renshumondai-kaito.dvi
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- よしたか わしあし
- 5 years ago
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3 E 326 F E 330 F 470 E F E F E F E E F 0.341, E / = / = / = ( )/3= / = z = 1 i = z 1 i x)/s = (x 1 ( ) x i x =0 s 1 i z) (z 2 = 1 i z 2 = 1 (x i x) 2 /s 2 = s 2 /s 2 = (2.2) (2.12) x = (1/) k f im i = 1 (1 207 ( 9.05) +3 ( 7.05) +12 ( 5.05) +29 ( 3.05) +53 ( 1.05) ) = s 2 = (1/) k f i(m i x) 2 =(1/) k f im 2 i x 2 = (1 ( 9.05)2 +3 ( 7.05) ( 5.05) ( 3.05) ( 1.05) ) ( ) 2 = s = =
4 x = k m i(f i/) = = 712 s 2 = k (m i x) 2 (f i/) = k m 2 i (f i/) x 2 = = s = = m i f i m if i m i x (m i x) 2 (m i x) 2 f i x = 1 m if i = 30 k 20 =1.5 s2 = 1 (m i x) 2 f i = 1776 k 20 = x = k m i(f i/) = = s 2 = k m 2 i (f i/) x 2 = = s = = x = ( )/6 = s 2 =( )/ = ( )/2 = s = =12.6 4
5 2.10 x i x i x (x i x) 2 x 2 i x = 125/5 =25 s 2 = 1 (x i x) 2 = 34 =6.8 s = s 2 = 1 x 2 i x 2 = 3159/ = (1+2+3)/3 =2 2 (3+4+2)/3 =3 3 ( )/3 2 2 = 2/3 4 ( )/3 3 2 =2/3 5 ( )/3 2 3 = 1/3 6 1/3 = /3 2/ x i y i x i x y i y (x i x) 2 (y i y) 2 (x i x)(y i y) x =0 y =2 s 2 x =10/5 =2 s 2 y =14/5 =2.8 s xy =0 r = s xy/(s xs y)=0/ = A = { } A c = { } P(A) =P(A c )=1/2 P(A) =1/2 5
6 P(A c 1 A 2)=P(A c 1)P(A 2)=1/ N = (A) +(A c ) N (3.9) (A) < = (B) N (3.10) 3.3 A 1 = {20 } A 2 = { 30 } A 3 = {40 } B 1 = { } B 2 = { } B 3 = { } B 4 = { } P(A 1)=0.3 P(A 2)=0.5 P(A 3)= P(B 2 A 2)=P(B 2 A 2)P(A 2)= = B 1 B 3 P(A 3 (B 1 B 3)) = P(B 1 B 3 A 3)P(A 3)=(P(B 1 A 3)+P(B 3 A 3))P(A 3)= = P((A 1 B 4) (A 2 B 3)) = P(B 4 A 1)+P(B 3 A 2)= = B 1 A 1 P(A 1 B 1)=P(A 1 B 1)/P(B 1) P(A 1 B 1)=P(B 1 A 1)P(A 1)=0.123 A 1, A 2, A 3 P(B 1)=P((B 1 A 1) (B 1 A 2) (B 1 A 3)) = P(B 1 A 1)+P(B 1 A 2)+P(B 1 A 3) P(B 1 A 1) P(B 1 A 2)=0.165 P(B 1 A 3)=0.054 P(A 1 B 1)=0.123/( ) = A B = {1, 2, 3, 4, 8} 2 A B = {2, 3} 3 A B = {1, 4} 4 A A c = φ A = {2, 4, 6} P(A) =1/2 2 B = {3, 6} P(B) = 1/3 3 A B = {2, 3, 4, 6} P(A B) =2/3 4 A B = {6} P(A B) =1/ P(E)=300/( )=0.6 2 P(J)=200/( )=0.4 3 P(M E)=30/100 = P(M J)=20/100 = P((E J) M)= P(M)=( )/500= P((E J) M c )=1 P((E J) M)= P((E J) M)=P(M)= P((E J) M c )=1 P((E J) M)= P(E M)=9/13= P(J M)=4/13= X X =2, 3,, 12 1/36, 2/36, 3/36, 4/36, 5/36, 6/36, 5/36, 4/36, 3/36, 2/36, 1/36 ( =1) 6
7 E[aX + b] = ae[x] +b V(aX + b) = E[(aX + b E[aX + b]) 2 ]=E[(a(X E[X])) 2 ]=E[a 2 (X E[X]) 2 ]= a 2 E[(X E[X]) 2 ] = a 2 V(X) 4.4 Z = (X μ)/σ E[Z] =E[(X μ)/σ] =(E[X] μ)/σ =0, V(Z) =E[(Z E[Z]) 2 ]= E[Z 2 ]=E[((X μ)/σ) 2 ]=E[(X μ) 2 ]/σ 2 = σ 2 /σ 2 =1 4.3 E[X] =7 V(X) =35/6 E[Y ] = 3E[X]+5 = 26 V(Y )=3 2 V(X) = A B C F A B C F f(0) = 1/8 = f(1) = 3/8 = f(2) = 3/8 = f(3) = 1/8 = f(4) = 0 6 F ( 1) = 0 7 F (1.9) = f(0) + f(1) = F (2) = f(0) + f(1) + f(2) = F (5) = f(0) + f(1) + f(2) + f(3) = P(X =2)= 2 2 f(x)dx =0 2 P(2 <X<5) = P(X <5) P(X <2) = F (5) F (2) E[X] = =4.6 2 V(X) =( ) = V(X) = 0.64 = E[Y ]=0.5E[X]+3= = V(Y )=0.5 2 V[X] = V(Y )= 0.16 = ( ) = = = = = ( ) = E[X] =2 ( )+4 ( ) = 3.2 E[Y ]=0 ( ) + 1 ( ) = 0.7 Cov(X, Y )=E[XY ] E[X]E[Y ]= = = V(X) =2 2 ( )+4 2 ( ) =0.96 V(Y ) = 0 2 ( ) ( ) = 0.21 ρ(x, Y ) = Cov(X, Y ) 0.16 = = P(Y =0 X =4)=0.1/(0.1+ V(X) V(Y ) ) = 1/6 5 P(Y =1 X =4)=0.5/( ) = 5/6 6 E[X + Y ]= 7
8 E[X]+E[Y ]= =3.9 7 V(X+Y )=V(X)+2Cov(X, Y )+X(Y )= = E[X+Y ]=E[X]+E[Y ]=3+2=5 2 ρ(x, Y )= Cov(X, Y ) V(X) V(Y ) 0.5 = Cov(X, Y ) Cov(X, Y )=1 3 Cov(X, Y )=E[XY ] 4 1 E[X]E[Y ] E[XY ]=10+2 4=18 4 V(X) =E[X 2 ] (E[X]) 2 = =2 5 Cov(X, Y )=E[XY ] E[X]E[Y ]=E[X]E[Y ] E[X]E[Y ]= 0 6 Cov(X, Y )=E[XY ] E[X]E[Y ]= = 36 7 V(X +Y )= V(X)+V(Y )= E[X] =E[X 1 + +X 5]/5 =(E[X 1]+ +E[X 5])/5 =(3+ + 3)/5 =3 2 V(X) =V(X X 5)/5 2 =(V(X 1)+ +V(X 5))/5 2 = (5 + +5)/5 2 = E[X] = = 3.9 V(X) = E[X 2 ] (E[X]) 2 = = E[X] =0 p +1 2p p +3 3p 2 =11p 2 +6p V(X) = E[X 2 ] (E[X]) 2 =0 2 p p p p 2 (11p 2 +6p) 2 = (29p 2 +12p) (11p 2 +6p) 2 1 p+2p 2 +3p+3p 2 =1 5p 2 +4p 1=(5p 1)(p +1)=0 p =1/5 p = 1 0 < = p < = 1 p =1/5 =0.2 E[X] V(X) E[X] = =1.64 V(X) =( ) = f(x)dx =1 f(x)dx = a(1 x)dx=a [ x x 2 /2 ] 1 0 = a(1 1/2) = a/2 =1 a =2 2 E[X] = xf(x)dx = (x x 2 )dx = 2 [ x 2 /2 x 3 /3 ] 1 = 2(1/2 1/3) = 0 1/3 E[X2 ] = x 2 f(x)dx =2 1 0 (x 2 x 3 )dx =2 [ x 3 /3 x 4 /4 ] 1 =2(1/3 1/4) = 1/6 V(X) =E[X 2 ] (E[X]) 2 =1/6 (1/3) 2 =1/18 3 F (x) =0.0 0 < = x<1 F (x) =P(X < = x)= 0 x 0 f(t)dt = x<0 x 0 2(1 t)dt =2[t t 2 /2] x 0 =2x x 2 1 < = x F (x) =1.0 4 m P(X < = m)=p(x>m)=1/2 P(X < = m)=2m m 2 =1/2 m =(2± 4 2)/2 =(2± 2)/2 =1± 2/2 0 < = m < = 1 m =1 2/
9 P(X =3)= 4C = P(2 < = X < = 3) = P(X = 2) + P(X =3)= 4C C = = P(X < = 3) = 1 P(X =4)=1 4C = a =a +0.9 =1 a =0.1 2 X Y P(X = x) P(Y = y) P(X =1,Y =1)=0.1 P(X =1)P(Y =1)= E[X] = 2 x=1 xp(x = x) = =1.6 E[X 2 ]= 2 x=1 x 2 P(X = x) = =2.8 V(X) =E[X 2 ] (E[X]) 2 = = Y P(Y X =1) E[Y X =1]= = 2 E[Y 2 X =1]= = 4.5 V(Y X =1)=E[Y 2 X =1] (E[Y X = 1]) 2 = =0.5 5 X Y P(X = x, Y = y) Z =2X + Y Z Z P(Z = z)
10 E[Z] = =5.1 E[Z 2 ]= = 27.5 V(Z) = E[Z 2 ] (E[Z]) 2 = = c = =0.3 2 μ x =E[X] = =1.3 3 V(X) =( 1) = μ y =E[Y ]= = V[Y ]= = Cov(X, Y )=E[XY ] μ xμ y =E[X(2X +5)] μ xμ y =2E[X 2 ] + 5E[X] μ xμ y =2 (( 1) ) = Cov(X, Y ) 7.62 = =1 V(X)V(Y ) a = 1 ( ) = X Y P (X = x i,y = y j)=p (X = x i)p (Y = y j) X Y P (X = 1,Y = 1) = 0.1 P (X = 1)P (Y = 1) = =0.06 P (X = 1,Y = 1) \= P (X = 1)P (Y = 1) X Y 4 P (X 2 =0,Y 2 =0)=P (X =0,Y =0)=0.1 P (X 2 =0,Y 2 =1)= P (X =0,Y = 1) + P (X =0,Y =1)=0.1 P (X 2 =1,Y 2 =0)= P (X = 1,Y =0)+P (X =1,Y =0)=0.3 P (X 2 =1,Y 2 =1)= P (X =1,Y =1)+P (X =1,Y = 1) + P (X = 1,Y =1)+P (X = 1,Y = 1) = 0.5 X 2 Y E[X 2 Y 2 ]= =0.5 E[X 2 ]= 0 ( )+1 ( ) = 0.8 E[Y 2 ]=0 ( )+1 ( ) = 0.6 Cov(X 2,Y 2 )=E[X 2 Y 2 ] E[X 2 ]E[Y 2 ]= = E[X] = =7/2 E[X 2 ]= =91/6 V(X) =E[X 2 ] (E[X]) 2 =91/6 49/4 =35/12 10
11 2 Y X / /6 2 1/6 1/6 0 1/6 1/6 0 3 Y /3 2/3 E[Y 2 ]=1 1/3+4 2/3 =3 4 E[X]=7/2 E[Y ]=1 1/3 +2 2/3=5/3 V(X)=35/12 V(Y )= E[Y 2 ] (E[Y ]) 2 =3 25/9=2/9 Cov(X, Y )=E[XY ] E[X]E[Y ]=1 3 1/ / / / / /6 (7/2)(5/3) = 1/3 ρ(x, Y )= Cov(X, Y ) 1/3 = = 6/35 V(X)V(Y ) (35/12)(2/9) μ =E[X] = x xp (X = x) = = 1 6 V(X) = x (x μ) 2 P (X = x) = x x 2 P (X = x) μ 2 = ( 1 6 )2 = x 1 x 2 x 3 f(x 1,x 2,x 3) y ( 1 6 )0 ( 5 6 )3 = ( 1 6 )1 ( 5 6 )2 = ( 1 6 )1 ( 5 6 )2 = ( 1 6 )2 ( 5 6 )1 = ( 1 6 )1 ( 5 6 )2 = ( 1 6 )2 ( 5 6 )1 = ( 1 6 )2 ( 5 6 )1 = ( 1 6 )3 ( 5 6 )0 = X 1 Y f(x 1,y) 11
12 X 1 Y E[X 1]= 1, V(X1) = 5 E[Y ]= 3 15, V(Y )= Cov(X1,Y)= E[X 1Y ] E[X 1]E[Y ] E[X 1Y ]= x1 x 1yf(x 1,y)= y = Cov(X1,Y) = = 30 ρ = Cov(X 1,Y ) 30/216 = = V(X1 ) 3 (4) x1 =1 Y V(Y ) 5/36 15/36 3 f(y x 1 =1) y f(y X 1 =1) E[Y X 1 =1] E[Y X 1 =1]= y yf(y x 1 =1)= = E[X 1 + X 2]=E[X 1]+E[X 2]=2+3=5 V(X 1 + X 2)=V(X 1)+ 2Cov(X 1,X 2)+V(X 2)= = E[ˆσ 2 ]= 1 E[ (X i X) 2 ]= 1 E[ ((X i μ) (X μ)) 2 ]= ) 1 E[ ((X i μ) 2 2(X μ)(x i μ)+(x μ) 2 ]= 1 E[ (X i μ) 2 (X μ) 2 ]= 1 E[(X i μ) 2 ] E[(X μ) 2 ]=σ 2 σ2 = 1 σ μ =E[X] = =0.2 σ 2 =V(X) =(1 0.2) (0 0.2) =0.16 E[(X μ) 3 ]=(1 0.2) (0 0.2) =0.096 E[(X μ) 3 ]/σ 3 =0.096/0.16 3/2 = f(x) = P(X = x) = 3C x0.5 3 P(X =0)=1/8 P(X =1)=3/8 P(X =2)=3/8 P(X =3)=1/8 E[X] =0 1/8 +1 3/8 +2 3/8 +3 1/8 =1.5 V(X) = 0 2 1/ / / / =
13 2 a f(2) = 3/8 b f(2.5) = 0 c F ( 0.1) = 0 d F (2.99) = f(0) + f(1) + f(2) = 7/8 3 Y X /8 0 1/8 1 1/8 0 3/8 0 4 Cov(X, Y )=E[XY ] E[X]E[Y ] E[XY ]=( 1) 1 3/8+( 1) 3 1/ / /8 =0 E[X] =3/2 V(X) =3/4 E[Y ]=0 V(Y )=( 1) 2 1/ /2 0 2 =1 Cov(X, Y )=E[XY ] E[X]E[Y ]= 0 3/2 0=0 ρ(x, Y )= c = =0.6 E[X] = = 3.2 V(X) = =5.36 X 2 X 2 Prob(X < = 2.1) = =0.7 5 ( ) X N(2, 9) Z =(X 2)/3 N(0, 1) X N(10, 4 2 ) Z =(X 10)/4 N(0, 1) P(X >12) = P(Z >0.5) = = X N(70, 12 2 ) Z =(X 70)/12 N(0, 1) x 0 P(X <x 0)=P(Z<(x 0 70)/12) = 0.33 x 0 1 P(Z < 0.44) = 0.33 (x 0 70)/12 = 0.44 x 0 = X X N(4400, ) 10 x 1 90 x 2 P(X < = x 1)=0.1 P(X < = x 2)=0.9 x 1 x 2 Z =(X 4400)/469 N(0, 1) P(Z < = (x )/469) = P(Z < = 1.28) = 0.1 P(Z < = (x )/469) = 13
14 P(Z < = 1.28) = 0.9 x 1 = 3800 g x 2 = 5000 g cm cm P(Z > 0) = P(Z < 2.22) = P(Z > 2.22) = P(Z =1.0) = 0 4 P( 0.3 <Z<0.5) = 1 P(Z >0.5) P(Z < 0.3) =1 P(Z >0.5) P(Z >0.3) = = P(X > 2) = P((X +2)/4 > 1) = P(Z > 1) = P(X < 2) = P((X +2)/4 > 0) = P(Z >0) = P( 3 <X<1) = P( 0.25 < (X +2)/4 < 0.75) = P( 0.25 <Z<0.75) = 1 P(Z >0.75) P(Z < 0.25) = 1 P(Z >0.75) P(Z >0.25) = = P( X < 0.4) = P( 0.4 <X<0.4) = P(0.4 < (X +2)/4 < 0.6) = P(0.4<Z<0.6)=P(Z >0.4) P(Z >0.6)= = P( X <x)=p( x <X<x)=P(( x 3)/5 < (X 3)/5 < (x 3)/5) = P(( x 3)/5 <Z<(x 3)/5) = 1 P(Z >(x 3)/5) P(Z < ( x 3)/5) = 1 P(Z >(x 3)/5) P(Z >(x +3)/5) x (x 3)/5 P(Z>(x 3)/5) (x +3)/5 P(Z>(x +3)/5) 1 P(Z>(x 3)/5) P(Z>(x +3)/5) x = P(X <x)=p((x 3)/5 < (x 3)/5) = P(Z <(x 3)/5) = P(Z < 1.96) = (x 3)/5 = 1.96 x = X N(5, 2 2 ), Z =(X 5)/2 N(0, 1) 1 P(X > = 4) = P((X 5)/2 > = (4 5)/2) = P(Z > = 0.5) = P(X < = 5) = P((X 5)/2 < = (5 5)/2) = P(Z < = 0) = P(X < = 3) = P((X 5)/2 < = (3 5)/2) = P(Z < = 1) = P(3.5 < = X < = 4.5) = P((3.5 5)/2 < = (X 5)/2 < = (4.5 5)/2) = P( 0.75 < = Z < = 0.25) = = P( X 4 > 0.5) = P(X 4 > 0.5, X 4 < 0.5) = P(X > 4.5, X<3.5) = P(X >4.5) + P(X <3.5) = P(Z >(4.5 5)/2) + P(Z < (3.5 5)/2) = P(Z > 0.25) + P(Z < 0.75) = = X N(68, 8 2 ) P(X < 60) = P((X 68)/8 < (60 68)/8) = P(Z < 1) = % 2 14
15 P(X > = 78) = P((X 68)/8 > = (78 68)/8) = P(Z > = 1.25) = % 3 P(X > = x 0)=0.05 P((X 68)/8) > = (x 0 68)/8) = z 0.05 = (x 0 68)/8 = x 0 = % X N(3, 5 2 ) P( X 3 < 5 = x 3 ) = 0.1 Z = 5 X 3 N(0, 1) P(Z < 5 = z)=0.1 z z = x 3 5 P(Z < = ) = = x 3 x = P(0.0 <Z<2.0) = P(Z >0.0) P(Z >2.0) = = E[W ]=E[X]+ 1 E[Y ]= =1.0 V(W )=V(X)+ 3 1 W 1.0 V(Y )= =4.0 W N(1.0, 4.0) 3 Z = N(0, 1) P( W 1.0 > 2.0) = P(W 1.0 < 2.0, W 1.0 > 2.0) = P(W < 1.0) + P(W >3.0) = P( W 1.0 < )+P( < W 1.0 ) = P(Z< 1.0)+P(Z >1.0) = 2P(Z >1.0) = = P(X >Y)=P(X Y > 0) X Y X N(0, 11/25) Y N( 1, 1) X Y N(1, 36/25) X Y 1 N(0, 1) P(X Y > 0) = P( X Y 1 > 36/25 36/ /25 )=P(Z> 5/6) = = (6.4) 7 (6.7) 26/9 ((4 3)/(4 1))(26/3)=26/9 (6.7) X X N(480, /64) Z = (X 480)/(320/ 64) N(0, 1) P(450 < = X < = 500) = P( 0.75Z < = 0.5)= P(X > = 520)=P(Z > = ( )/(320/ )) < = 0.05 P(Z > = ) = 15
16 0.05 ( )/(320/ ) > = > = = S P(S > = 400) < = 0.05 ( 1)S 2 / P(( 1)S 2 /320 2 > = ( 1)400 2 /320 2 ) < = 0.05 = /320 2 = P(20S 2 /320 2 > = χ (20)) = 0.05 χ (20) P(20S 2 /320 > = 31.25) > P(20S 2 /320 > = 31.41) = 0.05 =22 P(21S 2 /320 > = 32.81) < 0.05 = P(S1/S > = 2.5) < = () S 1/S (8, 1) F =18 P(S1/S > = 2.5) > =19 P(S1/S > = 2.5) < 0.05 = x = x = P(x <X<28.85) = P(X >x) P(X >28.85) = P(X >x) = x = (X E[X])/ V(X) = (X 1)/ 2/2500 N(0, 1) N(0, 1) 2 P(X < x)=p((x 1)/ (2/2500) < (x 1)/ (2/2500)) = P(Z < 1.96) = (x 1)/ (2/2500) = 1.96 x= P(0.98 < X<1.05)=P((0.98 1)/ (2/2500) < (X 1)/ (2/2500) < (1.05 1)/ (2/2500)) = P( 0.71 <Z<1.77) = 1 P(Z >0.71) P(Z >1.77) = = [ ] E[X] =E X i = 1 i]= E[X 1 9 3=3 2 V(X) = 9 ( ) ( ) 1 2 ( ) V X i = V(X i)= 25 = P(X <0) = 9 P((X 3)/ 25/9 < 3/ 25/9) = P(Z < 1.8) = P(Z >1.8) = P(X >x)=p((x 3)/ 25/9 > (x 3)/ 25/9), P(Z >1.282) = 0.1 (x 3)/ 25/9 =1.282 x = x = x = x = x = P(0.0 <X)=0.5 6 P( 1.35 <X)= N(μ, σ 2 ) 2 N(μ, σ 2 /) 3 N(0, 1/) 4 χ 2 () 5 χ 2 ( 1) 6 N(0, 1) 7 t( 1) 8 χ 2 (1) 6.11 X i N(35, 10 2 ) X N(35, (10/ 25) 2 )=N(35, 2 2 ) 1 P(X 16
17 > = 34.6) = P((X 35)/2 > = ( )/2) = P(Z > = 0.2) = = P(X < = 34.2) = P((X 35)/2 < = ( )/2) = P(Z < = 0.4) = P(34 < = X < = 35.5) = P((34 35)/2 < = (X 35)/2 < = ( )/2) = P( 0.5 < = Z < = 0.25) = 1 ( ) = X N(μ, ) =81 X N(μ, (7.2/ 81) 2 )=N(μ, ) P( X μ > 1) = P(X μ>1, X μ< 1) = P(X μ> 1) + P(X μ < 1) = P((X μ)/0.8 > 1/0.8) + P((X μ)/0.8 < 1/0.8) = P(Z >1.25) + P(Z < 1.25) = = X N(μ, ) =36 X N(μ, (4.2/ 36) 2 )=N(μ, ) P( (X μ)/0.7 < 1.96) = 0.95 P( X μ < ) = P( X μ < 1.372) = P( X μ < 1) = 0.95 P( X μ /(σ/ ) < 1.96) = 0.95 P( X μ < 1.96σ/ )= (σ/ )=1 =1.96σ σ =4.2 =( ) 2 = P(118 < = X < = 125) = P( < 6 = X 720 < = 125 )= X 6 P( < = 6 < 720 X = 6 )=P( 0.2 < = < = 0.5) = = X t(8) P(X < = x)=0.05 x P(3X +2< = )=0.05 y = = (7.213, 9.187) (7.024, 9.376) (8.634, ) (8.213, ) ( / 4271, / 4271) = (703.0, 721.0) ( / 4271, / 4271) = (701.3, 722.7) 7.4 x = s 2 = ( / 12, 17
18 / 12) = ( 1.365, 1.115) ( / 12, / 12) = ( 1.644, 1.394) (1.565, 6.740) (1.405, 8.063) 7.6 (( 1)s 2 /χ 2 α/2( 1) ( 1)s 2 /χ 2 1 α/2( 1)) = 12 s 2 = ( /19.68, /4.57) = (3.196, ) ( /21.92, /3.82) = (2.870, ) (0.593, 0.747) (0.578, 0.762) L = λ exp( x i ) 2 λ L λ λ + x i λ =0 λ ˆλ = 1 x 2 i = x E[X]=E[(1/3)X 1 +(1/3)X 2 +(1/3)X 3]=(1/3)E[X 1]+(1/3)E[X 2] + (1/3)E[X 3] = μ 2 E[ X] = E[(1/6)X 1 + (1/2)X 2 + (1/3)X 3] = (1/6)E[X 1]+(1/2)E[X 2]+(1/3)E[X 3]=μ 3 E[ ˆX] =E[(1/2)X 1 + (1/4)X 2 +(1/5)X 3]=(1/2)E[X 1]+(1/4)E[X 2]+(1/5)E[X 3]=(19/20)μ 4 V(X) = V((1/3)X 1 +(1/3)X 2 +(1/3)X 3)=(1/3) 2 V(X 1)+(1/3) 2 V(X 2) +(1/3) 2 V(X 3)=(1/3)σ 2 5 V( X) = V((1/6)X 1 +(1/2)X 2 +(1/3)X 3) =(1/6) 2 V(X 1)+(1/2) 2 V(X 2)+(1/3) 2 V(X 3)=(7/18)σ E[X] =μ 2 E[ X] =μ 3 E[ ˆX] \= μ 4 V(X) < V( X) ω i =1 ω ix i μ E[S 2 ]=σ 2 E[S 2 ]=(( 1)/)σ 2 \= σ 2 2 E[S 2 ]=σ X N(μ, 3 2 /25) (X μ)/ 3 2 /25 N(0, 1) 2 P( X μ / 3 2 /25 <z 0.005) =0.99 (x z /25, x+z /25) = ( /25, /25) = ( , ) X N(μ, σ 2 /12) (X μ)/ σ 2 /12 N(0, 1) (X μ)/ S 2 /12 t(11) 2 P( X μ / S 2 /12 <t 0.05(11)) = 0.90 (x t 0.05(11) s 2 /12, x + t 0.05(11) s 2 /12) = ( /12, /12) = (11.045, ) 18
19 X N(μ, σ 2 /400) (X μ)/ σ 2 /400 N(0, 1) (X μ)/ S 2 /400 t(399) N(0, 1) 2 P( X μ / S 2 /400 < z 0.025) =0.95 (x z s2 /400, x + z s2 /400) = ( /400, /400) = (2.168, 2.952) ( 1)S 2 /σ 2 χ 2 (15) 2 P(χ (15)<( 1)S 2 /σ 2 <χ (15)) =P(7.26 < 15 S 2 /σ 2 < 25.00) = 0.90 (15 s 2 /25.00, 15 s 2 / 7.26) = ( /25.00, /7.26) = (0.864, 2.975) X N(μ, σ 2 /4) (X μ)/ σ 2 /4 N(0, 1) (X μ)/ S 2 /4 t(3) 2 P( X μ / S 2 /4 <t 0.05(3)) = 0.90 (x t 0.05(3) s 2 /4, x + t 0.05(3) s 2 /4) x =( )/4 = 1.4 s 2 = (1/3)( ) = ( /4, /4) = ( 0.62, 3.42) (ˆp p)/ ˆp(1 ˆp)/ N(0, 1) 2 P( ˆp p / ˆp(1 ˆp)/ < z 0.025) =0.95 (ˆp z ˆp(1 ˆp)/, ˆp + z0.025 ˆp(1 ˆp)/) = ( ( )/536, ( )/536) E[Y ]=E[aX 1 + bx 2]=aE[X 1]+bE[X 2]=aμ + bμ =(a + b)μ Y a + b =1 2 V(Y )= E[(aX 1 +(1 a)x 2 μ) 2 ]=E[(a(X 1 μ)+(1 a)(x 2 μ)) 2 ]=a 2 E[(X 1 μ) 2 ]+(1 a) 2 E[(X 2 μ) 2 ]=σ 2 (a 2 +(1 a) 2 ) a a dv(y )/da = σ 2 (2a 2(1 a)) = σ 2 (4a 2) dv(y )/da =0 a = b =1/ x ± z α/2 σ/ x =61.2 σ =5, =49 90% 61.2 ± / 49 (60.025, ) 95% 61.2 ± / 49 (59.80, 62.60) 7.20 (( 1)s 2 /χ 2 α/2( 1) ( 1)s 2 /χ 2 1 α/2( 1)) =15 s 2 = % (14 3.6/23.68, /6.57) = (2.128, 7.671) 95% (14 3.6/26.12, /5.63) = (1.930, 8.952) 7.21 x ± t α/2 (15)s/ x =10.2 s 2 =8.4 (s =2.8982) =16 90% 10.2 ± / 16 (8.930, ) 95% 10.2 ± / 16 (8.656, ) 7.22 ˆp ± z α/2 ˆpˆq/ ˆp =45/300 = 0.15 =
20 90% 0.15 ± /300 (0.116, 0.184) 95% 0.15 ± /300 (0.110, 0.190) μ σ2 3 X X 4 X σ Z X Y Z = ax + by + c μ μ =E[Z] =E[aX + by + c] =ae[x]+be[y ]+c =(a + b)μ + c a + b =1 c =0 Z = ax +(1 a)y a V(Z) =a 2 V(X)+(1 a) 2 V(Y )=σ 2 (2a 2 2a +1)=2σ 2 (a 1 2 ) σ2 a = 1 V(Z) μ 2 Z = 1 X + 1 Y ( /81, /81) = ( , ) H 0 : μ = 120, H 1 : μ>120 α =0.01 x = 135 < 143 H 0 α =0.05 x <136 H 0 α =0.1 x >132 H β =P(X<136 H 1)=P((X μ 1)/(σ/ ) < ( )/(20/2)) = P(Z < 0.4) 0.34 α =0.01 β =P(Z < ( )/(20/2)) = P(Z < 0.3) 0.62 β t =5.67 μ = 740 t =1.67 μ = 630 t =2.33 μ = μ 95% (623, 687) H 0 : μ = μ H 0 : μ > = 80.9 H 1 : μ<80.9 H 0 5% x =62.5 < (18/10) = 77.9 H 0 1% H μ J μ U H 0 : μ J = μ U, H 1 : μ J \= μ U 1 = 2 =11, x J = 20
21 1.218, x U =3.055, s 2 J =( )/10 = 2.182, s 2 U = ( )/10 = % x J x U =1.837 > z / /11 = = 1.48 H0 2 H 0 : μ J > = μ U, H 1 : μ J <μ U 5% x J x U = < = H p H 0 : p =0.727 ( p < = 0.727) H 1 : p>0.727 ˆp = 7500/10000 = 0.75 z z =( )/ /10000 = 5.16 z 0.01 = z =5.16 >z % H 0 : μ =90 2 (x μ 0)/ σ 2 / = (101 90)/ 10 2 /4= >z = <z = H 0 : p =0.5 H 1 : p<0.5 2 ( )/ 0.5(1 0.5)/100 = N(0, 1) < z 0.05 = H 0 : μ =3.5 H 1 : μ<3.5 2 ( )/ 2.812/11 = t(10) < t 0.01(10) = H 0 : μ =3.5 H 1 : μ \=3.5 2 ( )/ 2 2 /25 = N(0, 1) > z = / 361/ /225 = N(0, 1) < z = z =(x μ 0)/(σ/ ) x =20 =36 z =(20 18)/(7/ 36) = > = z 0.05 H 0 2 z = < (= z 0.01) H 0 3 N(μ, σ 2 ) α P((X μ 0)/(σ/ ) >z α)=p(x > μ 0 + z ασ/ )=α X >μ 0 + z ασ/ μ 0 =18 σ =7 =36 z 0.05 =1.645 z 0.01 = X> / 36 = X> / 36 =
22 X 0.05 P(X >19.92) = P((X 22)/(7/ 36) > ( )/(7/ 36)) = P(Z > 1.78) = P(X > 20.71) = P((X 22)/(7/ 36) > ( )/(7/ 36)) = P(Z > 1.11) = x =87 s =7 =12 x±t α/2 s/ =87± / 12 (82.552, ) 2 H 0 : μ =83 H 1 : μ>83 t =(x 83)/(7/ 12) = (87 83)/(7/ 12) = > = t 0.05(11) H H 0 : μ 1 =μ 2 H 1 : μ 1 \=μ 2 Z =(X 1 X 2)/ S1/ S2/ 2 2 N(0, 1) x 1 =33 s 2 1 =10 1 = 160 x 2 =33.6 s 2 2 =12 2 = 180 α =0.05 z =( )/ 10/ /180 = 0.6/0.359 = > 1.96 = z H H 0 : p =1/6 (=0.167) H 1 : p>1/6 ˆp =54/240 = α =0.01 z =(ˆp 0.167)/ /240 = ( )/ = > = z 0.01 H H 0 : p =0.7 H 1 : p<0.7 ˆp = 2070/3000 = 0.69 α = 0.05 z =(ˆp 0.7)/ /3000 = ( )/ = > = z 0.05 H 0 70% 8.18 H 0 : μ =60 H 1 : μ \=60 x = ( )/5 = 295/5 =59 s 2 = ((59 59) 2 +(56 59) 2 +(62 59) 2 +(61 59) 2 + (57 59) 2 )/4=( )/4 =26/4 =6.5 s =2.55 α =0.05 t = (x μ 0)/(s/ ) = (59 60)/(2.55/ 5) = 1/1.14 = > = t 0.025(4) H =1 2 t 4/ (15) = =2.5 2 t 4/ (90) = t 0.025(100) = t 0.025(99) = t0.05(15) = / =2 2 z 0.01 = (1 0.5)/
23 (5 + 6) = ( ) = (( )2 +(6 5.3) 2 +(9 5.3) 2 +(6 5.3) 2 +(4 5.3) 2 +(3 5.3) 2 +(1 5.3) 2 +(4 5.3) 2 +(5 5.3) 2 +(7 5.3) 2 )= ( ) = ( /10, /10) = (3.58, 7.02) = t 5.789/ (9) = ( ( 1)S 2 /σ 2 χ 2 ( 1) P χ 2 1 α/2( ) 1) < ( 1)S 2 /σ 2 <χ 2 α/2( 1) =1 α α =0.05 =20 ( χ 2 1 α/2( 1) = 8.91 χ 2 α/2( 1) = P 19S 2 /32.85 < ) σ 2 < 19S 2 /8.91 =0.95 σ (19s 2 /32.85, 19s 2 /8.91) = (121.46, ) 2 19S 2 /190 χ 2 (19) 19s 2 /190 = 21 χ (19) = χ (19) = < 21 < x = ( )/9 =5 s 2 = ((11 5) 2 +(3 5) 2 +(4 5) 2 +(1 5) 2 +(5 5) 2 +(4 5) 2 +(3 5) 2 + (8 5) 2 +(6 5) 2 )/8 =9 2 Z = (X μ)/σ N(0, 1) 0.95 ( α =0.05 ) z x =5 =9 s 2 =9 σ 2 =2.25 = x±z α/2 σ/ =5± /3 =5± , T = (X μ)/s 1 t 8(=9 1 ) 0.95 ( α =0.05 ) t α/2 (8) t x =5 =9 s 2 =9=3 2 x ± t α/2 (8)s/ =5± /3 =5± (2.694, 7.306) 4 H 0 X>6+z /9 = =6.823 H 1 P(X >6.823) = P( X 7 > )= 2.25/9 2.25/9 P(Z > 0.36) = 1 P(Z >0.36) = = x = 1 x i =64/16 = 4 s 2 = 1 1 (x i x) 2 = 1 ( 1 x 2 i 23
24 x 2 )= 1 15 ( )=4 2 (X μ)/s t( 1) t 0.025(15) = P( < (X μ)/s < ) = 0.95 P(X S/ <μ<x S/ )=0.95 (x s/, x s/ )=( / 16, / 16) = (2.9343, ) 3 (X 3)/S t( 1) t 0.05(15) = < (x 3)/s =4(4 3)/2 =2 5% 4 ( 1)S 2 /σ 2 χ 2 ( 1) ( 1)S 2 /5 χ 2 ( 1) χ (15) = 7.26 < ( 1)s 2 /5=15 4/5 = / ( 9.1 r = (X t X)(Y t Y ) (X t X) 2 (Y t Y ) 2 = (X t X) (Y t Y ) ˆβ / ) (X t X) 2 (X t X) 2 / (Y t Y ) 2 (X t X) 2 / (Y t Y ) 2 = (2.16) r = s xy/(s xs y) (9.9) ˆβ = s xy/s 2 x r = ˆβ(s x/s y) 9.2 Y )= (Y t Y ) 2 = ((Y t Ŷt)+(Ŷt Y ))2 = e 2 t +2 e (Ŷt Y )2 e t =0 e tx t =0 t(ŷt Y )+ e t(ŷt e t(ˆα + ˆβX t Y )=(ˆα Y ) e t + ˆβ e tx t =0 (Ŷt Y )2 = e 2 t + (Ŷt Y ) ˆα =0.5 ˆβ =0.9 2 s 2 = Se(ˆα) =0.506 Se( ˆβ) = ˆα ˆβ t H 0 : α =0 5% H 0 : β =0 1% (5) R 2 = H 0 : β =1 ( ˆβ 1)/Se( ˆβ) 24
25 2 t ˆβ =0.8 ˆβ t ˆβ/Se( ˆβ) =2.5 Se( ˆβ) =0.32 ( ˆβ 1)/Se( ˆβ) = H 0 : β =1 5% log X t = x t log Y t = y t x =6.08 y =5.46 x ty t = t x 2 t = t x ty t xy t = x 2 t x 2 = t ˆβ =0.3498/ = ˆα = s 2 = Se(ˆα) = Se( ˆβ) = ˆα ˆβ t H 0 : α =0 H 0 : β =0 1% 5 R 2 = β GDP GDP 1% 0.320% 1 GDP 9.6 t Y t X t Xt 2 X ty t Yt 2 Ŷ t e t e 2 t X =1 Y =2 1 ˆβ = 2 R 2 =1 1 2 X ty t XY = Xt 2 X =1 ˆα=Y ˆβX =2 1 1=1 e 2 t Y 2 =1 t Y 2 e 2 t = = 4 4 Se( ˆβ)= = 5 7 = s2 = s 2 X 2 t X 2 = 4/ = 25
26 ( 2 15 =0.365 Se(ˆα)= 1 s2 + X 2 ) ( ) 4 1 = Xt 2 X = = t0.025(5 2) = α (ˆα t0.025( 5 2) Se(ˆα), ˆα + t 0.025( 2) Se(ˆα)) = ( /5, /5) = ( 1.01, 3.01) β ( ˆβ t0.025( 2) Se( ˆβ), ˆβ+t 0.025( 2) Se( ˆβ)) = ( /15, /15) = ( 0.16, 2.16) 6 =5 s 2 =4/3 χ 0.005(5 2) = χ 0.995(5 2) = (( 2)s 2 /χ 0.005( 2), ( 2)s 2 /χ 0.995( 2)) = (4/12.84, 4/0.0717) =(0.312, 55.79) 7 10% H 0 : β =0 H 1 : β \=0 t ˆβ = ˆβ/Se( ˆβ) =1/ 2/15 = >t 0.05(5 2) = % H 0 : β =0 X Y x i = X i X y i = Y i Y ˆβ (9.9) ˆβ = x iy i x 2 i = β + x iu i x 2 i = β + w iu i w i = x i/ x 2 i x i =0 E[ ˆβ] =E[β + w iu i]=β + w ie[u i]=β w i =0 V( ˆβ) =V(β + w iu i)=v( w iu i)= V(w iu i)= w 2 i V(u i)= σ 2 w 2 i = σ 2 / x 2 i w 2 i =1/ x 2 i 2 ( ˆβ β)/se( ˆβ) t( 2) ( ˆβ 1)/Se( ˆβ) t( 2) ( ˆβ 1)/Se( ˆβ) =(4.5 1)/2 =1.75 >t 0.05(18) = % ˆβ1 = (X i X)(Y i Y ) X i Y i XY = (X i X) 2 Xi 2 X2 = =2.0 26
27 ˆβ 0 = Y ˆβ 1X = = s = = 10 = Se( ˆβ 1)= s i (X i X) 2 = s = i Xi 2 X2 t = 2.0 =8.0 >t0.05(8) = t = =4.0 >t0.10(8) = /4 10 = % 3 R 2 = ˆβ 1 2 (X i X) 2 ˆβ 1 2 ( Xi 2 X2 ) 2 = = 2 10 (Y i Y ) 2 =8/ Yi 2 Y R 2 = =0.619 DW = t=2 e 2 t (e t e t 1) 2 e 2 t /( k) ( Y 2 t Y 2 )/( 1) =1 4/(5 2) ( )/(5 1) = = (0 1)2 +(1 ( 1)) 2 +(( 1) ( 1)) 2 +(( 1) 1) 2 4 =9/4= β 1 γ 1 β 1 > 0 γ 1 < 0 2 a Y t 1 e r t 1 b c I t β 1 f % g I t γ 1 d h 10 3 R 2 R
28 DW DW = =28 k = k 1=2 dl = <dl 4 H 0 : β 1 =0 H 1 : β 1 \=0 t t 2.5 % H 0 H 0 : γ 1 =0 H 1 : γ 1 \=0 t t 2.5 % H 0 5 DW s a Y t 1 e r t 1 b % c I t β 2 f % g I t 100 γ 2 d % h % 2 t 0.025(25) = < 7.06 β 2 \=0 β 2 > 0 γ 2 =0 γ 2 3 DW ( ˆβ 0 β 0)/Se( ˆβ 0) t( 2) =7 ˆβ 0/Se( ˆβ 0) t(5) 5/ 4=2.5 <t 0.025(5) = ( ˆβ 1 β 1)/Se( ˆβ 1) t( 2) =7 ( ˆβ 1 1)/Se( ˆβ 1) t(5) (3 1)/ 1.0 =2<t 0.05(5) =
29 (11.3)
30 ( t ) (CHOCO t ) CHOCO t = (345.6) (9.053) t, R 2 = ( ) t R = GDP t = (32.14) (15.04) t ( 7.54) t 2, R 2 = GDP t t ( ) t R 2 ( e t ) e t =GDP t t t
31 11.5 X t =0.7X t 1 + u t =0.7(0.7X t 2 + u t 1)+u t =0.7 2 X t 2 + u t + 0.7u t 1 X t 2, X t 3, X t = u t +0.7u t u t u t 3 + = 0.7 i u t i i=0 E[X t]= i=0 0.7 i E[u t i] =0 φ(0) = E[(X t 0) 2 ]= E[Xt 2 ]=( ) 1=1/( ) φ(s) =Cov(X t,x t s) =E[X tx t s] =(0.7 s +0.7 s s+4 + ) 1= 0.7 s ( ) 1 =0.7 s /( ) ρ(s) =φ(s)/φ(0) = 0.7 s 11.6 E[X t]=2 φ(0) = E[(X t 2) 2 ]=E[(u t + 0.8u t 1 0.3u t 2) 2 ]=( ) 1=1.73 φ(1) = Cov(X t,x t 1) =E[(X t 2)(X t 1 2)] = E[(u t+0.8u t 1 0.3u t 2)(u t u t 2 0.3u t 3)] = = =0.56 φ(2) = Cov(X t,x t 2) = E[(X t 2)(X t 2 2)] = E[(u t +0.8u t 1 0.3u t 2)(u t u t 3 0.3u t 4)] = 0.3 s > = 3 φ(s) =0 ρ(1) = 0.56/1.73 = ρ(2) = 0.3/1.73 = s > = 3 ρ(s) =0 31
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50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
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No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
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1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1.
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1 1 f (t) lim t a f (t) = 0 f (t) t a 1.1 (1) lim(t 1) 2 = 0 t 1 (t 1) 2 t 1 (2) lim(t 1) 3 = 0 t 1 (t 1) 3 t 1 2 f (t), g(t) t a lim t a f (t) g(t) g(t) f (t) = o(g(t)) (t a) = 0 f (t) (t 1) 3 1.2 lim
[1][2] [3] *1 Defnton 1.1. W () = σ 2 dt [2] Defnton 1.2. W (t ) Defnton 1.3. W () = E[W (t)] = Cov[W (t), W (s)] = E[W (t)w (s)] = σ 2 mn{s, t} Propo
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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
/ n (M1) M (M2) n Λ A = {ϕ λ : U λ R n } λ Λ M (atlas) A (a) {U λ } λ Λ M (open covering) U λ M λ Λ U λ = M (b) λ Λ ϕ λ : U λ ϕ λ (U λ ) R n ϕ
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β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E
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