, p, Excel. 1-503, http://hig3.net. ( )... 3.5(1-539) 4(1-502), Math - (1-614) 2018-02-05, 06 (=4 ) 2018-02-17 (4 ), (3 ) 2018-02-20 ( 2 ) 2018-06-17 (2 ) ( ) L14, p, I(2017) 19 / 20
, p, p, Excel... Excel.,., Trial. :. 2018-01-18. (L05, ) (L06, ) N(µ, σ 2 ), (L09) (L10) (L10,L11) (L10,L13) (L10,L11) t (L12) (L13) (p,, ) (L10,L12, ) (,, t, ) ( ) ( ) L14, p, I(2017) 20 / 20
I L15(2018-01-17 Wed) : Time-stamp: 2018-01-17 Wed 11:52 JST hig http://hig3.net ( ) L15 I(2017) 1 / 4
L14-Q1 Quiz : 1 α = 0.05,. 2, σ 2, 2 2 2 2. 3 n s 2, Y = (n 1) s2 2 2, n 1.. 4 Y = (n 1) s2 = (9 1) 16 = 32. 2 2 2 2 5, Y > χ 2 α(n 1) = 15.5073,. 2 2. ( ) L15 I(2017) 2 / 4
,. 2 2. L14-Q2 Quiz : 1 α = 0.05,. 2, σ 2, 2 2 2 2. 3 n s 2, Y = (n 1) s2 2 2, n 1.. 4 Y = (n 1) s2 = (7 1) 1 = 3 2 2 2 2 2. 5, Y < χ 2 1 α (n 1) = 1.6354,. 2 2. ( ) L15 I(2017) 3 / 4
,. 2 2. L15-Q3 Quiz : 4 ( ) L15 I(2017) 4 / 4