80000 400 200 X i X1 + X 2 + X 3 + + X n i= 1 x = n n x n x 17 + 15 + 18 + 16 + 17 + 16 + 14 + 17 + 16 + 15 + 18 + 16 = 12 195 = = 1625. ( ) 12

X X n i = = 1 n i= 1 X f i f Xf = f n i= 1 X f ( Xf). i i i i (X) (f) (Xf) 20 10 200 22 12 264 24 25 600 26 30 780 30 18 540 32 15 480 33 10 330 120 3,194 Xf X = f = 3194, 120 = 26. 61( ) X Xf = f Xf X = f fi Wi = X Xf i Xf = f

X = X W i i : f = f = f = f 1 2 n 0 Xf f X X = = = f nf n X 2 () X f Xf 50 60 55 10 550 60 70 65 10 650 70 80 75 30 2,250 80 90 85 10 3,400 90 100 95 10 950 100 7,800 Xf 7, 800 X = = = 78( ) f 100

1 1 1 + + + 1 X1 X2 Xh = XH n n n : XH = = ( 103 ) 1 1 1 1 + + X X X X 1 2 h n 3 X H = = = 1 0. 71( / ) 1 1 1 + + X 1. 0 0. 8 0. 50 m1 m 2 m3 mn + + + + 1 X1 X2 X3 Xn = XH m + m + m + + m X H 1 2 3 m1 + m 2 + m 3 + + m n = m1 m2 m3 m n + + + + X X X X 1 2 3 n n n i 1 = = n i= 1 m i m X i i m Xf Xf XH = = = = X( m = Xf ) m Xf f X X Xh X h 4 mi i= 1 50, 000 = 4 = = 4102. ( ) mi 1219, X i= 1 1

/ X m m X 35 10,000 286 40 20,000 500 45 15,000 333 50 5,000 100 50,000 1,219 X n n = X X X = X G 1 2 n XG 1 log XG = (log X1 + log x2 + Λ + log Xn 1 + log Xn) n 1 = log X n k XG = fi X f X f X f X f i= 1 1 1 2 2 3 3 k k = n + 1 2

( ) 50 60 24 60 70 48 70 80 105 80 90 60 90 100 27 110 120 21 100 110 12 120 130 3 300 f 300 = = 150 2 2 n X d1 M0 = L + i d + d 1 2

d2 M0 = U i d + d 1 2 A. D.= X X N 2 ( X X) σ 2 = σ n X

2 ( X X) σ = n σ i X i ( ) X X ( X X) i 2 X i 2 1 113 0 0 12,769 2 121 8 64 14,641 3 113 0 0 12,769 4 114 1 1 12,996 5 113 0 0 12,769 6 114 1 1 12,996 7 115 2 4 13,225 8 106 7 49 11,236 9 111 2 4 12,321 10 110 3 9 12,100 1,130 0 132 127,822 1130, X = = 113( ) 10 ( X X) i σ2 = ( X X) ( Xi X) σ = n n 2 132 = = 132. 10 2 = 132. = 363. ( ) 2 2 2 ( X X) [ X 2XX + ( X) ] = n n X 2X X = + ( X) n n 2 X = 2( X) + ( X) n 2 2 X = ( X) n 2 2 2 2

σ 2 2 X = n X n 2 2 2 X σ = X n n 2 2 ( Xi X) f σ = f σ = ( ) 2 Xi X f f X σ A. D. VA. D. = 100% X σ Vσ = 100% X

X = 1 0 X x1x2λ Λ xkλ Λ P p p Λ Λ p Λ Λ 1 2 k : F( x) = P( X < x) = p( x = x i ) x < x i

x i x i P(X=x i ) 0 1 1/50 1 2 2/50 2 4 4/50 3 3 3/50 4 6 6/50 5 8 8/50 6 10 10/50 7 7 7/50 8 5 5/50 9 3 3/50 10 1 1/50 50 1 E( X) = xip( X = xi) = µ

σ D( X) = E{[X E( X)] } = σ 2 2 σ = D( X) 2 D( X) = E[( X µ ) ] 2 = ( x µ ) P( X = x ) i 2 2 D( X) = E( X ) [ E( X)] 2 D( X) = E{[X E( X)] } 2 2 = E{X 2XE( X) + [ E( X)] } = E( X ) [ E( X)] 2 2 10 1 µ = xi P x = xi = 0 + + i 0 50 1 2 4 ( ) 2 50 50 3 6 8 + 3 + 4 + 5 + 50 50 50 8 10 50 7 5 3 + 7 + 8 + 9 + = 50 50 50 10 1 5. 38 50 1 2 2 i 2 2 2 σ = ( x µ ) P ( X = xi ) = ( 0 5. 38) + ( 1 5. 38) 50 50 2 4 2 3 2 6 + ( 2 5. 38) + ( 3 5. 38) + ( 4 5. 38) 50 50 50 2 8 2 10 2 7 + (5 5. 38) + ( 6 5. 38) + ( 7 5. 38) 50 50 50 2 5 2 3 2 1 + (8 5. 38) + ( 9 5. 38) + ( 10 5. 38) 50 50 50 = 51956. i

a P X a b P X b { = 1} =, { = 0} = + a + b P{ X = 1} = p( 0 < p < 1( p ) P{ X = 0} = 1 p A 1A 2 A 3A 1A 2A 3A 1A 2A 3 P{ 2 } = P( A A A ) + P( A A A ) + P( A A A ) 1 2 3 1 2 3 1 2 3 2 2 2 = p ( 1 p) + p ( 1 p) + p ( 1 p) 2 2 2 = 3p ( 1 p) = C p ( 1 p) 3 k k n k P( X = k) = C p q ( k = 01,, 2, Λ, n) n

P { k x k } λ k! e λ = = ( k = 0, 1, 2, Λ Λ, λ > 0) k n k CmC N M P( X = k) = n ( k = 01,, 2, Λ min( M, n)) C N f x 0 : ( x) dx = 1 f ( x) F( x) : x F( x) = f( x) dx

P( a x b) = f( x) dx = F( b) F( a) a b E( x) f ( x) dx x 2 D( X) = E{[X E( X)] } 2 = [ x = E( X)] f( x) dx 1 a x b b a f( x) = 0 1 0 2 a + b ( b a) 2 12 0 x a x a F( x) = a < x < b b a λx λe x 0 f( x) = ( λ > 0) 0 x < 0 λ

0 x 0 F( x) = λx 1 e x > 0 2 ( x µ ) 1 2 2σ f( x) = e ( < X < + ) 2πσ σ σ 1 f( µ ) = 2πσ 1 f( µ ) = σ 2πσ 2 ( x µ ) 1 x 2 2σ F( x) = e dx 2πσ 2 1 x f( x) = e 2π 2 x2 1 x dx 2 F( X) = φ( X) = e 2π

σ x x µ x x σ 2 x σ 2 x σ x σ

σ 2 1 σ 2 2 x1 x 2 σ 2 σ 2 1 2 + n1 n 1 p( 1 p) n

Y = E( Y X = x0) ( Y X = x 0 ) E ( Y X = x 1 ) Y = E( YX = x) Y f x E Y X x 1 lxy xiyi ( xi )( yi ) n r = = l xxl yy 1 1 2 2 2 2 [ xi ( xi ) ][ yi ( yi ) ] n n l xy b = l xx

bl xx r = l l xx yy 2 r F = n 2 ( 2) 1 r y n SS = ( y y) 2 i= 1 i n 1 y = = yi n i 1 2 SS y y e n i= 1 i

2 SS y y R n i= 1 i SS = SS + SS, n 2 i : ( y y ) = + ( y) y) : SS = l = ( y y ) R i i= 1 i= 1 R e yy n = ( a + bx a bx) = b ( x x) 2 2 i 2 = b l = bl xx i xy 2 2 2 SS R SS 2 SS bl l R xy xy 2 2 2 r SSR r SS r l yy SS l l l yy xx yy 2 2 SS SS SS SS l r yyl 1 r e e R yy l gg SS r SS 2 R 1 2 y y i 1 y y y a bx) y i y 2 SSR / 1 F = SS / n 2 e i 2

2 2 SS rl n R / 1 yy r F = n SS n = ( 2) 2 = ( 2) / 2 ( 1 r ) l 2 2 r e yy y a b 0 y a bx 0 y 0 y i 2 ( yi y i) Syx = (5) n 2 y 2 i a yi b xi yi Syx = ( 6) n 2 1 ( x0 x) y 0 t a ( n 2)Syx 1+ + n ( x x) 2 i 2 y y + t ( n 2)S 1 ( x0 x0) 2 1+ + t a t n ( x x) 2 i 2 2 2 0 0 a xy

y i b 0 b1x1ib 2x2i y i nb b x x b x 0 1 i 1i 2 2i 2 x y b x b x b x x 2i i 0 1i 1 1i 2 2i 1i 2 x y b x b x x b x 1i i 0 2i 1 2i 1i 2 2i y b0 b1x1b 2x 2b 3x3 y nb b x b x b x 0 1 1 2 2 3 3 2 x y b x b x b x x b x x 1 0 1 1 2 1 2 3 1 3 2 2 2 x y b x b x x b x b x x 2 0 1 1 2 2 3 2 3 3 2 x y b x b x x b x x b x 3 0 3 1 1 2 3 2 3 3 y b0 b1xlb 2x2 bmxm y nb b x b x b x 0 1 1 2 2 m m 2 x y b x b x + b x x b x 1 0 1 1 1 2 1 2 m 2 x y = b x + b x x + b x x + Λ Λ + b x m Μ Μ 0 m 1 1 2 m 2 m m n 2 ( y i y i ) i= 1 S = n m 1 C jjoc jj L 1 L 1 m m

l11 l12λ Λ l1 Λ Λ Λ Λ Λ Λ l 21 l22 Λ Λ l 2 Λ Λ Λ Λ Λ Λ L = Μ Μ Λ Λ Λ Λ Λ Λ Μ Μ Λ Λ Λ Λ Λ Λ l 1 l 2 Λ Λ l m m m m mm n l = l = ( x x )( x xj) i, j = 1, 2Λ Λ, m ij ij it i jt t= 1 n 1 xi x j = xjt n t= 1 n 1 x j x j = x jt n t= 1 t j b j = j = 1, 2, Λ Λ, m s c jj t a 2 t > t j a 2 y ` = ab x

lg y = lga + xlgb y ' = lg y, a' = lg a, b' = lg b y' a' b' x 3 y a bx cx dx 2 2 3 x1 x x2 x x3 x y a bx cx 2dx3 ξη

19.00 12.00 1.583

GNP

Qd = f P T Y Ps Pc Pe

ϑ 0 0 1 50 50 50 2 150 75 100 3 300 100 150 4 400 100 100 5 480 96 80 6 540 90 60 7 580 83 40 8 610 76 30 9 610 68 030 10 580 58 30

MRTS LN / L dn / dl L MP N MP L N

R C 1 r r = R - C C

1. 2. 3. 4. 5. 6. 7. 8. 9. 10 ( ) ( ) ( )

( ) () 20 15 10 ( ) 5 5 10 15 ( )

PUV A ( ) B C 30 25 20 15 10 1 3 1 2. 5 4 30 75 20 25 40 3 5 2 1 0.5 100 190 275 340 90 125 40 15 5 4 3 4.5 2.0 2.5 20,000 21,000 26 500 120 75 90 30 25

500 P U V 250 0 C B A

C Q = E P V C + 2 Q1 = P C A S A A Μ A Μ A S S S S P(S )P(S )P(S )P(S ) V V V V V V V V n AiS j ( i = 12,, Λ, m) j= 1

n Q x1 x 1 0 1 i=1 n i i 1 2 2 n n i= 1 g ( x ) = g( x ) + g ( x ) + Λ Λ + g ( x ) n fn ( Q) = Max gi ( xi ) x i 0Λ Λ ( 2) i= 1 g n ( X n ) + fn 1( Q X n )

Max f Q X Q g X f n( ) = [ n( n) + n ) ( )] 0 0 1 Λ Λ 3 n f Q g Q 4 1 1 Max f Q X Q g X f Q x 2( ) = [ 2 ( 2) + 1( 2 ) Λ Λ (5)] 0 2 g x f Q - x g 0 f Q 0 2 2 i 2 2 1 g 1 f Q -1 1 1 g 2 f Q - 2 Μ 2 2 Μg Q f O g x 2 f Q x 2 1 2 2 1 2 f Q x 2 2 Max f3( Q) = [ 0 X Q g X f Q x 3( 3) + 3( 3) Λ Λ ( 6)] n (Q) 1 2 3 4 5 A g (Q) 15 36 78 90 100 B g (Q) 16 35 70 92 110 C g (Q) 20 40 80 90 110

f Max g f 2 ( 0) + 1 ( 1) = 15 2 ( 1) = g 2 ( 1) + f1 ( 0) = 16 g2 ( 0) + f1( 2) = 36 f2( 1) = Max g2 ( 1) + f1( 1) = 31 g2 ( 2) + f1( 0) = 35 g2 ( 0) + f1( 3) = 78 g2 ( 1) + f1( 2) = 31 f2 ( 3) = Max g2 ( 2) + f1 ( 1) = 52 g2 ( 3) + f1 ( 0) = 70 g3 ( 0) + f1( 4) = 90 g3( 1) + f2( 3) = 94 f2( 4) = Max g3 ( 2) + f 2( 2) = 71 g3( 3) + f 2( 1) = 85 g3( 4) + f 2( 0) = 92 g3( 0) + f1(5) = 100 g3( 1) + f 2 ( 4) = 106 g3( 2) + f2 ( 3) = 113 f2 (5) = Max g3( 3) + f2 ( 2) = 106 g3( 4) + f2 ( 1) = 107 g3(5) + f 2 ( 0) = 110

f Max g f 3 ( 0) + 2 ( 1) = 16 3( 1) = g3 ( 1) + f2 ( 0) = 20 g3 ( 0) + f2( 1) = 36 f3( 2) = Max g3 ( 1) + f2( 0) = 36 g3 ( 2) + f 2( 0) = 40 g 3( 0) + f2 ( 3) = 78 g 3( 1) + f2 ( 2) = 56 f3( 3) = Max g 3( 2) + f2 ( 1) = 56 g 3( 2) + f2 ( 0) = 80 g3( 0) + f 2( 4) = 94 g3( 1) + f 2( 3) = 98 f3( 4) = Max g3( 2) + f2( 2) = 76 g3( 3) + f2( 1) = 96 g3( 4) + f2( 0) = 90 g3 ( 0) + f2 (5) = 113 g3 ( 1) + f2 ( 4) = 114 g3 ( 2) + f 2 ( 3) = 76 f3(5) = Max g3 ( 3) + f2 ( 2) = 116 g3 ( 4) + f 2 ( 1) = 106 g3 (5) + f2 ( 0) = 110

/ / / / /

+ k d [ ] 1

K e R f R m R f x K( e) = 8% + 12. ( 12% 8%) = 12. 8%

2000 A + 2000 0 = 400+ 2 + 3 10925. (1.925) ( 10925. ) = 4000+ 1831+ 1676 + 0 = 493 500 500 6000 B = 4000 + + 10925. ( 10925. ) ( 10925. ) = 4000+ 458 419 + 4601 = 640 2 3

1 2 3 4 5

1 2 1 2

E D C A B

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 1 3 1 2 3 5 10 6 4 1% 2% 5% 10% 15% 27% 30% 10% 2% 4% 10% 29% 20% 18% 16% 4% 4% 8% 15% 35% 20% 10% 5% 3% 2% 8% 15% 30% 20% 12% 12% 4% 10% 10% 24% 39% 10% 8% 1% 1% 15% 20% 25% 37% 3% 38% 40% 15% 8% 2%

1 2 3 4 5 6 7 8 63 82 48 45 67 33 55 41 28 16 38 39 31 42 27 52 9 2 14 16 2 25 18 7 60 58 34 57 55 31 28 38 38 34 33.5 39 25 51 34 57 2 6 32.5 4 20 18 38 5