(Frequecy Tabulatios)
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- やすもり のしろ
- 7 years ago
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2 (Frequecy Tabulatios) ( ) ( ) χ t F
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4 (Frequecy Tabulatios) (samplig) (sample) : (frequecy table) x i x i (frequecy) f i f 1 + f + + f k =
5 4 1 x 1 f 1 x f.. x k f k x i a b a b (class iterval) a b b a (class iterval width) a + b (midpoit) f i / (relative frequecy) f i F i = f 1 + f + + f i (frequecy distributio) : (histogram) (cumulative distributio fuctio)
6 1.1. (Frequecy Tabulatios) 5 1.1: 1.: Sturges = 1 + log 10 log 10 (1.1) 50 k k = 1 + log log 10 = log = (1.699) = ( - )/ ( )/6.64 = : x x 1, x,..., x (total) T = x 1 + x + x = x i x = x 1 + x + x = 1 x i = T (mea) x x 1, x,..., x k f 1, f,..., f k T = x 1 f 1 + x f + + x f
7 6 1 x = x 1f 1 + x f + + x f = 1 k x i f i (media) x i m i (mode) (1.1) ( 1.1)
8 1.1. (Frequecy Tabulatios) (g/cm )
9 (dispersio) 5 f i f i / F i F i / T x fi x i s s s s : x i (i = 1,,..., k) f i x x : s = (x 1 x) f 1 + (x x) f + + (x k x) f k = 1 k (x i x) f i (variace) : s = s = 1 k (x i x) f i (stadard deviatio) s = (x 1 x) f 1 + (x x) f + + (x k x) f k 1 1 k = (x i x) f i 1
10 1.. 9 x x 1, x,..., x s = (x 1 x) + (x x) + + (x k x) = 1 (x i x) 1. s = 1 s = 1 x i (x) (x i x) = 1 (x i x i x + x ) ( = 1 x i x x i + ( = 1 ) x i xx + x = 1 x i x ) x x s s x 1.3 x 67 s x 8.5 ȳ 53 s y 1.6 A A z i = x i x s
11 10 1 {z i } 0 s 1 z eglish = z math = = = A. (x, y) (x 1, y 1 ), (x, y ),, (x, y ) (covariace) (correratio coefficiet) s xy = 1 (x i x)(y i y) = 1 r = s xy s x s y s x x s y y x i y i xy 1. p/100 x p 5% Q 1 1, 50% Q, 75% Q
12 : x y x y
13 x i, y i x y (x i, y i ) (correratio diagram) x i y i x i y i (positive correratio) x i y i x i y i (egative correratio) x x 1, x,..., x y y 1, y,..., y : (x 1, y 1 ), (x, y ),..., (x, y ) y ( (liear regressio)): y = ax + b x i y ( ŷ i ) ŷ i = ax i + b x i ( ) y i y i ŷ i d i d i = y i y i = y i ax i b d i = (y i ax i b)
14 a, b (method of least square) (y i ax i b) a, b a, b (y i ax i b) F (a, b) = (y i ax i b) y = f(x) y = f (x) = 0 a b F a F b = = [(y i ax i b)( x i )] = (y i ax i b)x i [(y i ax i b)( 1)] = (y i ax i b) x = P xi x i y i a x i b x i = 0 y i a x i b = 0 P, y = yi x i y i a x i bx = 0 y ax b = 0 y ax b = 0 x i y i a x i (y ax)x = 0 ( ) x i y i xy = a x i x 1 ( 1 x i y i xy = a ) x i x s xy a x as x s xy = as x
15 14 1 a a = s xy as x y ax b = 0 b b = y ax = y s xy as x x x y y y = s xy s (x x) x {(x i, y i ) (i = 1,,..., ) y x a yx = s xy s x = T xy T x T y x i T x y x l y y = a xx (x x) x,y y y 3 x 1 : x : x 3 : y 3 x 1, x, x 3 y x 1, x, x 3 y = b 0 + b 1 x 1 + b x + b 3 x 3
16 : x y x y
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18 17.1 x 1, x,..., x X X = x i p i X X P (X = x i ) X x F (x) X.1 F (x) = P (X x) 6 E = 1 P (E) = 1 6 X = E X ( ) ( ) i ( ) 6 i p i = P (X = i) = i 6 6 X (biomial distributio) X B(6, 1 6 ) X p i X P (X = i) = ( ) p i (1 p) i i X X B(.p) X X X X x 1, x,..., x (X = x i ) p 1, p,..., p P (X = x i ) = p i (i = 1,,..., ) pi = 1, (p i 0) X f
19 18 X x i x 1 x x P (X = x i ) = p i = f(x i ) p 1 p p X x 1 < x < < x F (x r ) f F 1. 0 p i = f(x i ) 1 (i = 1,,..., ) F (x r ) = P (X x r ) = p 1 + p + + p r =. F (x ) = P (X x ) = p 1 + p + + p = 1 3. P (a < X b) = F (b) F (a) 4. a < b = F (a) < F (b) X ( ) µ = E(X) = k x i p i r σ = V (X) = E ( (X µ) ) = E(X ) E(X). E(X) = k x ip i, E(Y ) = l j=1 y jq j E(X + Y ) = E(X) + E(Y ) P (X = x i, Y = y j ) p ij { l j=1 p k ij = p i p ij = q j k l j=1 p ij = k p i = l j=1 q j = 1 p i E(X + Y ) = = = k l (x i + y j )p ij j=1 k l l k (x i p ij ) + (y j p ij ) j=1 j=1 k l x i p i + y j q j = E(X) + E(Y ) j=1.3 E((X µ) ) = E(X ) (E(X))
20 E((X µ) ) = E(X Xµ + µ ) = E(X ) µe(x) + µ E(1) = E(X ) E(X)E(X) + E(X) = E(X ) E(X) a, b (a < b) P r (a X b) P r (a X b) = b a f(x)dx f(x) (, ) f(x) (probability desity fuctio) f(x) 0 X < X x f(x)dx = 1 F (x) = P r (X x) F (x) X (probability distributio) X ( ) X µ = E(X) = σ = V (X) = E ( (X µ) ) = g(x) = 1 πσ EXP [ xf(x)dx (x µ) f(x)dx ] (x µ) σ, < x < X X N(µ, σ ) X N(3, )
21 0 (ormalizatio) X E(X) 0 V (X) 1 Z = X E(X) V (X) E(Z) = 0, V (Z) = 1 P r (Z z) P r (Z z) = P r (Z 0) + P r (0 < Z z) P r (Z 0) 0.5 P r (0 < Z z).1:.4 X N(60.9,.9 ) (1) P (X 63.8) () P (6.3 < X 63.0)
22 .1. 1 (1) () P (X 63.8) = P ( X = P (Z ).9 ) = P (Z 1) = P (Z 0) + P (0 Z 1) = = P ( < X ) = P ( < Z.1.9 = P (0.48 < Z 0.7) = P (0 Z 0.7) P (0 Z 0.48) = =
23 4 1. X N(80, 6 ) (a) P r (X 90) (b) P r ( X 80 1). A N(10, 1 ) ( ) 50(l/ )
24 (populatio) 6 (sample) 6 Π 6 X (Π, X) (x 1, x,..., x ) x i X X i (X 1, X,..., X ) (X 1, X,..., X ) X i (Π, X) X X 1, X,..., X X i (i = 1,,..., ) E(X i ) = µ, V (X i ) = σ X 1, X,..., X X 1, X,..., X X = 1 S = 1 X i (X i X) X µ S σ
25 V (X) = E(X ) E(X) = E( 1 ( Xi ) µ = 1 E(X X + (X 1 X + + X 1 X )) µ = 1 E(Xi ) + E(X i X j ) µ 1 i,j = 1 (σ + µ ) + ( )µ µ = σ σ 3.1 ( ) X µ σ λ > 1 3. P ( X µ λσ) 1 λ P ( X µ < λσ) 1 1 λ µ, σ (Π, X) X X µ σ P ( X µ < σ 5 ) 0.9 X σ P ( X µ < λσ ) 1 1 λ λ = 1 5 P ( X µ < σ 5 ) X 1, X, X 3,..., X
26 X = 1 [X 1 + X + X X ] S = 1 [(X 1 X) + (X X) + + (X X) ( ) ( ) 1 θ {x 1, x,..., x } T (x 1, x,..., x ) (X 1, X,..., X ) ˆθ = T (X 1, X,..., X ) θ θ ˆθ = T (X 1, X,..., X ) ˆθ θ E(ˆθ) = E(T (X 1, X,..., X )) = θ N(µ, σ ) U 3.3 X = 1 X E(X + Y ) = E(X) + E(Y ) E( X) = µ. =1 X i, U = 1 1 (X i X) E( X) = E(X 1 ) + E(X ) + + E(X ) = µ σ U S = 1 (X i X) = 1 U U S σ
27 x 1, x,..., x X 1, X,..., X (maximum likelihood) x 1, x, x 3 µ x 1, x, x 3 X 1, X, X 3 P (X 1 = x 1, X = x, X 3 = x 3 ) L X 1, X, X 3 L = P (X 1 = 1)P (X = x )P (X 3 = x 3 ) = e µ µx 1 x 1! µx µx3 e µ e µ x! x 3! = e 3µ µx1+x+x3 x 1!x!x 3! µ L µ x 1, x, x 3 µ L = L(µ) dl dµ = 3e 3µ µx1+x+x3 x 1!x!x 3! dl dµ = 0 3µ µx1+x+x3 1 + (x 1 + x + x 3 )e x 1!x!x 3! = 3L + µ 1 (x 1 + x + x 3 )L = L µ ( 3µ + x 1 + x + x 3 ) = 0 µ = 1 3 (x 1 + x + x 3 ) L 3.5 N(µ, σ ) x 1, x,..., x σ µ N(µ, σ ) f(x) = 1 (x µ) exp{ πσ σ } ( 1 L = P (X 1 = x 1 ) P (X = x ) = exp { (x 1 µ) + + (x µ) }) πσ σ x 1, x,..., x, σ dl dµ = 1 σ {(µ x 1) + (x µ) + + (µ x )}L = 1 σ {µ (x 1 + x + + x )}L = 0
28 3.. 7 µ = 1 (x 1 + x + + x ) = x
29 , 11, 133, 14, 16, 118, 11, 15, 131, 10(cm)
30 θ [θ 1, θ ] θ θ 1, θ α (0 < α < 1) P r (θ 1 < θ < θ ) = 1 α (θ 1, θ ) θ θ 1, θ 100(1 α)% [θ 1, θ ] θ [θ 1, θ ] θ 1 α N(µ, σ ) σ µ X 1, X,..., X X i N(µ, σ ) µ (σ ) S = 1 S = E(X) = µ, V (X) = σ X N(µ, σ ) (X i X) E(S ) = 1 σ 1 S E(S ) = σ α = % X N(µ, σ ) Z = X µ N(0, 1) σ / P r ( Z z α ) = 1 α = 0.95 z α P r (Z z α ) = α
31 30 3 z α α = 0.05 z α z α = 1.96 X µ Z = σ / z α µ σ σ X z α µ µ X + z α 3.1: 3.6 8, 4, 31, 7,. 95%. σ = 6.5 X i X i N(µ, 6.5) X N(µ, σ /5) X X = 1 13 [ ] = 5 5 = 6.4 Z = X µ N(0, 1) σ /5 95% P r ( Z z α ) = z 0.05 = X z α σ 5 µ X + z α σ 5
32 /5 µ /5 4.1 µ 8.59 σ (σ ) α = % µ, σ σ σ S σ T = X µ S / 1 t t 1,α/ P r ( T t 1,α/ )) = 1 α P r (T t 1,α/ ) = α t 1,α/ t α = 0.05 = 10 t 9,0.05/ µ X t 1,α/ S t 9,0.05/ =.6 X µ t 1,α/ S / µ X + t 1,α/ S
33 BOD(ppm) σ = 6.5(ppm) = 15 X = 7.ppm 95% 145.3, 145.1, 145.4, %
34 3.4. ( ) ( ) A p A p (X 1,..., X ) { 1 A X i = 0 Ā X = X X X A X A X p p (X 1,..., X ) X = X X X B(, p) X ( ) N(p, p(1 p)) X = ˆp N p, p(1 p) ˆp p p(1 p) P ˆp p p(1 p) N(0, 1) z α = 1 α p(1 p) p(1 p) ˆp z α p p + z α p ˆp p 100(1 α)% ( ˆp z α ˆp(1 ˆp), ˆp + z α ˆp(1 ˆp) ) 3. ( ) X B(, p) X N(p, p(1 p)) 3.7
35 p 95% ˆp = = 0.18 z α = z 0.05 = 1.96 ( ) (0.18)(0.8) (0.18)(0.8) , (0.15, 0.1)
36 3.5. ( ) ( ) p = P (A) A ( ) x p 100(1 α)% (p 1, p ) ( ) m 1 f, 1 f 1 + m 1 f + m 1 = ( x + 1), = x ( 1, ) F F f 1 F P (F > f 1 ) = α
37 % %
38 X, Y N(µ 1, σ1), N(µ, σ) ax + by N(aµ 1 + bµ, a σ 1 + b σ ) 3.3 X 1, X,..., X N(µ, σ ) X = X 1 + X + + X N(µ, σ ) X 1 + X + + X N(µ, σ ) 3.8 X = 1 (X 1 + X + + X ) N(µ, σ σ ) = N(µ, ) 10cm 4.5cm cm X i X i N(10, 4.5 ) 50 X 3.3 X N(10, ) P (X > 10.5) = P (Z > ) = P (Z > 0.948) = 0.5 P (0 < Z < 0.958) = X i [ ] X 1, X,..., X µ σ X = X 1 + X + + X > 100 N(µ, σ )
39 χ χ χ () χ f (x) Γ(x) 1 f (x) = Γ( ) x 1 e 1 x x > 0 0 x 0 Γ(x) = χ 0 t x 1 e t dt (x > 0) 3.4 X 1, X,..., X N(0, 1) χ = X 1 + X + + X χ E(χ ) =, V (χ ) = 3.5 (χ ) χ, χ m,m χ χ = χ +χ m + m χ S 3.6 N(µ, σ ) {X 1, X,..., X } Y = 1 σ (X i X) = S σ 1 χ P (S 1.5) X i N(µ, 1) Y = 0S 1 = 0 (X i X)
40 3.7. χ χ P (S 1.5) = P (0S 8.5) = P (χ ) χ P (χ 19 > 7.0) = 0.10 P (χ 19 > 30.14) = 0.05 P (χ ) = ( )
41 t 3.1 f (x) f (x) = +1 γ( ) πγ( x +1 + )(1 ) ( 1) T t 3 E(T ) = 0, V (T ) = 3.7 X 1, X,..., X N(µ, σ ) U U = S = 1 1 (X i X) T = X µ U/ 1 t 3.8 Z χ χ Z χ T = Z χ m / t t t() t t P r (T c) P r (Z c) χ χ / 1 T T
42 3.9. F F 3. f m, (x) f m, (x) = 0, γ( m+ ) γ( m )γ( )mm x m 1 (mx+) m+ (x > 0) F m, (m, ) F E(F m, ) = m ( > ) V (F m, ) = (m + ) m( ) ( 4) ( > 4) 3.9 X 11, X 1,..., X 11 N(µ 1, σ 1) 1 X 1 U 1 X 1,..., X N(µ, σ ) X U X 1 = X = 1 1 i X 1i 1 i X i, U 1 = 1 1 1, U = 1 1 (X 1i X 1 ) i (X i X ) i F = U 1 /σ 1 U /σ = σ U 1 σ 1 U F ( 1 1, 1) 3.10 F 1 (0.05) = F 5 10(0.05) F 5 10(0.05) = F 1 (1 α) 3.11 F 1 (1 α) = 1 F 1 (α) F 5 11(1 0.05) F 5 11(1 0.05) = 1 5 (0.05) = = 0.3 F 11
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44 ESP p 5 X X B(5, p) p = 0. p > H 0 : p = 0. ( ) P r (X 3) = P r (X = 3) + P r (X = 4) + P r (X = 5) ( ) ( ) ( ) = (0.) 3 (0.8) + (0.) 4 (0.8) + (0.) 5 = H 0 H 0 H 0 (sigificace level) α α 0.05, H 0 ( ) H 0 X P r (X 4) = X 4 (critical regio) H 0 : p = 0. (ull hypothesis) H 1 : p > 0. (alterative hypothesis)
45 44 4 θ H 0 : θ = θ α H 1 : θ > θ 0, H 1 : θ < θ 0, H 1 : θ θ 0 θ (1) N(µ, σ ) (X 1, X,..., X ) X N(µ, σ ) µ (a) σ µ ( α) Z = X µ N(0, 1) σ / (b) σ µ ( α) () T = X µ t( 1) S / N(µ, σ ) (X 1, X,..., X ) σ (a) µ σ ( α) χ = 1 σ (X i µ) χ α, (b) µ σ ( α) χ = S σ χ α, 1
46 (a) σ µ ( α) H 0 : µ = µ 0 3 (1) H 1 : µ > µ 0 () H 1 : µ < µ 0 (3) H 1 µ µ 0 ( µ = µ 0 ) Z 0 = X µ 0 N(0, 1) σ / α P r (Z 0 > z α ) = α z α α (1) Z 0 > z α (1) Z 0 > z α () Z 0 < z α (3) Z 0 > z α , % 1 µ H 0 : µ = 64.5 H 1 : µ > 64.5 X i X i N(µ, 0) X = 1 [ ] = 533/8 = X N(µ, 0/8). Z 0 = /8 = = 1.8
47 : Z 0 1
48 mm 0.0mm mm α = 0.05 µ 95% 1 ( mm) α = % 4. X µ 1 σ 1 X 1, X,..., X 1 X S 1 Y µ σ Y Y 1, Y,..., Y Ȳ S 1. µ 1 µ (a) σ1, σ X Y N X N(µ 1, σ 1 1 ), Ȳ N(µ, σ ) ( µ 1 µ, σ σ ) Z = ( X Ȳ ) (µ 1 µ ) σ 1 / 1 + σ / N(0, 1) 4. A 30 B cm 146.4cm 4.8cm 0.05 A N(µ 1, 4.8 ), B N(µ, 4.8 ) 1 = 30, = 50 X N(µ 1, ), Ȳ N(µ, ) H 0 : µ 1 = µ H 1 : µ 1 µ α = 0.05
49 48 4 H 0 z 0.05/ = 1.96 Z = ( X Ȳ ) (µ 1 µ ) σ 1 / 1 + σ / Z 0 = N(0, 1) / /50 = Z 0 = 1.6 < z 0.05/ = 1.96 H 0 (b) σ 1, σ σ 1 = σ X 1,..., X 1, Y 1,..., Y S 1, S S 1 = (X i X), S = 1 (Y i 1 Ȳ ) i ˆσ = ( 1 1)S 1 + ( 1)S 1 + S1 = 1 (X i X), S = 1 (Y i 1 Ȳ ) i ˆσ = 1S 1 + S 1 + ˆσ (a) T = ( X Ȳ ) (µ 1 µ ) ˆσ / 1 + ˆσ / t( 1 + ) i i 1 + t 4.3 A,B. 5 X A = 97.5%, XB = 95.3% S A = 1.3%, S B = 1.56% 0.05 µ A %, µ B % H 0 : µ 1 = µ H 1 : µ 1 µ
50 α = 0.05 T = ( X Ȳ ) (µ 1 µ ) ˆσ / 1 + ˆσ / t( 1 + ), ˆσ = 1S 1 + S 1 +. H 0 ˆσ = 5(1.3)+5(1.56) 5+5 = 1.748% T 0 = / /5 =.634 t 0.05/,8 =.3060 T 0 = < t 0.05/,8 =.3060 H 0 (c) σ 1, σ T = ( X Ȳ ) (µ 1 µ ) S 1 /( 1 1) + S /( 1) t(ϕ), 1 ϕ = c (1 c) , 1 c = 1 + ( 1 1)S ( 1)S1. σ 1/σ 1 S 1 σ 1 χ ( 1 1), S σ χ ( 1) F = σ S 1 σ1 S F ( 1 1, 1) 1. H 1 : σ 1 σ. H 1 : σ 1 > σ 3. H 1 : σ 1 < σ W = {F : F > F (α )} {F : F < F (1 α )} W = {F : F > F (α)} W = {F : F < F (1 α)} F (1 α) F (1 α) = 1 F (α)
51 A,B g.48g 5% A = 10, S A = 5.3, S = 10 9 S A B = 16, S B =.4, S = S B H 0 : σ 1 = σ H 1 : σ 1 σ α = 0.05 H 0 F 9 15(0.05) = 3.17 F = σ S 1 σ1 S F ( 1 1, 1) F 0 = 10(5.3) 9 16(.4) 15 =.1967 F 0 =.1967 < F 9 15(0.05/) = 3.17 H 0
52 A B A 10 B 1 A B A B N(µ 1, σ 1 ), N(µ, σ ) H 0 : µ 1 = µ 5% A B A 10 B 1 A B A B N(µ 1, σ 1 ), N(µ, σ ) 5% H 0 : σ 1 = σ 4.3 ( ) A p A p (X 1,..., X ) { 1 A X i = 0 Ā X = X X X A X A p p 0 (0 p 0 1) H 0 : p = p 0 H 1 : p p 0 p (X 1,..., X ) X = X X X B(, p) X ( ) N(p, p(1 p)) X = ˆp N p, p(1 p) Z = ˆp p p(1 p) N(0, 1)
53 p 1 6 5% H 0 : 1 p = 1 6 H 1 : p 1 6 α = 0.05 H 0 ˆp = = 0.18 Z = ˆp p p(1 p) N(0, 1) Z 0 = (1 1 6 ) 600 = Z 0 < Z 0.05 = 1.96 H 0 A, B 1 C p 1, p 1, X 1, X H 0 : p 1 = p H 1 : p 1 p p 1, p p 1 = p = p, 1 p = q 1, X 1, X (N( 1 p, 1 pq), N( p, pq) X 1 = X 1 1 X = X (N(p, pq 1 ) (N(p, pq ) X 1 X N(0, ( )pq) 1 1 Z = X 1/ 1 X / N(0, 1) ( )p(1 p)
54 p 4.6 p = X 1 + X % p 1 p p 1 = , p = H 0 : p 1 = p H 1 : p 1 p α = 0.05 Z = X 1/ 1 X / N(0, 1) ( )p(1 p) H 0 p = = 1 3 Z 0 = 10/ /500 ( ) 1 3 (1 1 3 ) = Z 0 < Z 0.05 = 1.96 H 0
55 % 3,000 5% 5% % 4.4 (goodess of fit test) X (1) k A 1, A,..., A k A i P (A i ) P (A i ) = p i p 1 + p + + p k = 1 A 1, A,..., A k 1,,..., k ( ) p 1 1 1,,..., p p k k =! k 1!! k! p 1 1 p p k k k = (multiomial distributio) A i X i 4.7 P (X 1 = 1, X =,..., X k = k ) =! 1!! k! p1 1 p p k k ( 6 1,1,1,1,1,1) ( ) 6 1, 1, 1, 1, 1, 1 ( 1 6 )6 = 6! 6 6 = = 5 34 A 1, A,..., A k P (A i ) = p i A i p i = m i A i X i m 5 χ = (X 1 m 1 ) m 1 + (X m ) m + + (X k m k ) m k
56 χ (k 1) m i X i i χ χ m i H 0 : p 1 = p 10, p = p 0,..., p k = p k0 (p i0 p 10 + p p k0 = 1 ) H 1 : p 1 = p 11, p = p 1,..., p k = p k1 (p 11, p 1,..., p k1 ) (p 10, p 0,..., p k0 ) H 0 A i m i m i = p i0 i m i 5 A i x i k χ (x i p i0 ) = > χ α,k 1 H 0 p i H 0 : (p 1, p, p 3, p 4, p 5, p 6 = 1 6, 1 6, 1 6, 1 6, 1 6, 1 6 ) H 1 : (p 1, p, p 3, p 4, p 5, p 6 1 6, 1 6, 1 6, 1 6, 1 6, 1 6 ) α = 0.05 H 0 χ = 6 (X i p i ) p i = 6 X i p i χ 0 = = = 5.56 χ 0.05,6 1 = 1.83 χ 0 = 5.56 < χ 0.05,5 = H 0
57 A : B : C : D = 9 : 3 : 3 : 1 5% A B C D () H 0 : D D θ 1, θ,..., θ i µ, σ A 1, A,..., A k (X 1, X,..., X k ) (x 1, x,..., x k ) θ i θ i = ˆθ i (x 1, x,..., x k ) (i = 1,,..., l) θ i A 1, A,..., A k m 1, m,..., m k m i = p i0. k χ (X i m i ) = χ χ k l 1 H % H 0 : P (λ) α = 0.05 P (λ) λ k p k kp k = E(X) = λ k=0 m i
58 k f k kf k p k m k p k f k k kf k λ λ λ 1 k kf k = 1 00 = 0.61 k x k m k k 3 m k 5 χ m i 5 k 1 H 0 χ = (x i m i ) i=0 χ 0 = ( ) = m i ( ) (6 5) 5 χ 0.05,3 1 1 = 3.84 χ 0 = < χ 0.05,1 = 3.84 H 0 λ = 1 (3) A, B A, B A 1,..., A k B 1,..., B l A i B j x ij B 1 B B l A 1 x 11 x 1 x 1l x 1 A x 1 x x l x A A k x k1 x k x kl x k
59 58 4 x i., x.j k l (cotigecy table) A B A i, B j X ij A i, B j p i, q j A i, B j P ij A, B A, B H 0 P ij = P r (A i B j ) = P r (A i )P r (B j ) = p i q j p i, q j ˆp i = x i., ˆq j = x.j H 0 χ = k j=1 l (X ij P ij ) (k 1)(l 1) x ij χ P ij χ 0 = = k j=1 k j=1 l (x ij ˆp i ˆq j ) l ˆp i ˆq j { x ij ˆp i ˆq j x ij + ˆp i ˆq j } k = l j=1 x ij 1 x i. x.j
60 % 350 A 1, A, A 3 3 B 1, B, B 3, B 4 4 5% B 1 B B 3 B 4 A A A N(µ, σ ) σ Z = X µ N(0, 1) σ / N(µ, σ ) σ σ S T = X µ t( 1) S / N(µ, σ ) µ χ = 1 σ (X i µ) χ () N(µ, σ ) µ χ = S σ χ ( 1)
61 60 4 N(µ 1, σ 1),N(µ, σ ) σ 1, σ Z = ( X Ȳ ) (µ 1 µ ) σ 1 / 1 + σ / N(0, 1) N(µ 1, σ 1),N(µ, σ ) σ 1, σ T = ( X Ȳ ) (µ 1 µ ) ˆσ / 1 + ˆσ / t( 1 + ), ˆσ = 1S 1 + S 1 +. N(µ 1, σ 1),N(µ, σ ) F = σ S 1 σ1 S F ( 1 1, 1) A X X B(, p) X N(p, p(1 p)) ( ). p Z = X p p(1 p) N(0, 1) p = X 1+X 1 + A, B 1 C p 1, p 1, X 1, X Z = X 1/ 1 X / N(0, 1) ( )p(1 p) A 1, A,..., A k P (A i ) = p i A i p i = m i A i X i m 5 χ = k (X i p i ) p i = k X i p i χ (k 1) H 0 : D D θ 1,..., θ l A 1, A,..., A k (X 1,..., X k ) (x 1,..., x k ) θ i θ i A i m 1,..., m k χ = k (X i m i ) χ (k l 1) p i
62 A i, B j X ij A i, B j p i, q j A i, B j P ij P ij 5 χ = k l j=1 X ij P ij P ij χ ((k 1)(l 1)), P ij = p i q j
63
64 = 1 + log 100 log = = = = : x = 1 [ ] 100 = =
65 : 5.:. T x = 841 T y = 806 x = y = T xx = x i = T yy yi = s x = 4 T xx (x) = 5.88 s y = 4 T yy (y) = 0.34 T xy = 4 x i y i = 3719 s xy = 1 T xy T x T y = = r = s xy = s x s y = 0.71
66 65 3 = 1 + log log 4 log = 1 + log = = 5.58 x = = 17.0 x 17 y 65 5 y 10 = = : x y T x = 841 T y = 806 x = y = T xx = x i = T yy yi = s x = 4 T xx (x) = 5.88 s y = 4 T yy (y) = T xy = x i y i = 3719 s xy = 1 T xy T x = 1 4 T y = r = s xy = s x s y = 0.71
67 66 5 x y y = (x 35.04) = 0.56(x 35.04) y = 0.56x (a) P r (X 90) = P r ( X 80 6 ) = P r (Z 1.67) = P r ( < Z < 0) + P r (0 Z 1.67) = 0.95 (b) P r ( X 80 1) = P r ( X ) 6 = P r ( Z ) = P r (0 Z ) = (0.477) = 0.95 () X X N(10, 1 ) ( ) 50(l/ ) P r (X 50) ( ) X P r (X 50) = P r = P r (Z 1.90) = 1 P r(0 < Z < 1.96) = = 0.05 X = 1 ( ) = 1(cm) 10 U = 1 ( (110 1) + (11 1) ) ) 9 = (cm) S = 9 10 U = S = BOD X X N(µ, 6.5) 15 X = 7. X N(µ, )
68 67 X Z = X µ = 7.5 µ P r ( Z z α ) = 0.95 z α z α = % Z = 7.5 µ µ σ = 6.5 X i X i N(µ, 6.5) X N(µ, σ /5) X X = 1 13 [ ] = 5 5 = 6.4 Z = X µ N(0, 1) σ /5 95% P r ( Z z α ) = z 0.05 = X z α σ 5 µ X + z α σ /5 µ /5 4.1 µ µ = 146 σ X i X i N(146, σ ) X N(146, σ /4) X X = 1 58 [ ] = 4 4 = S T = X µ t 1,α/ S /4
69 68 5 S S = 1 3 [( ) + ( ) + ( ) + ( ) ] = 1 ( ) = % P r ( T t 1,α/ ) = t 3,0.005/ = X t 3,0.05/ S 4 µ X + t 3,0.05/ S /4 µ / µ ˆp = = 0. z α = z 0.05 = 1.96 X N(p, pq ) α P ( X p pq z α ) = 1 α 4 p(1 p) p(1 p) X z α X p p p X + z α p(1 p) p(1 p) ( p z α, p + z α ( ) (0.)(1 0.) (0.)(1 0.) , (0.174, 0.6) 1 ˆp = = 0.63 z α = z 0.05 = 1.96 X N(p, pq ) α P ( X p pq z α ) = 1 α p(1 p) p(1 p) X z α X p p p X + z α p(1 p) p(1 p) ( p z α, p + z α
70 69 ( ) (0.63)(1 0.63) (0.63)(1 0.63) , (0.568, 0.678) 1 X X N(µ, 0.0 ) 16 X X = 7.09, X N(µ, 0.0 /16) H 0 : µ = 7 H 1 : µ 7 α = 0.05 σ H 0 Z 0 = Z = X µ σ / N(0, 1) /16 = 4(0.09) = : z 0.05 = 1.96 H 0 95% µ
71 µ 7.19 X X N(µ, ) 8 X X = 1 ( ) = α = 0.05 µ σ H 0 : σ = H 1 : σ > χ = S σ χ α, 1 H 0 S = 1 8 [( ) + + ( ) ] = χ 0 = 8(0.001) = 10.5 χ 0 > χ 0.05,7 = H 0
72 71 5.4: 95% χ /, σ χ 0.05/, σ σ σ 1 = σ A = 10, X = 8, S A = B = 1, Ȳ = 76, S B = H 0 : µ 1 = µ H 1 : µ 1 µ α = 0.05 T = ( X Ȳ ) (µ 1 µ ) t A + B ˆσ A + ˆσ B
73 7 5 H 0 t 0.05/,0 =.3 T 0 = ˆσ = AS A + BS B A + B = = / / T 0 = 1.77 < t 0.05/,0 =.09 H 0 II. A = 10, X = 8, SA = 54.41, S A = B = 1, Ȳ = 76, S B = 59.17, S B = σ 1 < σ H 0 : σa = σ B H 1 : σa < σ B α = 0.05 H 0 F ,10 1,1 1 = F = σ B S A σa S F A 1, B 1 B F 0 = = = 1 F 0.05,1 1, = 0.3 F 0 = > F ,10 1,1 1 = 0.3 H 0 F α,1, = 1 F 1 α,, 1 1 A : B : C : D = 9 : 3 : 3 : 1 H 0 : (p 1, p, p 3, p 4 = 9 16, 3 16, 3 16, 1 16 ) H 1 : (p 1, p, p 3, p , 3 16, 3 16, 1 16 ) α = 0.05
74 73 H 0 χ = 4 (X i p i ) p i = 4 X p i χ 0 = = = χ 0.05,4 1 = 7.81 χ 0 = < χ 0.05,3 = 7.81 H 0 1 H 0 : P (λ) α = 0.05 P (λ) λ k p k kp k = E(X) = λ k=0 k f k kf k p k m k p k f k k kf k λ λ λ 1 k kf k = = 0.77 k f k m k
75 74 5 k = 4 m k 5 χ m i 5 k 3 1 H 0 χ 0 = ( ) = χ = 3 (x i m i ) i=0 (99 99) 99 m i + ( ) (13 1, 78) 1.78 χ 0.05,3 1 1 = 3.84 χ 0 = < χ 0.05,1 = 3.84 H 0 λ = 1 H 0 : H 1 : α = 0.05 H 0 χ = 350 (X ij P ij ) P ij χ 0 = 1 350[ 159 ( ) ( ) ( )] 1 = (3 1) (4 1) = 6 χ 0.05,6 = χ 0 = > χ 0.05,6 = 1.59 H 0
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No1 1 (1) 2 f(x) =1+x + x 2 + + x n, g(x) = 1 (n +1)xn + nx n+1 (1 x) 2 x 6= 1 f 0 (x) =g(x) y = f(x)g(x) y 0 = f 0 (x)g(x)+f(x)g 0 (x) 3 (1) y = x2 x +1 x (2) y = 1 g(x) y0 = g0 (x) {g(x)} 2 (2) y = µ
,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.
9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,
x y 1 x 1 y 1 2 x 2 y 2 3 x 3 y 3... x ( ) 2
1 1 1.1 1.1.1 1 168 75 2 170 65 3 156 50... x y 1 x 1 y 1 2 x 2 y 2 3 x 3 y 3... x ( ) 2 1 1 0 1 0 0 2 1 0 0 1 0 3 0 1 0 0 1...... 1.1.2 x = 1 n x (average, mean) x i s 2 x = 1 n (x i x) 2 3 x (variance)
n Y 1 (x),..., Y n (x) 1 W (Y 1 (x),..., Y n (x)) 0 W (Y 1 (x),..., Y n (x)) = Y 1 (x)... Y n (x) Y 1(x)... Y n(x) (x)... Y n (n 1) (x) Y (n 1)
D d dx 1 1.1 n d n y a 0 dx n + a d n 1 y 1 dx n 1 +... + a dy n 1 dx + a ny = f(x)...(1) dk y dx k = y (k) a 0 y (n) + a 1 y (n 1) +... + a n 1 y + a n y = f(x)...(2) (2) (2) f(x) 0 a 0 y (n) + a 1 y
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黒住英司 [ 著 ] サピエンティア 計量経済学 訂正および練習問題解答 (206/2/2 版 ) 訂正 練習問題解答 3 .69, 3.8 4 (X i X)U i i i (X i μ x )U i ( X μx ) U i. i E [ ] (X i μ x )U i i E[(X i μ x )]E[U i ]0. i V [ ] (X i μ x )U i i 2 i j E [(X i