renshumondai-kaito.dvi

3 1 13 14 1.1 1 44.5 39.5 49.5 2 0.10 2 0.10 54.5 49.5 59.5 5 0.25 7 0.35 64.5 59.5 69.5 8 0.40 15 0.75 74.5 69.5 79.5 3 0.15 18 0.90 84.5 79.5 89.5 2 0.10 20 1.00 20 1.00 2 1.2 1 16.5 20.5 12.5 2 0.10 2 0.10 8.5 12.5 4.5 3 0.15 5 0.25 0.5 4.5 3.5 6 0.30 11 0.55 7.5 3.5 11.5 6 0.30 17 0.85 15.5 11.5 19.5 3 0.15 20 1.00 20 1.00 1

2 1.3 1 2 9.05 10.05 8.05 0 0.0000 0 0.0000 7.05 8.05 6.05 0 0.0000 0 0.0000 5.05 6.05 4.05 4 0.1111 4 0.1111 3.05 4.05 2.05 8 0.2222 12 0.3333 1.05 2.05 0.05 8 0.2222 20 0.5556 0.95 0.05 1.95 11 0.3056 31 0.8611 2.95 1.95 3.95 4 0.1111 35 0.9722 4.95 3.95 5.95 1 0.0278 36 1.0000 6.95 5.95 7.95 0 0.0000 36 1.0000 8.95 7.95 9.95 0 0.0000 36 1.0000 36 1.0000 1.4 1 125 2 375 3 625 4 875 5 1125 6 1375 7 0.0331 8 0.2758 9 0.3463 10 0.2133 11 0.0946 12 0.0369 13 137 14 1280 15 2715 16 3599 17 3991 18 4144 19 0.0331 20 0.3089 21 0.6552 22 0.8685 23 0.9631 24 1.0000 25 4144 26 1.0000 2

2 26 27 2.1 1 E 326 F 644 2 E 330 F 470 E 12384 F 19884 E 111.283 F 141.011 3 E 700 3.361 F 1000 2.525 E 700 4 E F 0.341, 0.219 E 2.2 552221.8/542249.0=1.01839 572618.1/ 552221.8=1.03693 574707.3/572618.1=1.00365 (1.01839+1.03693+1.00365)/3=1.01966 3 574707.3/542249.0 =1.01957 2.3 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 =1 1 2.4 2.5 (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) +65 0.95 +33 2.95 +9 4.95 +1 6.95 +1 8.95) = 0.07415 s 2 = (1/) k f i(m i x) 2 =(1/) k f im 2 i x 2 = 1 207 (1 ( 9.05)2 +3 ( 7.05) 2 +12 ( 5.05) 2 +29 ( 3.05) 2 +53 ( 1.05) 2 +65 0.95 2 +33 2.95 2 +9 4.95 2 +1 6.95 2 +1 8.95 2 ) ( 0.07415) 2 =7.53082 s = 7.53082 = 2.74423 3

2.6 1.6 100 225 275 325 375 425 475 525 575 625 675 725 775 850 950 1125 1375 2000 0.0101 0.0220 0.0281 0.0426 0.0543 0.0719 0.0707 0.0679 0.0707 0.0709 0.0604 0.0660 0.0524 0.0850 0.0695 0.0918 0.0358 0.0297 x = k m i(f i/) = 100 0.0101 +225 0.0220 +275 0.0281 +325 0.0426 +375 0.0543 +425 0.0719 +475 0.0707 +525 0.0679 +575 0.0707 +625 0.0709 +675 0.0604 +725 0.0660 +775 0.0524 +850 0.0850 +950 0.0695 +1125 0.0918 +1375 0.0358 +2000 0.0297 = 712 s 2 = k (m i x) 2 (f i/) = k m 2 i (f i/) x 2 = 100 2 0.0101 +225 2 0.0220 +275 2 0.0281 +325 2 0.0426 +375 2 0.0543 +425 2 0.0719 +475 2 0.0707 +525 2 0.0679 +575 2 0.0707 +625 2 0.0709 +675 2 0.0604 +725 2 0.0660 +775 2 0.0524 +850 2 0.0850 +950 2 0.0695 +1125 2 0.0918 +1375 2 0.0358 +2000 2 0.0297 712 2 = 127744 s = 127744 = 357.4 2.7 m i f i m if i m i x (m i x) 2 (m i x) 2 f i 20.5 12.5 16.5 2 33.0 18 324 648 12.5 4.5 8.5 3 25.5 10 100 300 4.5 3.5 0.5 6 3.0 2 4 24 3.5 11.5 7.5 6 45.0 6 36 216 11.5 19.5 15.5 3 46.5 14 196 588 20 30.0 1776 x = 1 m if i = 30 k 20 =1.5 s2 = 1 (m i x) 2 f i = 1776 k 20 =88.8 2.8 1 x = k m i(f i/) = 125 0.0331 +375 0.2758 +625 0.3463 +875 0.2133 +1125 0.0946 +1375 0.0369 = 667.8 2 s 2 = k m 2 i (f i/) x 2 = 125 2 0.0331 +375 2 0.2758 +625 2 0.3463 +875 2 0.2133 +1125 2 0.0946 +1375 2 0.0369 667.8 2 = 81418.16 3 625 4 s = 81418.16 = 285.34 2.9 1 x = (47 + 61 + 77 + 74 + 60 + 43)/6 =60.33 2 s 2 =(47 2 +61 2 + 77 2 +74 2 +60 2 +43 2 )/6 60 2 6 60.33 2 = 157.6 3 (61 + 60)/2 =60.5 4 s = 157.6 =12.6 4

2.10 x i x i x (x i x) 2 x 2 i 28 3 9 784 23 2 4 529 26 1 1 676 27 2 4 729 21 4 16 441 125 0 34 3159 x = 125/5 =25 s 2 = 1 (x i x) 2 = 34 =6.8 s =2.608 26 5 s 2 = 1 x 2 i x 2 = 3159/5 25 2 =6.8 2.11 1 (1+2+3)/3 =2 2 (3+4+2)/3 =3 3 (1 2 +2 2 +3 2 )/3 2 2 = 2/3 4 (3 2 +4 2 +2 2 )/3 3 2 =2/3 5 (1 3+2 4+3 2)/3 2 3 = 1/3 6 1/3 = 0.5 7 2/3 2/3 2.12 x i y i x i x y i y (x i x) 2 (y i y) 2 (x i x)(y i y) 2 4 2 2 4 4 4 1 1 1 1 1 1 1 0 0 0 2 0 4 0 1 1 1 1 1 1 1 2 4 2 2 4 4 4 0 10 0 0 10 14 0 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/ 2 2.8 =0 3 43 44 3.1 A = { } A c = { } P(A) =P(A c )=1/2 P(A) =1/2 5

P(A c 1 A 2)=P(A c 1)P(A 2)=1/4 1 2 1 2 3.2 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)= 0.2 1 P(B 2 A 2)=P(B 2 A 2)P(A 2)=0.35 0.5 =0.175 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)=0.54 0.2 =0.108 3 P((A 1 B 4) (A 2 B 3)) = P(B 4 A 1)+P(B 3 A 2)=0.066+0.115 = 0.181 4 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/(0.123 + 0.165 + 0.054) = 0.360 3.4 1 A B = {1, 2, 3, 4, 8} 2 A B = {2, 3} 3 A B = {1, 4} 4 A A c = φ 3.5 1 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) =1/6 3.6 1 P(E)=300/(300+200)=0.6 2 P(J)=200/(300+200)=0.4 3 P(M E)=30/100 = 0.3 4 P(M J)=20/100 = 0.2 5 P((E J) M)= P(M)=(0.3 300+0.2 200)/500=0.26 6 P((E J) M c )=1 P((E J) M)=0.74 7 P((E J) M)=P(M)=0.26 8 P((E J) M c )=1 P((E J) M)=0.74 9 P(E M)=9/13=0.692 10 P(J M)=4/13=0.308 4 64 70 4.1 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

4.2 4.3 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) = 52.5 4.4 A B C F A 0.05 0.06 0.07 0.12 0.30 B 0.02 0.10 0.14 0.14 0.40 C 0.02 0.01 0.05 0.12 0.20 F 0.01 0.03 0.04 0.02 0.10 0.10 0.20 0.30 0.40 1 4.5 1 f(0) = 1/8 =0.125 2 f(1) = 3/8 =0.375 3 f(2) = 3/8 = 0.375 4 f(3) = 1/8 =0.125 5 f(4) = 0 6 F ( 1) = 0 7 F (1.9) = f(0) + f(1) = 0.5 8 F (2) = f(0) + f(1) + f(2) = 0.875 9 F (5) = f(0) + f(1) + f(2) + f(3) = 1 4.6 1 P(X =2)= 2 2 f(x)dx =0 2 P(2 <X<5) = P(X <5) P(X <2) = F (5) F (2) 4.7 1 E[X] =3 0.2 +5 0.8 =4.6 2 V(X) =(3 2 0.2 +5 2 0.8) 4.6 2 =0.64 3 V(X) = 0.64 = 0.8 4 E[Y ]=0.5E[X]+3= 0.5 4.6+3 = 5.3 5 V(Y )=0.5 2 V[X] =0.16 6 V(Y )= 0.16 = 0.4 4.8 1 1 (0.2+0.1+0.4) = 0.3 2 2 3 4 4 0.2+0.3 =0.5 5 0.1+0.4 =0.5 6 0 7 1 8 0.2+0.1 =0.3 9 0.3+0.4 =0.7 4.9 1 1 (0.2+0.2+0.5) = 0.1 2 E[X] =2 (0.2+0.2)+4 (0.1+0.5) = 3.2 E[Y ]=0 (0.2+0.1) + 1 (0.2+0.5) = 0.7 Cov(X, Y )=E[XY ] E[X]E[Y ]=2 0 0.2+2 1 0.2+4 0 0.1+4 1 0.5 3.2 0.7 = 2.4 2.24 = 0.16 3 V(X) =2 2 (0.2+0.2)+4 2 (0.1+0.5) 3.2 2 =0.96 V(Y ) = 0 2 (0.2 +0.1) + 1 2 (0.2 +0.5) 0.7 2 = 0.21 ρ(x, Y ) = Cov(X, Y ) 0.16 = =0.356 4 P(Y =0 X =4)=0.1/(0.1+ V(X) V(Y ) 0.96 0.21 0.5) = 1/6 5 P(Y =1 X =4)=0.5/(0.1+0.5) = 5/6 6 E[X + Y ]= 7

E[X]+E[Y ]=3.2+0.7 =3.9 7 V(X+Y )=V(X)+2Cov(X, Y )+X(Y )= 0.96 + 2 0.16 + 0.21 = 1.49 4.10 1 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 = 11 3 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 ]=20 7 8 = 36 7 V(X +Y )= V(X)+V(Y )=8 4.11 1 E[X] =E[X 1 + +X 5]/5 =(E[X 1]+ +E[X 5])/5 =(3+ + 3)/5 =3 2 V(X) =V(X 1 + + X 5)/5 2 =(V(X 1)+ +V(X 5))/5 2 = (5 + +5)/5 2 =1 4.12 E[X] = 0 0.4 +2 0.1 +5 0.2 +9 0.3 = 3.9 V(X) = E[X 2 ] (E[X]) 2 =0 2 0.4+2 2 0.1+5 2 0.2+9 2 0.3 3.9 2 =14.49 4.13 E[X] =0 p +1 2p 2 +2 3p +3 3p 2 =11p 2 +6p V(X) = E[X 2 ] (E[X]) 2 =0 2 p +1 2 2p 2 +2 2 3p +3 2 3p 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] =11 0.2 2 +6 0.2 =1.64 V(X) =(29 0.2 2 +12 0.2) 1.64 2 = 0.8704 1 1 1 4.14 1 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] = 2 1 0 1 0 0 0 1 0 0 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 2/2 0.293 8

4.15 1 P(X =3)= 4C 30.7 3 0.3 1 =0.4116 2 P(2 < = X < = 3) = P(X = 2) + P(X =3)= 4C 20.7 2 0.3 2 + 4C 30.7 3 0.3 1 =0.2646 + 0.4116 = 0.6762 3 P(X < = 3) = 1 P(X =4)=1 4C 40.7 4 0.3 0 =0.7599 4.16 1 0.1+0.2+a +0.3+0.1+0.2 =a +0.9 =1 a =0.1 2 X Y 1 2 3 P(X = x) 1 0.1 0.2 0.1 0.4 2 0.3 0.1 0.2 0.6 P(Y = y) 0.4 0.3 0.3 1.0 P(X =1,Y =1)=0.1 P(X =1)P(Y =1)=0.16 3 E[X] = 2 x=1 xp(x = x) =1 0.4+2 0.6 =1.6 E[X 2 ]= 2 x=1 x 2 P(X = x) =1 2 0.4+2 2 0.6 =2.8 V(X) =E[X 2 ] (E[X]) 2 =2.8 1.6 2 =0.24 4 Y 1 2 3 P(Y X =1) 0.25 0.5 0.25 E[Y X =1]=1 0.25 + 2 0.5 +3 0.25 = 2 E[Y 2 X =1]=1 2 0.25 + 2 2 0.5+3 2 0.25 = 4.5 V(Y X =1)=E[Y 2 X =1] (E[Y X = 1]) 2 =4.5 2 2 =0.5 5 X Y P(X = x, Y = y) Z =2X + Y 1 1 0.1 3 1 2 0.2 4 1 3 0.1 5 2 1 0.3 5 2 2 0.1 6 2 3 0.2 7 Z Z 3 4 5 6 7 P(Z = z) 0.1 0.2 0.4 0.1 0.2 9

E[Z] =3 0.1 +4 0.2 +5 0.4 +6 0.1 +7 0.2 =5.1 E[Z 2 ]= 3 2 0.1 +4 2 0.2 +5 2 0.4 +6 2 0.1 +7 2 0.2 = 27.5 V(Z) = E[Z 2 ] (E[Z]) 2 =27.5 5.1 2 =1.49 4.17 1 c =1 0.3 0.4 =0.3 2 μ x =E[X] = 1 0.3+1 0.4+ 4 0.3 =1.3 3 V(X) =( 1) 2 0.3+1 2 0.4+4 2 0.3 1.3 2 =3.81 4 μ y =E[Y ]=2 1.3 +5 = 7.6 5 V[Y ]=2 2 3.81 = 15.24 6 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) 2 0.3+1 2 0.4+4 2 0.3) + 5 1.3 1.3 7.6 =7.62 7 Cov(X, Y ) 7.62 = =1 V(X)V(Y ) 3.81 15.24 4.18 1 a = 1 (0.1 + 0.0 + 0.1 + 0.2 + 0.1 + 0.1 + 0.0 + 0.3) = 0.1 2 X 1 0 1 Y 1 0 1 0.3 0.2 0.5 0.2 0.4 0.4 3 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) = 0.3 0.2 =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 2 0 1 0 0.1 0.1 1 0.3 0.5 5 E[X 2 Y 2 ]=0 0 0.1+0 1 0.1+1 0 0.3+1 1 0.5 =0.5 E[X 2 ]= 0 (0.1+0.1)+1 (0.3+0.5) = 0.8 E[Y 2 ]=0 (0.1+0.3)+1 (0.1+0.5) = 0.6 Cov(X 2,Y 2 )=E[X 2 Y 2 ] E[X 2 ]E[Y 2 ]=0.5 0.8 0.6 =0.02 4.19 1 E[X] =1 1 6 +2 1 6 +3 1 6 +4 1 6 +5 1 6 +6 1 6 =7/2 E[X 2 ]=1 2 1 6 +22 1 6 +32 1 6 +42 1 6 +52 1 6 +62 1 6 =91/6 V(X) =E[X 2 ] (E[X]) 2 =91/6 49/4 =35/12 10

2 Y X 1 2 3 4 5 6 1 0 0 1/6 0 0 1/6 2 1/6 1/6 0 1/6 1/6 0 3 Y 2 1 4 1/3 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+1 6 1/6+2 1 1/6+2 2 1/6+2 4 1/6+2 5 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) 4.20 1 μ =E[X] = x xp (X = x) =1 1 6 +0 5 6 = 1 6 V(X) = x (x μ) 2 P (X = x) = x x 2 P (X = x) μ 2 =1 2 1 6 +02 5 6 ( 1 6 )2 = 5 36 2 x 1 x 2 x 3 f(x 1,x 2,x 3) y 0 0 0 ( 1 6 )0 ( 5 6 )3 = 125 216 0 0 1 ( 1 6 )1 ( 5 6 )2 = 25 216 0 1 0 ( 1 6 )1 ( 5 6 )2 = 25 216 0 1 1 ( 1 6 )2 ( 5 6 )1 = 5 216 1 0 0 ( 1 6 )1 ( 5 6 )2 = 25 216 1 0 1 ( 1 6 )2 ( 5 6 )1 = 5 216 1 1 0 ( 1 6 )2 ( 5 6 )1 = 5 216 1 1 1 ( 1 6 )3 ( 5 6 )0 = 1 216 0 1 1 2 1 2 2 3 X 1 Y f(x 1,y) 11

X 1 Y 0 1 2 3 0 125 216 1 0 50 216 25 216 5 216 10 216 0 1 216 3 E[X 1]= 1, V(X1) = 5 E[Y ]= 3 15, V(Y )= Cov(X1,Y)= 6 36 6 36 E[X 1Y ] E[X 1]E[Y ] E[X 1Y ]= x1 x 1yf(x 1,y)=0 0 125 y 216 +0 1 50 216 +0 2 5 25 +0 3 0+1 0 0+1 1 216 216 +1 2 10 216 +1 3 1 216 = 48 48 Cov(X1,Y) = 216 216 1 3 = 30 ρ = 6 6 216 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 0 1 2 3 f(y X 1 =1) 0 25 36 E[Y X 1 =1] E[Y X 1 =1]= y yf(y x 1 =1)=0 0+1 25 10 +2 +3 1 = 4 36 36 36 3 4.21 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)=1+2 2+5=10 4.22 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 σ2 4.23 μ =E[X] =1 0.2+0 0.8 =0.2 σ 2 =V(X) =(1 0.2) 2 0.2+(0 0.2) 2 0.8 =0.16 E[(X μ) 3 ]=(1 0.2) 3 0.2+(0 0.2) 3 0.8 =0.096 E[(X μ) 3 ]/σ 3 =0.096/0.16 3/2 =1.5 4.24 1 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 1/8+1 2 3/8+2 2 3/8+3 2 1/8 1.5 2 =0.75 10 36 1 36 12

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 0 1 2 3 1 0 3/8 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+1 0 1/8+1 2 3/8 =0 E[X] =3/2 V(X) =3/4 E[Y ]=0 V(Y )=( 1) 2 1/2+1 2 1/2 0 2 =1 Cov(X, Y )=E[XY ] E[X]E[Y ]= 0 3/2 0=0 ρ(x, Y )=0 4.25 c =1 0.1 0.2 0.1 =0.6 E[X] =1 0.1+2 0.6+5 0.2+9 0.1 = 3.2 V(X) =1 2 0.1+2 2 0.6+5 2 0.2+9 2 0.1 3.2 2 =5.36 X 2 X 2 Prob(X < = 2.1) = 0.1+0.6 =0.7 5 ( 79 81 ) 5.1 1 0.0582 2 0.9099 3 0.5895 4 0.2437 5 0.3980 5.2 X N(2, 9) Z =(X 2)/3 N(0, 1) 1 0.1151 2 0.9962 3 0.4452 4 0.3159 5 0.2579 5.3 X N(10, 4 2 ) Z =(X 10)/4 N(0, 1) P(X >12) = P(Z >0.5) = 0.3085 0.3085 1200 = 370.2 370 5.4 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 =64.72 65 5.5 X X N(4400, 469 2 ) 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 1 4400)/469) = P(Z < = 1.28) = 0.1 P(Z < = (x 2 4400)/469) = 13

P(Z < = 1.28) = 0.9 x 1 = 3800 g x 2 = 5000 g 10 52 cm 90 57 cm 5.6 1 P(Z > 0) = 0.5 2 P(Z < 2.22) = P(Z > 2.22) = 0.0132 3 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) = 1 0.3085 0.3821 = 0.3094 5.7 1 P(X > 2) = P((X +2)/4 > 1) = P(Z > 1) = 0.1587 2 P(X < 2) = P((X +2)/4 > 0) = P(Z >0) = 0.5 3 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) = 1 0.2266 0.4013 = 0.3721 4 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)=0.3446 0.2743=0.0703 5.8 1 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) 9.4 1.28 0.1003 2.48 0.0066 0.8931 9.5 1.30 0.0968 2.50 0.0062 0.8970 9.6 1.32 0.0934 2.52 0.0059 0.9007 9.7 1.34 0.0901 2.54 0.0055 0.9044 x =9.6 0.9 1 P(X <x)=p((x 3)/5 < (x 3)/5) = P(Z <(x 3)/5) = P(Z < 1.96) = 0.025 (x 3)/5 = 1.96 x = 6.8 5.9 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) = 0.6915 2 P(X < = 5) = P((X 5)/2 < = (5 5)/2) = P(Z < = 0) = 0.5 3 P(X < = 3) = P((X 5)/2 < = (3 5)/2) = P(Z < = 1) = 0.1587 4 P(3.5 < = X < = 4.5) = P((3.5 5)/2 < = (X 5)/2 < = (4.5 5)/2) = P( 0.75 < = Z < = 0.25) = 0.4013 0.2266 = 0.1747 5 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) = 1 0.1747 = 0.8253 5.10 1 X N(68, 8 2 ) P(X < 60) = P((X 68)/8 < (60 68)/8) = P(Z < 1) = 0.1587 15.9 % 2 14

P(X > = 78) = P((X 68)/8 > = (78 68)/8) = P(Z > = 1.25) = 0.1057 10.6% 3 P(X > = x 0)=0.05 P((X 68)/8) > = (x 0 68)/8) = 0.05 0.05 z 0.05 =1.6449 (x 0 68)/8 =1.6449 x 0 =81.1592 5% 82 5.11 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 < = 1.2816) = 0.1 1.2816 = x 3 x = 3.408 5 5.12 1 P(0.0 <Z<2.0) = P(Z >0.0) P(Z >2.0) = 0.5000 0.0228 =0.4772 2 E[W ]=E[X]+ 1 E[Y ]=0.0+1.0 =1.0 V(W )=V(X)+ 3 1 W 1.0 V(Y )=1.0+3.0 =4.0 W N(1.0, 4.0) 3 Z = 9 2.0 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 < 1.0 1.0 )+P( 3.0 1.0 < 2.0 2.0 2.0 W 1.0 ) = P(Z< 1.0)+P(Z >1.0) = 2P(Z >1.0) = 2 0.1587 = 0.3174 2.0 5.13 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 1 36/25 )=P(Z> 5/6) = 1 0.2033 = 0.7967 6 100 103 6.1 6.2 7 26 (6.4) 7 (6.7) 26/9 ((4 3)/(4 1))(26/3)=26/9 (6.7) 6.3 1 X X N(480, 320 2 /64) Z = (X 480)/(320/ 64) N(0, 1) P(450 < = X < = 500) = P( 0.75Z < = 0.5)=0.4649 2 P(X > = 520)=P(Z > = (520 480)/(320/ )) < = 0.05 P(Z > = 1.6449) = 15

0.05 (520 480)/(320/ ) > = 1.6449 > = 173.1645 = 174 6.4 S P(S > = 400) < = 0.05 ( 1)S 2 /320 2 1 2 P(( 1)S 2 /320 2 > = ( 1)400 2 /320 2 ) < = 0.05 =21 20 400 2 /320 2 =31.25 2 P(20S 2 /320 2 > = χ 2 0.05(20)) = 0.05 χ 2 0.05(20) 31.41 P(20S 2 /320 > = 31.25) > P(20S 2 /320 > = 31.41) = 0.05 =22 P(21S 2 /320 > = 32.81) < 0.05 =22 6.5 P(S1/S 2 2 2 > = 2.5) < = 0.05 2 () S 1/S 2 2 2 (8, 1) F 6.4 1 =18 P(S1/S 2 2 2 > = 2.5) > 0.05 1 =19 P(S1/S 2 2 2 > = 2.5) < 0.05 =20 6.6 1 x =15.09 2 x =2.70 3 P(x <X<28.85) = P(X >x) P(X >28.85) = P(X >x) 0.025 = 0.925 x =7.96 6.7 1 (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) = 0.025 (x 1)/ (2/2500) = 1.96 x=0.94456 3 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) = 1 0.2389 0.0384 = 0.7227 [ ] 1 6.8 1 E[X] =E X i = 1 i]= E[X 1 9 3=3 2 V(X) = 9 ( ) ( ) 1 2 ( ) 2 1 1 V X i = V(X i)= 25 = 9 9 25 3 P(X <0) = 9 P((X 3)/ 25/9 < 3/ 25/9) = P(Z < 1.8) = P(Z >1.8) = 0.0359 4 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 =5.137 6.9 1 x =3.365 2 x = 2.262 3 x = 2.120 4 x = 2.015 5 P(0.0 <X)=0.5 6 P( 1.35 <X)=0.9 6.10 1 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

> = 34.6) = P((X 35)/2 > = (34.6 35)/2) = P(Z > = 0.2) = 1 0.4207 = 0.5793 2 P(X < = 34.2) = P((X 35)/2 < = (34.2 35)/2) = P(Z < = 0.4) = 0.3446 3 P(34 < = X < = 35.5) = P((34 35)/2 < = (X 35)/2 < = (35.5 35)/2) = P( 0.5 < = Z < = 0.25) = 1 (0.3085 + 0.4013) = 0.2902 6.12 X N(μ, 7.2 2 ) =81 X N(μ, (7.2/ 81) 2 )=N(μ, 0.8 2 ) 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) = 2 0.1057 = 0.2114 6.13 1 X N(μ, 4.2 2 ) =36 X N(μ, (4.2/ 36) 2 )=N(μ, 0.7 2 ) P( (X μ)/0.7 < 1.96) = 0.95 P( X μ < 1.96 0.7) = P( X μ < 1.372) = 0.95 0.95 1.372 1.372 0.05 2 P( X μ < 1) = 0.95 P( X μ /(σ/ ) < 1.96) = 0.95 P( X μ < 1.96σ/ )=0.95 1.96(σ/ )=1 =1.96σ σ =4.2 =(1.96 4.2) 2 =67.77 68 6.14 P(118 < = X < = 125) = P(118 720 < 6 = X 720 < 720 6 = 125 )= 6 118 720 X 6 P( < 720 125 720 1 5 = 6 < 720 X 6 6 720 1 5 = 6 )=P( 0.2 < 6 6 720 1 5 = 720 6 < 6 6 720 1 5 = 0.5) = 1 6 6 0.4207 0.3085 = 0.2708 6.15 X t(8) P(X < = x)=0.05 x 1.8595 P(3X +2< = 1.8595 3+2)=0.05 y = 1.8595 3+2= 3.5785 7 122 126 7.1 0.9 0.95 (7.213, 9.187) (7.024, 9.376) 7.2 0.9 0.95 (8.634, 12.366) (8.213, 12.787) 7.3 0.9 0.95 (712 1.6449 357.4/ 4271, 712+1.6449 357.4/ 4271) = (703.0, 721.0) (712 1.9600 357.4/ 4271, 712 + 1.9600 357.4/ 4271) = (701.3, 722.7) 7.4 x = 0.125 s 2 =2.3913 2 0.9 0.95 ( 0.125 1.7959 2.3913/ 12, 17

0.125 + 1.7959 2.3913/ 12) = ( 1.365, 1.115) ( 0.125 2.2010 2.3913/ 12, 0.125 + 2.2010 2.3913/ 12) = ( 1.644, 1.394) 7.5 0.9 0.95 (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 = 2.3913 0.9 0.95 (11 2.3913 2 /19.68, 11 2.3913 2 /4.57) = (3.196, 13.764) (11 2.3913 2 /21.92, 11 2.3913 2 /3.82) = (2.870, 16.466) 7.7 0.9 0.95 (0.593, 0.747) (0.578, 0.762) 7.8 1 L = λ exp( x i ) 2 λ L λ λ + x i λ =0 λ ˆλ = 1 x 2 i = x 7.9 1 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 2 7.10 1 E[X] =μ 2 E[ X] =μ 3 E[ ˆX] \= μ 4 V(X) < V( X) 5 3 3 ω i =1 ω ix i μ 6 7.11 1 E[S 2 ]=σ 2 E[S 2 ]=(( 1)/)σ 2 \= σ 2 2 E[S 2 ]=σ 2 7.12 1 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 0.005 32 /25, x+z 0.005 32 /25) = (8.05 2.5758 3 2 /25, 8.05 + 2.5758 3 2 /25) = (6.50452, 9.59548) 7.13 1 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) = (12.6 1.7959 9/12, 12.6+1.7959 9/12) = (11.045, 14.155) 18

7.14 1 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 0.025 s2 /400, x + z 0.025 s2 /400) = (2.56 1.960 4 2 /400, 2.56 + 1.960 4 2 /400) = (2.168, 2.952) 7.15 1 ( 1)S 2 /σ 2 χ 2 (15) 2 P(χ 2 0.95(15)<( 1)S 2 /σ 2 <χ 2 0.05(15)) =P(7.26 < 15 S 2 /σ 2 < 25.00) = 0.90 (15 s 2 /25.00, 15 s 2 / 7.26) = (15 1.44/25.00, 15 1.44/7.26) = (0.864, 2.975) 7.16 1 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 =(3.9 +1.1 +0.1 +0.5)/4 = 1.4 s 2 = (1/3)(3.9 2 +1.1 2 +0.1 2 +0.5 2 4 1.4 2 ) = 2.947 (1.4 2.3534 2.947/4, 1.4+2.3534 2.947/4) = ( 0.62, 3.42) 7.17 1 (ˆp p)/ ˆp(1 ˆp)/ N(0, 1) 2 P( ˆp p / ˆp(1 ˆp)/ < z 0.025) =0.95 (ˆp z 0.025 ˆp(1 ˆp)/, ˆp + z0.025 ˆp(1 ˆp)/) = (0.345 1.960 0.345(1 0.345)/536, 0.345+1.960 0.345(1 0.345)/536) 0.385 3 0.305 7.18 1 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 =1/2 7.19 x ± z α/2 σ/ x =61.2 σ =5, =49 90% 61.2 ± 1.6449 5/ 49 (60.025, 62.375) 95% 61.2 ± 1.9600 5/ 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 = 3.6 90% (14 3.6/23.68, 14 3.6/6.57) = (2.128, 7.671) 95% (14 3.6/26.12, 14 3.6/5.63) = (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 ± 1.7531 2.8982/ 16 (8.930, 11.470) 95% 10.2 ± 2.1314 2.8982/ 16 (8.656, 11.744) 7.22 ˆp ± z α/2 ˆpˆq/ ˆp =45/300 = 0.15 = 300 19

90% 0.15 ± 1.6449 0.15 0.85/300 (0.116, 0.184) 95% 0.15 ± 1.9600 0.15 0.85/300 (0.110, 0.190) 7.23 1 5 4 μ 2 5 9 σ2 3 X X 4 X 5 1 2 σ2 7.24 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 )2 + 1 2 σ2 a = 1 V(Z) μ 2 Z = 1 X + 1 Y 2 2 7.25 (45 2.5758 2 2 /81, 45 + 2.5758 2 2 /81) = (44.4276, 45.5724) 8 155 162 8.1 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 0 8.2 β =P(X<136 H 1)=P((X μ 1)/(σ/ ) < (136 140)/(20/2)) = P(Z < 0.4) 0.34 α =0.01 β =P(Z < (143 140)/(20/2)) = P(Z < 0.3) 0.62 β 8.3 1 t =5.67 μ = 740 t =1.67 μ = 630 t =2.33 μ = 690 2 μ 95% (623, 687) H 0 : μ = 630 8.4 μ H 0 : μ > = 80.9 H 1 : μ<80.9 H 0 5% x =62.5 < 80.9 1.6449(18/10) = 77.9 H 0 1% H 0 8.5 1 μ J μ U H 0 : μ J = μ U, H 1 : μ J \= μ U 1 = 2 =11, x J = 20

1.218, x U =3.055, s 2 J =(38.14 11 1.218 2 )/10 = 2.182, s 2 U = (117.10 11 3.055 2 )/10 = 1.444 1% x J x U =1.837 > z 0.005 2.182/11 + 1.444/11 = 2.5758 0.574 = 1.48 H0 2 H 0 : μ J > = μ U, H 1 : μ J <μ U 5% x J x U = 1.837 < 1.6449 0.574 = 0.944 H 0 8.6 p H 0 : p =0.727 ( p < = 0.727) H 1 : p>0.727 ˆp = 7500/10000 = 0.75 z z =(0.75 0.727)/ 0.727 0.273/10000 = 5.16 z 0.01 =2.3263 z =5.16 >z 0.01 1% 8.7 1 2 3 4 8.8 1 H 0 : μ =90 2 (x μ 0)/ σ 2 / = (101 90)/ 10 2 /4=2.2 3 3 4 2.2 >z 0.025 =1.960 5 2.2 <z 0.005 =2.576 8.9 1 H 0 : p =0.5 H 1 : p<0.5 2 (0.41 0.5)/ 0.5(1 0.5)/100 = 1.8 3 N(0, 1) 4 1.8 < z 0.05 = 1.645 8.10 1 H 0 : μ =3.5 H 1 : μ<3.5 2 (1.218 3.5)/ 2.812/11 = 5.12 3 t(10) 4 5.12 < t 0.01(10) = 2.7638 8.11 1 H 0 : μ =3.5 H 1 : μ \=3.5 2 (2.9 3.5)/ 2 2 /25 = 1.5 3 N(0, 1) 4 1.5 > z 0.025 = 1.96 8.12 1 642 637 / 361/289 + 961/225 = 2.13 2 N(0, 1) 3 2.13 < z 0.005 =2.576 8.13 1 z =(x μ 0)/(σ/ ) x =20 =36 z =(20 18)/(7/ 36) = 1.714 > 1.645 = z 0.05 H 0 2 z = 1.714 < 2.326 (= 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 =2.326 0.05 0.01 0.05 X>18 + 1.645 7/ 36 = 19.92 0.01 X>18 + 2.326 7/ 36 = 20.71 21

X 0.05 P(X >19.92) = P((X 22)/(7/ 36) > (19.92 22)/(7/ 36)) = P(Z > 1.78) = 0.9625 0.01 P(X > 20.71) = P((X 22)/(7/ 36) > (20.71 22)/(7/ 36)) = P(Z > 1.11) = 0.8665 8.14 1 x =87 s =7 =12 x±t α/2 s/ =87±2.201 7/ 12 (82.552, 91.448) 2 H 0 : μ =83 H 1 : μ>83 t =(x 83)/(7/ 12) = (87 83)/(7/ 12) = 1.979 > 1.796 = t 0.05(11) H 0 8.15 H 0 : μ 1 =μ 2 H 1 : μ 1 \=μ 2 Z =(X 1 X 2)/ S1/ 2 1 + 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 =(33 33.6)/ 10/160 + 12/180 = 0.6/0.359 = 1.671 > 1.96 = z 0.025 H 0 8.16 H 0 : p =1/6 (=0.167) H 1 : p>1/6 ˆp =54/240 = 0.225 α =0.01 z =(ˆp 0.167)/ 0.167 0.833/240 = (0.225 0.167)/ 0.0241 = 2.407 > 2.326 = z 0.01 H 0 6 8.17 H 0 : p =0.7 H 1 : p<0.7 ˆp = 2070/3000 = 0.69 α = 0.05 z =(ˆp 0.7)/ 0.7 0.3/3000 = (0.69 0.7)/0.00837 = 1.195 > 1.645 = z 0.05 H 0 70% 8.18 H 0 : μ =60 H 1 : μ \=60 x = (59+56+62+61+57)/5 = 295/5 =59 s 2 = ((59 59) 2 +(56 59) 2 +(62 59) 2 +(61 59) 2 + (57 59) 2 )/4=(0+9+9+4+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 = 0.877 > 2.776 = t 0.025(4) H 0 8.19 1 2.5 2 =1 2 t 4/16 0.025(15) = 2.1314 3 8.20 1 2.5 2 =2.5 2 t 4/100 0.025(90) = 1.9867 t 0.025(100) = 1.9840 t 0.025(99) 1.9840 3 8.21 1 4.5 13 15 = 1.7778 2 t0.05(15) = 1.7531 3 2 /16 0.51 0.5 8.22 1 =2 2 z 0.01 =2.3263 3 0.5(1 0.5)/10000 22

8.23 1 1 1 (5 + 6) = 5.5 2 (8+6+9+6+4+3+1+4+5+ 2 10 7) = 5.3 3 1 ((8 10 1 5.3)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 )= 1 10 1 (82 +6 2 +9 2 +6 2 +4 2 +3 2 +1 2 +4 2 +5 2 +7 2 10 5.3 2 ) = 5.789 4 (5.3 2.2622 5.789/10, 5.3+2.2622 5.789/10) = (3.58, 7.02) 5 5.3 7 = 2.234 6 t 5.789/10 0.05(9) = 1.8331 7 ( 8.24 1 6.3 ( 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) = 32.85 P 19S 2 /32.85 < ) σ 2 < 19S 2 /8.91 =0.95 σ 2 0.95 (19s 2 /32.85, 19s 2 /8.91) = (121.46, 447.81) 2 19S 2 /190 χ 2 (19) 19s 2 /190 = 21 χ 2 0.025(19) = 32.85 χ 2 0.975(19) = 8.91 8.91 < 21 < 32.85 8.25 1 x = (11+3+4+1+5+4+3+8+6)/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 0.025 1.96 x =5 =9 s 2 =9 σ 2 =2.25 = 1.5 2 x±z α/2 σ/ =5±1.96 1.5/3 =5±0.98 4.02 5.98 0.95 4.02, 5.98 3 T = (X μ)/s 1 t 8(=9 1 ) 0.95 ( α =0.05 ) t α/2 (8) t 2.306 x =5 =9 s 2 =9=3 2 x ± t α/2 (8)s/ =5± 2.306 3/3 =5± 2.306 2.694 7.306 0.95 (2.694, 7.306) 4 H 0 X>6+z 0.05 2.25/9 =6+1.645 0.5 =6.823 H 1 P(X >6.823) = P( X 7 > 6.823 7 )= 2.25/9 2.25/9 P(Z > 0.36) = 1 P(Z >0.36) = 1 0.3594 = 0.64 8.26 1 x = 1 x i =64/16 = 4 s 2 = 1 1 (x i x) 2 = 1 ( 1 x 2 i 23

x 2 )= 1 15 (316 16 42 )=4 2 (X μ)/s t( 1) t 0.025(15) = 2.1314 P( 2.1314 < (X μ)/s < 2.1314) = 0.95 P(X 2.1314 S/ <μ<x+2.1314 S/ )=0.95 (x 2.1314 s/, x+2.1314 s/ )=(4 2.1314 2/ 16, 4+2.1314 2/ 16) = (2.9343, 5.0657) 3 (X 3)/S t( 1) t 0.05(15) = 1.7531 < (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 2 0.95(15) = 7.26 < ( 1)s 2 /5=15 4/5 =12 9 187 191 / ( 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 )2 9.3 1 ˆα =0.5 ˆβ =0.9 2 s 2 =0.486 3 Se(ˆα) =0.506 Se( ˆβ) = 0.090 4 ˆα ˆβ t 0.988 10.0 H 0 : α =0 5% H 0 : β =0 1% (5) R 2 =0.935 9.4 H 0 : β =1 ( ˆβ 1)/Se( ˆβ) 24

2 t ˆβ =0.8 ˆβ t ˆβ/Se( ˆβ) =2.5 Se( ˆβ) =0.32 ( ˆβ 1)/Se( ˆβ) = 0.625 H 0 : β =1 5% 9.5 1 log X t = x t log Y t = y t x =6.08 y =5.46 x ty t = t 930.08 x 2 t = t 1036.64 x ty t xy t =0.3498 x 2 t x 2 = t 1.0941 ˆβ =0.3498/1.0941 = 0.320 ˆα =3.516 2 s 2 =0.007922 3 Se(ˆα) =0.5177 Se( ˆβ) =0.0851 4 ˆα ˆβ t 6.79 3.76 H 0 : α =0 H 0 : β =0 1% 5 R 2 =0.3519 β 2 0.320 GDP 0.320 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 1 4 3 9 12 16 4 0 0 2 1 1 1 1 1 0 1 1 3 0 0 0 0 0 1 1 1 4 1 1 1 1 1 2 1 1 5 4 2 4 8 16 3 1 1 10 5 15 20 34 0 4 X =1 Y =2 1 ˆβ = 2 R 2 =1 1 2 X ty t XY = 20 5 1 2 Xt 2 X 2 15 5 1 1 =1 ˆα=Y ˆβX =2 1 1=1 e 2 t Y 2 =1 t Y 2 e 2 t = 4 5 2 = 4 4 Se( ˆβ)= 3 4 34 5 2 2 = 5 7 =0.714 3 s2 = s 2 X 2 t X 2 = 4/3 15 5 1 2 = 25

( 2 15 =0.365 Se(ˆα)= 1 s2 + X 2 ) ( ) 4 1 = Xt 2 X 2 3 5 + 1 2 15 5 1 2 2 = =0.632 5 t0.025(5 2) = 3.182 α (ˆα t0.025( 5 2) Se(ˆα), ˆα + t 0.025( 2) Se(ˆα)) = (1 3.182 2/5, 1+3.182 2/5) = ( 1.01, 3.01) β ( ˆβ t0.025( 2) Se( ˆβ), ˆβ+t 0.025( 2) Se( ˆβ)) = (1 3.182 2/15, 1+3.182 2/15) = ( 0.16, 2.16) 6 =5 s 2 =4/3 χ 0.005(5 2) = 12.84 χ 0.995(5 2) = 0.0717 (( 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 = 2.739 >t 0.05(5 2) = 2.353 10% H 0 : β =0 X Y 9.7 1 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 2 4.3 3 4.8 4 4.3 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) = 1.7341 5% 9.8 1 ˆβ1 = (X i X)(Y i Y ) X i Y i XY = (X i X) 2 Xi 2 X2 = 30 10 2.0 0.5 50 10 4 =2.0 26

ˆβ 0 = Y ˆβ 1X =0.5 2.0 2.0 = 3.5 5 2 s = = 10 = 10 8 16 4 Se( ˆβ 1)= s i (X i X) 2 = s = i Xi 2 X2 t = 2.0 =8.0 >t0.05(8) = 1.8595 0.25 t = 2.0 1.0 =4.0 >t0.10(8) = 1.3968 0.25 10/4 10 =0.25 10% 3 R 2 = ˆβ 1 2 (X i X) 2 ˆβ 1 2 ( Xi 2 X2 ) 2 = = 2 10 (Y i Y ) 2 =8/9 47.5 10 0.5 0.5 Yi 2 Y 2 10 217 219 10.1 R 2 =1 13 21 =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) (34 5 2 2 )/(5 1) = = (0 1)2 +(1 ( 1)) 2 +(( 1) ( 1)) 2 +(( 1) 1) 2 4 =9/4=2.25 10.2 1 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 2 0 1 1 0 0.8539 0.8422 27

DW 0 4 0 4 2 DW =0.3082 =28 k = k 1=2 dl =1.255 0.3082 <dl 4 H 0 : β 1 =0 H 1 : β 1 \=0 t 5.978 25 t 2.5 % 2.0595 H 0 H 0 : γ 1 =0 H 1 : γ 1 \=0 t 0.378 25 t 2.5 % 2.0595 H 0 5 DW 0.3082 s 10.3 1 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) = 2.0595 < 7.06 β 2 \=0 β 2 > 0 γ 2 =0 γ 2 3 DW 0.2778 10.4 1 ( 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) = 2.5706 2 ( 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) = 2.0150 28

11 247 249 11.1 1 2 3 11.2 2 12 (11.3) 6 6 2007 1 2008 1 366.0 2009 1 375.4 2 2 366.6 2 376.0 3 3 368.6 3 375.7 4 4 370.7 4 375.3 5 5 373.5 5 375.3 6 6 375.2 6 375.5 7 357.7 7 375.8 7 8 363.3 8 373.8 8 9 367.5 9 373.3 9 10 365.8 10 375.1 10 11 364.6 11 376.2 11 12 365.8 12 376.0 12 29

11.3 2007 7 1 2009 6 24 ( t ) (CHOCO t ) CHOCO t = 362.87 (345.6) + 0.6653 (9.053) t, R 2 =0.7787 ( ) t R 2 1 0.665 12 = 7.98 11.4 2 GDP t = 257646 (32.14) + 18524 (15.04) t 300.4 ( 7.54) t 2, R 2 =0.9747 GDP t t 1980 1 2008 29 ( ) t R 2 ( e t ) e t =GDP t 257646 18524 t + 300.4 t 2 7 7 10 1980 8505.4 1990 22308.3 2000 11053.8 1981 2760.5 1991 25565.6 2001 15732.9 1982 4258.2 1992 18337.5 2002 19417.0 1983 11305.7 1993 8721.5 2003 16678.6 1984 13036.7 1994 2940.5 2004 6418.3 1985 7374.0 1995 2588.8 2005 562.6 1986 12067.0 1996 6629.5 2006 8906.9 1987 11276.6 1997 6318.0 2007 19846.4 1988 2130.3 1998 11336.9 2008 11892.8 1989 10910.5 1999 18836.0 30

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 1 +0.7 2 u t 2 +0.7 3 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+0.7 2 +0.7 4 +0.7 6 + ) 1=1/(1 0.7 2 ) φ(s) =Cov(X t,x t s) =E[X tx t s] =(0.7 s +0.7 s+2 +0.7 s+4 + ) 1= 0.7 s (1+0.7 2 +0.7 4 ) 1 =0.7 s /(1 0.7 2 ) 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+0.8 2 +0.3 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 1+ 0.8u t 2 0.3u t 3)] = 0.8 0.3 0.8 =0.8 0.7 =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 2 +0.8u t 3 0.3u t 4)] = 0.3 s > = 3 φ(s) =0 ρ(1) = 0.56/1.73 = 0.3237 ρ(2) = 0.3/1.73 = 0.1734 s > = 3 ρ(s) =0 31