201711grade1ouyou.pdf
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- ふじよし そや
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1 A 1 2 Web Web
2 2
3 3
4 1 p p >2 2 A (a) B (b) A B (a) (b) 5 4
5 5
6 2 3 A B C n A B C A B C n A,n B,n C n A + n B + n C = n y A i (i =1,...,n A ) y B i (i =1,...,n B ) y C i (i =1,...,n C ) y A, y B, y C y = 1 n (n Ay A + n B y B + n C y C ) 1 SST SSB SSW μ A, μ B, μ C H 0 : μ A = μ B = μ C H n A = n B = n C =5 SSB = 1290 SSW = % 5 H 1 : μ A <μ B = μ C 3 4 y A =45, y B =63, y C =66 6
7 7
8 3 ( ) (( ) ( )) X X Y 2 N, Y X = x Y E[Y X = x] =α + βx α β 2 X =60 Y 60 P (Y 60 X = 60) 3 X 50 X ξ = E[X X 50] Z N(0, 1) ν = E[Z Z 0] = 2 0 z 1 2π e z2 /2 dz ν ξ =50+15ν 4 X 50 Y η = E[Y X 50] = α + βξ α β ξ 4 η 8
9 9
10 4 N U n y i A y i i A N U n S i p i ( p i =1) U i i U y i T = y i ˆT = 1 y i n p i i U i S V [ ˆT ]= 1 ( ) 2 yi p i T n p i i U 1 U n i y i y = 1 y i μ = T n N = 1 y i N i S i U V [y] σ 2 = 1 (y i μ) 2 N i U 2 U L N l l =1,...,L U l n l = n N l N l S l ỹ = 1 y i = 1 L y i μ n n i S l=1 i S l V [ỹ] l μ l = 1 y i,σl 2 = 1 (y i μ l ) 2 N l N l i U l i U l
11 4 N = 120, 000 S 2 T 3 S T A B X Y Z N l 60,000 60,000 40,000 40,000 40,000 μ l σl n l
12 5 2 A B (a + b) A a b (a + b +1) b a + b B a a + b 1 n A X 2 n A Y 1 n =5 X =3 P (X =3) X E[X] V [X] 2 n =5 Y =3 P (Y =3) Y E[Y ] V [Y ] 3 n n 2 E[X] E[Y ] V [X] V [Y ] n A B N(μ A, 20 2 ) N(μ B, 20 2 ) 4 n = 200 A 96 B 104 μ A 95% L A μ B 95% L B L A L B 5 1 n μ A 95% n 12
13 13
14 1 log (wage) =β 0 + β 1 (educ)+ɛ (1) wage 1 /hour educ 6+3+3= = β 0 β 1 b 0,b 1 b 0 =0.584 (0.0973), b 1 = ( ) R 2 = H 0 : β 1 =0 H 1 : β 1 0 5% 2 β 1 95% 3 2 (1) (1) 4 (1) β
15 15
16 2 2 x t, y t (t =1, 2,...,T) 2 3 ɛ t 1 y t = β 0 + β 1 x t + ɛ t 2 t x t,y t ry t = y t (a 0 + a 1 t), rx t = x t (c 0 + c 1 t) ry t = β 0 + β 1 rx t + ɛ t 3 x t, y t Δy t = y t y t 1, Δx t = x t x t 1 Δy t = β 0 + β 1 Δx t + ɛ t ( ) x t, y t 3 b 0,b 1 β 0,β b b
17 1 1 H 0 : β 1 =0 H 1 : β 1 0 5% 2 2 H 0 : β 1 =0 H 1 : β 1 0 5% 3 3 H 0 : β 1 =0 H 1 : β 1 0 5% β β 1 1 β
18 3 n x 1,...,x n X λ P oisson(λ) P oisson(λ) X f(x) =P (X = x) = λx x! e λ (x =0, 1, 2,...) 1 P oisson(λ) X E[X] V [X] 2 x 1,...,x n P oisson(λ) n λ x = 1 n x i n i=1 3 X =0 X 1 ZTP(λ) ZTP(λ) g(x) =P (X = x X 1) ξ = E[X X 1] τ 2 = V [X X 1] ZTP(λ) n λ 4 X( 1) 3 g(x) ξ h(x) =P (X = x) = xg(x) ξ (x =1, 2,...) h(x) h(x) f(x) 18
19 19
20 4 N U n y i A y i i A N U n S i p i ( p i =1) U i i U y i T = y i ˆT = 1 y i n p i i U i S V [ ˆT ]= 1 ( ) 2 yi p i T n p i i U 1 U n i y i y = 1 y i μ = T n N = 1 y i N i S i U V [y] σ 2 = 1 (y i μ) 2 N i U 2 U L N l l =1,...,L U l n l = n N l N l S l ỹ = 1 y i = 1 L y i μ n n i S l=1 i S l V [ỹ] l μ l = 1 y i,σl 2 = 1 (y i μ l ) 2 N l N l i U l i U l
21 4 N = 120, 000 S 2 T 3 S T A B X Y Z N l 60,000 60,000 40,000 40,000 40,000 μ l σl n l
22 5 2 A B (a + b) A a b (a + b +1) b a + b B a a + b 1 n A X 2 n A Y 1 n =5 X =3 P (X =3) X E[X] V [X] 2 n =5 Y =3 P (Y =3) Y E[Y ] V [Y ] 3 n n 2 E[X] E[Y ] V [X] V [Y ] n A B N(μ A, 20 2 ) N(μ B, 20 2 ) 4 n = 200 A 96 B 104 μ A 95% L A μ B 95% L B L A L B 5 1 n μ A 95% n 22
23 23
24 1 X α β Gamma(α, β) Gamma(α, β) 1 f(x) = β α Γ(α) xα 1 e x/β (x>0) 0 (x 0) Γ(α) = 0 t α 1 e t dt n X =(X 1,...,X n ) x =(x 1,...,x n ) x 1 = = x n 1 Gamma(α, β) X E[X] =αβ V [X] =αβ 2 2 x α, β l(α, β; x) 3 l(α, β; x) 0 ψ(α) log α =log x n x n, β = x n α x n = 1 n ( n n ) 1/n x i x n = x i ψ(α) i=1 i=1 1 24
25 α, β 5 α η(α) =ψ(α) log α η(α) α 5 α, β η(α) = ψ(α) log α α (0, ) (, 0) 25
26 2 {N(t), t 0} 3 λ(> 0) (1) N(0) = 0 (2) {N(t), t 0} (3) t λt s, t 0 λt (λt)n P (N(t + s) N(s) =n) =e, n =0, 1, 2,... n! t N(t) λ 1 λ Exp(λ) { λe λx (x 0) f(x) = 0 (x<0) Exp(λ) F (x) Exp(λ) f(x) 1 F (x) 2 X 1 X 1 Exp(λ) 3 t 0 MTTF (Mean Time To Failure) 95% t 0 1 MTTF 95% λ x log(1 x) x 26
27 27
28 3 2 T 1 T 2 1 f(t) = μ e t/μ (t 0) (1) 0 (t<0) 1 T (1) E[T ] t T>t ξ = E[T T >t] 2 T 1 = t 1 T 2 = t 2 μ l 1 (μ) μ ˆμ 3 T 1 = t 1 t(> t 1 ) 2 T 2 >t μ l 2 (μ) μ μ 4 μ (0) μ = μ (k) T 2 ξ (k) = E[T 2 μ (k),t 2 >t] (k =0, 1, 2,...) T 1 = t 1 T 2 = ξ (k) μ μ (k+1) μ (k+1) 5 4 μ (0),μ (1),μ (2),... 3 μ μ 28
29 29
30 4 (0, 1) F X f(x) θ = E[X] = xf(x)dx (i) F n x 1,...,x n ˆθ 1 = 1 n n x i i=1 (ii) (0, 1) n u 1,...,u n ˆθ 2 = 1 n u i f(u i ) n i=1 f(x) =12x(1 x) 2 (1) X a, b > 0 1 B(a, b) xa 1 (1 x) b 1 (0 <x<1) a, b Beta(a, b) B(a, b) = B(a, b) = 1 0 (a 1)!(b 1)! (a + b 1)! x a 1 (1 x) b 1 dx a, b 1 X Beta(a, b) E[X] = a a + b 2 1 f(x) x x max f(x) f max = f(x max ) 3 1 f(x) x f(x) (i) (ii) θ = E[X] ˆθ 1 ˆθ (i) (ii) 30
31 31
32 5 2 A B (a + b) A a b (a + b +1) b a + b B a a + b 1 n A X 2 n A Y 1 n =5 X =3 P (X =3) X E[X] V [X] 2 n =5 Y =3 P (Y =3) Y E[Y ] V [Y ] 3 n n 2 E[X] E[Y ] V [X] V [Y ] n A B N(μ A, 20 2 ) N(μ B, 20 2 ) 4 n = 200 A 96 B 104 μ A 95% L A μ B 95% L B L A L B 5 1 n μ A 95% n 32
33 33
34 1 T ( 0) df (t) F (t) =P (T t) f(t) = t 0 T dt S(t) =1 F (t) =P (T >t) 1 T 1 λ(t) = lim P (t T<t+Δ T t) Δ 0 Δ λ(t) = f(t) S(t) 2 T Λ(t) = log S(t) t 0 λ(u)du Λ(t) = 3 P (Λ(T ) >t)=exp( t) 4 Z p Z 1 λ(t Z) = lim P (t T<t+Δ T t, Z), Δ 0 Δ S(t Z) =P (T >t Z) T Cox λ(t Z) =λ 0 (t)exp(β Z) (1) λ 0 (t) β h(t) log( log S(t Z)) = h(t)+β Z h(t) 5 Cox (1) h(t) = β Z + ɛ(t) h(t) 4 ɛ(t) =log( log S(t Z)) T Cox ɛ(t ) P (ɛ(t ) x) =1 exp( exp(x)) 34
35 6 p =1 Z Z [40, 70] T λ 0 (t) = 3 1 t β = Cox 2 2 λ(t Z) = 3 ( ) Z t exp 2 2 T [0, 1] 35
36 2 2 2 H 0 5% H L t = L t 1000 Z(t) H 0 Z(t) B(t) = tz(t) H 0 E[B(t)] = 0 Cov[B(t 1 ),B(t 2 )] = min(t 1,t 2 ) 1 α α α(t) =0.05t 2 L = 400 α 2 Z(0.4) z 1 3 Z(0.4) Z(1.0) Cov[Z(0.4),Z(1.0)] 4 ( 0 0 ) ( 1 ρ ρ 1 ) 2 φ 2 (x 1,x 2 : ρ) Z(1.0) z 2 φ 2 5 L = α 2 α 36
37 37
38 3 T 1 n 1 r 1 T 1 (r 1 +1) 2 n 2 (n 1 + n 2 ) (r +1) T T π π 1 π 0 <π n 1 X 1 T π = π 0 X 1 r 1 2 T π = π 1 T 2 β n π X P (X = x) =b(x; π, n) 3 T π = π 0 T 1 α 4 π 1 =0.5 π 0 =0.1 n 1 =5 r 1 =0 n 2 =7 r =3 α β : x b(x;0.1, 5) b(x;0.1, 7) b(x;0.5, 5) b(x;0.5, 7) T π = π 0 N 38
39 39
40 4 k =1, 2, 3 McCullagh Nelder (1989) 5 k y 11k y 12k y 21k y 22k ψ ψ ψ 3 1 ψ 1,ψ 2,ψ y 11k E[y 11k ] V [y 11k ] N k = y 11k + y 12k + y 21k + y 22k E[y 11k ]= (y 11k + y 12k )(y 11k + y 21k ) N k V [y 11k ]= (y 11k + y 12k )(y 21k + y 22k )(y 11k + y 21k )(y 12k + y 22k ) Nk 2(N k 1) 1 log ψ k log ψ k = y 11k E[y 11k ] V [y 11k ] 3 ψ log ψ 1 V [ log ψ k ] = V [y 11k] ψ 40
41 4 log ψ ψ 95%
42 5 2 A B (a + b) A a b (a + b +1) b a + b B a a + b 1 n A X 2 n A Y 1 n =5 X =3 P (X =3) X E[X] V [X] 2 n =5 Y =3 P (Y =3) Y E[Y ] V [Y ] 3 n n 2 E[X] E[Y ] V [X] V [Y ] n A B N(μ A, 20 2 ) N(μ B, 20 2 ) 4 n = 200 A 96 B 104 μ A 95% L A μ B 95% L B L A L B 5 1 n μ A 95% n 42
43 43
44 44
45 1. Q(u) 0 u u u = Q(u) u = Q(u) =.0250 u 45
46 2. t =4 0 t ( ) ν ν t α t t α (ν) ν =20 5% (α =0.05) t 0.05 (20) = α 46
47 3. =5 0 2 ( ) ν ν α χ 2 χ 2 α(ν) ν =20 5% (α =0.05) χ (20) = α 47
48 4. F 1 =10 2 =20 0 F ( 1, 2) α =0.05 ν2 \ ν α =0.025 ν2 \ ν (ν1,ν2) F α F Fα(ν1,ν2) ν1 =5,ν2 =20 5% (α =0.05) F0.05(5, 20) =
49 5. x e x x e x x log 10 x x log 10 x :
50 50
51 51
52
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