分散分析・2次元正規分布

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1 2 II L10( Thu) : Time-stamp: Thu 13:55 JST hig F ( ) L10 2 II(2016) 1 / 24

2 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 /σ n 1, n 2, S 2 1, S2 2, F = S2 1 S 2 2, (n 1 1, n 2 1) F.. 5 F = 28 4 = 7. 6 F, F α/2 (10 1, 5 1) = 8, 905 > 7 = F., F 1 α/2 (10 1, 5 1) < 7... ( ) L10 2 II(2016) 2 / 24

3 F 2 F L09-Q2 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 n 1, n 2, S 2 1, S2 2, F = S2 1 S 2 2, (n 1 1, n 2 1) F.. 5 F = 28 4 = 7. 6 F, F α (10 1, 5 1) = < 7 = F.. 1. ( ) L10 2 II(2016) 3 / 24

4 F 2 F 3 F 2 F ( ) L10 2 II(2016) 4 / 24

5 F 2 F? i 1 79,80,80, [(79 80)2 + ] 2 78,86,81, ,81,80, , N(µ i, σ 2 ). ( i = 1, 2, 3).?,. ( ) L10 2 II(2016) 5 / 24

6 F 2 F (= or )? A 1 y 11, y 12,..., y 1r r y 1 j (y 1j y 1 ) 2 A 2 y 21, y 22,..., y 2r r y 2 j (y 2j y 2 ) 2. A l y l1, y l2,..., y lr r y l j (y lj y l ) 2 rl y. y i = 1 r r j=1 y ij. y = 1 rl l i=1 r j=1 y ij. Y ij N(µ + a i, σ 2 ),. i a i = 0. : Y ij = µ + a i + E ij, E ij N(0, σ 2 ) a 1 = a 2 = = a l = 0? ( ) L10 2 II(2016) 6 / 24

7 F 2 F L10-Q3 Example ( ),? r l y 12 y 1 y j (y 1j y 1 ) 2 ( ), ( ) 2 2 t ( ), 2 n ( ) L10 2 II(2016) 7 / 24

8 F 2 F (=) ( ), ()? a i 0. a i = l r l S A = (y i y ) 2 = r (y i y ) 2 χ 2 (l 1) i=1 j=1 i=1 E ij = S E = l i=1 j=1 r (y ij y i ) 2 χ 2 ((rl 1) (l 1)) = l r S T = (y ij y ) 2 χ 2 (rl 1) i=1 j=1 S A + S E = S T. (l 1) + (rl l) = rl 1. ( ) L10 2 II(2016) 8 / 24

9 F 2 F (ANOVA) or F. a i = 0, S A ϕ A = l 1 (*), S E ϕ E = rl l, F = V A VE = S A/(l 1) S E /(rl l) (l 1, rl l) F (**). a i 0, S A (*). F (**). F, F, a i 0. ( ) L10 2 II(2016) 9 / 24

10 F 2 F 1 F S A ϕ A = l 1 V A = S A /ϕ A V A /V E S E ϕ E = (rl 1) (l 1) V E = S A /ϕ A S T ϕ T = rl 1 Example (), α = 0.05 F. ( ) L10 2 II(2016) 10 / 24

11 F 2 F L10-Q4 Quiz(), 1. α = 0.05 F. A A A ( ) L10 2 II(2016) 11 / 24

12 F 2 F F k1, k2 F F, α = P (F > Fα(k1, k2)) Fα(k1, k2). F = Yk 1 /k1 Yk /k2 2 α = 0.05 k2\k α = k2\k Yk χ2 (k). ( ) L10 2 II(2016) 12 / 24

13 2 2 3 F 2 F ( ) L10 2 II(2016) 13 / 24

14 2 2 :2 I(2016)L01 P (X = x, Y = y) = f XY (x, y). y\x /8 0 1/ /8 1/3 1/12 y\x x 1 x 2 x 3 y 1 f XY (x 1, y 1 ) f XY (x 2, y 1 ) f XY (x 3, y 1 ) y 2 f XY (x 1, y 2 ) f XY (x 2, y 2 ) f XY (x 3, y 2 ) 2 E[ϕ(X, Y )] = a b ϕ(x i, y j )f XY (x i, y j ) i=1 j=1 ( ) L10 2 II(2016) 14 / 24

15 2 2 2 (2 ) f XY (x, y) 2 E[ϕ(X, Y )] = + dx + 2 ( ) dy ϕ(x, y)f XY (x, y) P (a X < b, c Y < d) =E[1 [a x<b,c y<d] (X, Y )] = b a dx d c dy f XY (x, y). ( ) L10 2 II(2016) 15 / 24

16 2 2 3 F 2 F ( ) L10 2 II(2016) 16 / 24

17 2 2 :1 f Z (z) = 1 2π e z N(0,1) N(3,2 2 ) 0.3 X = az + b., z z = x b, x µ σ II(2016)L06 N(µ, σ 2 ) f(x; µ, σ 2 ) = 1 (x µ)2 e 2σ 2. 2πσ 2 a = x p µ(= E[X]), σ 2 (= V[X]) x I(2015)L08 ( ) L10 2 II(2016) 17 / 24

18 2 2 2 ( X, Y ) f XY (x, y) = 1 2πσX 2 e (x µ 2 X) 2σ X 2 1 2πσY 2 e (y µ 2 Y) 2σ Y 2. E[X] = µ X, E[Y ] = µ Y, V[X] = σ 2 X, V[Y ] = σ 2 Y, C XY = Cov[X, Y ] = E[XY ] E[X]E[Y ] = = ( ) L10 2 II(2016) 18 / 24

19 2 2 M X (t) = e µt+ 1 2 σ2 t 2.. f(x; µ, σ 2 ): x 0 f(x; µ, σ 2 ) dx = x 1 f(x; µ, σ 2 ) dx = x 2 f(x; µ, σ 2 ) dx = (x µ) 2 f(x; µ, σ 2 ) dx = x 0 x 1 x 2 1 (x µ)2 e 2σ 2 dx = 1. 2πσ 2 1 (x µ)2 e 2σ 2 dx = µ. 2πσ 2 1 (x µ)2 e 2σ 2 dx = σ 2 + µ 2. 2πσ 2 (x µ) 2 1 (x µ)2 e 2σ 2 dx = σ 2 2πσ 2 ( ) L10 2 II(2016) 19 / 24

20 2 2 L10-Q5 Quiz(2 ) 2 2. f(x, y) = C e x2 4x 2y 2 +12y 5. 1 X, Y,,. 2 E[1] = 1 C. 2 ( I) ( ) L10 2 II(2016) 20 / 24

21 2 2 ( ) L10 2 II(2016) 21 / 24

22 2 2 L10-Q6 Quiz(2 ) 2 2. f(x, y) = C e 4x2 1 6 y2 +2y 1 X, Y,,. 2 E[1] = 1 C. ( ) L10 2 II(2016) 22 / 24

23 2 2, I. RaMMoodle II(2016) /Math ryukoku.ac.jp manaba ( ) L10 2 II(2016) 23 / 24

24 2 2,, (RQ=Research Question),,.. =.?. = II. Web manaba. 4 1, 2. manaba ,2,, 6 1,2, =. 7 ( ) L10 2 II(2016) 24 / 24

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