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1 1 ( S/E) (1 ) 01 Excel (-4 ) (5-7 ) Simplex 1 4 (shadow price) 14 5 (reduced cost) 14 3 (8-10 ) Excel 3 4 (11-13 )

2 0 (1 ) 0 (1 ) 01 Excel Excel ( ) = Excel Excel = =5-5 3 =5* 5 10 =5/ 5 5 =5^ 5 5 ( ), 0, Excel, Excel 13E E E E ( ),, 13E , (F) ( ),, ( ) ( ) $, =A1 =$A$1 ( ) ( ) =A$1 =$A1, F4, ( ),, (-4 ), (5-7 ), Excel,, (V), (I),

3 r, ( + ) 1% r = 001 x (x ) y, y 1 1 x = 10000, r = 01 (10%) 0 10,000 1,000 11, ,000 1,100 1,100 1,100 1,10 13, ,310 1,331 14, ,641 1,464 16, ,105 1,611 17, ,716 1,77 19, ,487 1,949 1, ,436,144 3, ,487, x = 10000, r = 01, y = 7 r, y, 1 70 bababababababababababababababab y r = 70 (r ) x, x r, 1 x(1 + r) + = x + xr = x (1 + r) 1 + r, 1 x (1 + r),, y x (1 + r) y y (= x), x (1 + r) y = x x, x (1 + r) y = y, y log(1 + r) = log

4 4 0 (1 ) log(1 + r),, log(1 + r) = r r log = 0693, yr = 0693 r ( ) 05 35%, ( ) r, 70 y r = 70 bababababababababababababababab 7 y r = 7 (r ) 6 9%, ( ) r, 7 y r = 7 7,, %, 01%, 0 3% 100 1, 1 1, 1 1, 1

5 5 1 (-4 ) , , , ( ), p = 1 0 a p, a,,, ( ) = ( ) = ( ) a, p,, p = 0 1 a ( ) = ( = (

6 6 bababababababababababababababab 1 (-4 ) = =,, ( ), p p = a X 1 0 X a 75, 75, 75 (6677) (75 (75 ) = ) 6677 (75 = ) 6677 bababababababababababababababab X = X X = X X 0 1, ,300 80, ,900 (006 13,860 ) 65, 794, %, % ( ) 65, 50% 1%

7 ( ) 100,000, 0% 100,000 1,000,000, 18% 1,000,000, 15%, 100 ( 15%) , 0% 1 31, 0% 15% = 5% (5 ), ( 15%) , 15% 1 31 (30 ) , , 170, 00 15%, 5 5, ( ) 1, 1 1 ( ) %,

8 8 1 (-4 ) 11 ( ) (0 ) 1 ( 1) 0 0, 1, % ( ) 01% 60 81, 40 ( 40 ) :, 13 ( ) 0, %, % = , 35 3%, 1,, = %, 1,, , 1 35, , 35 = 0

9 9 (5-7 ) x 4y + 6z = 1 (1) 3x + y + z = 11 () x 3y 3z = 14 (3), (1) x y + 3z = 6 (4) () (4) 3, () x, (3) (4), (3) x 7y 7z = 7 (5) y 6z = 0 (6) (5) 7 y z = 1 (7) (6) (7), (6) y, 7z = 1 (9) z = 3 (10) z z (7) y, (4) x y = (1) x = 1 (11)

10 10 (5-7 ) (1) x 4y + 6z = () 3x + y + z = (3) x 3y 3z = (4) (1) x y + 3z = (5) () (4) 3 7y 7z = (6) (3) (4) y 6z = (7) (5) 7 y z = (8) (4) + (7) x + z = (9) (6) + (7) 7z = (10) (9) ( 7) z = (11) (8) (10) x = (1) (7) + (10) y = , (1) x 4y + 6z = () 3x + y + z = (3) x 3y 3z = (4) (1) (5) () (4) (6) (3) (4) (8) (4) + (7) (7) (5) (9) (6) + (7) (11) (8) (10) x = (1) (7) + (10) y = (10) (9) ( 7) z = x 3y + 6z = 15 x + 4y + z = 10 x + y + 6z = 1

11 11 A B, X Y 1 X, A B, 1 Y, 6 A 1 B X Y A 60, B 0, X Y X x, Y y, A x + 6y A 60, x + 6y 60 B, x + y 0 x, y, x 0, y 0,,, p p = 9x + 16y y 0 y = x (6, 8) y = 1 3 x x,,, p (6, 8) p max p = = 18 18, X 6, Y 8

12 1 bababababababababababababababab (5-7 ) = A, B x + 6y = = 60 x + y = = 0, 3 Simplex,, ( ) Smplex X x, Y y, x + 6y 60 x + y 0, ( ) a 0, b 0, a, b, A, B x + 6y + a = 60 x + y + b = 0 (A) (B) p = 9x + 16y 9x 16y + p = 0 (P ) Simplex, 3 (A), (B), (P ) 1 ( ) x y a b p (A) (B) (P ) 1/3 1 1/ /3 0 1/ /3 0 8/ /5 1/5 0 8 (Y ) y a 1 5 b = /10 3/5 0 6 (X) x 1 10 a b = /10 11/ ( ) 3 10 a b + p = 18 Simplex, 3 10 a + 11 b + p = 18 ( ) 5

13 3 Simplex 13 p (a 0, b 0), a, b, p a = b = 0, p (X), (Y ), max p = 18 y a 1 5 b = 8 (Y ) x 1 10 a b = 6 (X) a = b = 0, X, Y x = 6, y = 8 a = b = 0,, ( ) ),, x y a b p (10) (0) /3 1 1/ (30) 5/3 0 1/ (6) 11/3 0 8/ /5 1/ /10 3/ /10 11/ ) ( 9, 16),, ( ) ), 3) ( 1 ), 1), ( 1 ) 4), 1), p

14 14 (5-7 ) 4 (shadow price) x, y, a, b p x + 6y + a = 60 x + y + b = 0 (A) (B) 9x 16y + p = 0 (P ) 3 10 a + 11 b + p = 18 ( ) 5 (x = 0, y = 0), (A), (B), a A, b B (P ) p = 0 a = 60, b = 0, p = 0 ( ), = , 11 5 bababababababababababababababab 3 10, 11 5,,, (shadow price) = (reduced cost) 3, X, Y 6, 0, X, Y p = 6x + 0y, (0, 10) p y 0 y = x + 0 (0, 10) y = 1 3 x x (x, y) = (0, 10) p max p = = = 60, = 10, B 10

15 5 (reduced cost) 15 Simplex 1 x y a b p (10) (0) /3 1 1/ /3 0 1/ /3 0 10/ , 3 x + 10 a + p = 00 ( ) 3 p, x a, x = 0, a = 0, p x = 0, a = 0, max p = 00, 1 3 x + y a = 10, 5 3 x 1 6 a + b = 10 x 0, a 0 x = 0, a = 0, y, b y = 10, b = 10, x = 0, y = 10 ( ), max p = 00 ( ), A, B 10 ( ) ( ), 10 3 (shadow price), = 00 3 (reduced cost) X 1, 3, X 1, 3 ( )

16 16 (5-7 ) 1 X Y, A, 1, B 1, A B 10, 150 X, Y, 3, Simplex X, Y, Z, A 1, B, 3, 0 C 1,, 1 A, B, C 10, 4, 16 X, Y, Z 3, 5,, X, Y, Z,, L 3 ( ) X Y, x, y Y 4 Z z, z = 3y Y C x, C y ( : ) C x = x x, C y = 03y 4z X, Y, 1 1, 101 x + y , C = C x + C y ( : x = 360, y = 650, C = 1435) p = 61, R ( : x = 35, y = 635, R = 13949)

17 17 3 (8-10 ) 31, 1 ( Excel ) (AVERAGE) x 1, x, x 3,, x,, x (VARP ) x = x 1 + x + + x x i x x i x,, V (x), (kg ) V (x) = (x 1 x) + (x x) + + (x x) (STDEVP ) V (x) σ(x) (kg) σ(x) = V (x) (MAX) (MI) (MAX()-MI() ) (MODE) (MEDIA), ( + 1)/, / ( +)/, V (x) = x ( x) bababababababababababababababab (x ) = (x ) (x ) x = xi

18 18 3 (8-10 ), (xi x) V (x) = (x = i x x i + x ) x = i x xi + x x = i x x + x = x x x i (i = 1,,, ), x, σ(x) x bababababababababababababababab z z, x ( ) z i = x i x σ(x) (x ), (11-14 ), 0, 1 z = 0, σ(z) = 1, x i t i t i = z i t i = x i x σ(x) 50, 10 t = 50, σ(t) = 10 3 ( ) x, y, x, y (covariance) Cov(x, y) = 1 (x i x)(y i ȳ) i=1 = (x 1 x)(y 1 ȳ) + (x x)(y ȳ) + + (x x)(y ȳ) x x Cov(x, y),, x Cov(x, x) = V (x) Cov(x, y) = xy x ȳ (x,y ) = (x y ) (x ) (y )

19 33 19 x, y ( Peason ) r Cov(x, y) r = σ(x) σ(y) (xi x)(y i ȳ) = (xi x) (yi ȳ) x i, y i, x y y i Cauchy-Schwarz pq p q, r bababababababababababababababab 1 1 (r) 1 33 x i, y i, x i y i,, x i y i, ŷ i, ŷ i = a + bx i, ŷ, a, b a, b,, E E = a 8ab + 11b + 4a 14b + 9 = (a 4ab + a) + 11b 14b + 9 = (a (b 1)) (b 1) + 11b 14b + 9 = (a b + 1) + 3b 6b + 7 = (a b + 1) + 3(b 1) + 4 0, E 4 min E = 4 E a b + 1 = 0, b 1 = 0, a = b = 1

20 0 3 (8-10 ) (1) ŷ i y i e i e i e i = y i ŷ i y = a + bx, E, a, b E = (y i ŷ i ) i=1 = (y i a bx i ) = a + ab x i + b x i a y i b x i y i + yi ( = a + b ) x i yi ( x + i ( xi ) ) xi y i b xi yi x i ( xi ) + Const = (a + bx y) + V (x) ( b ) Cov(x, y) + Const V (x) Const, a + b x ȳ = 0, b Cov(x, y) V (x), E Const bababababababababababababababab, = 0 ŷ = a + bx, b = Cov(x, y), a = ȳ b x V (x)

21 33 1 () E, E a E = (y i a bx i ) E a = (y i a bx i ) E, 0,, yi a b xi = 0 a = ȳ b x ( ) E b E b = x i (y i a bx i ) E 0, xi y i 0 = a xi x b i xi y i x = (ȳ b x) x b i ( ) ( ) xi y i x = x ȳ b i x = Cov(x, y) b V (x) a b a = ȳ b x, b = Cov(x, y) V (x) bababababababababababababababab ŷ, ŷ = y, ŷ y, ŷ = a + bx = a + b x = ȳ

22 3 (8-10 ) 34 ŷ i V (ŷ), y i V (y) R R = V (ŷ) V (y), bababababababababababababababab R 1 0 (R ) 1, y, 1, x ( R ) = 1 (ŷ ŷ) V (y) (a + bxi a b x) = 1 V (y) = 1 V (y) b (xi x) = 1 Cov(x, y) V (y) V (x) V (x) = Cov(x, y) V (x)v (y) = ( r), 35 3 x i, y i, z i, z i x i y i 1, z i ẑ i, ẑ i = a + bx i + cy i a b, c E,, E = (z i ẑ) a, = (z i a bx i cy i ) E a = (z i a bx i cy i ) = ( z a b x cȳ)

23 36 Excel 3 b, c, E b = x i (z i a bx i cy i ) ( ) = xz a x bx cxy E c = y i (z i a bx i cy i ) ) = (yz aȳ bxy cy E, 0, a, b, c 3 z a b x cȳ = 0 xz a x bx cxy = 0 yz aȳ byx cy = 0 z, xz, yz, z 1 x y a xz = x x xy b yz y yx y, a b, c a 1 x y b = x x xy c y yx y 1 c z xz yz y, m x j (j = 1,,, m), ŷ = a 0 + a 1 x 1 + a x + + a m x m, a j (j = 0, 1,,, m), a 0 1 x 1 x m a 1 = x 1 x 1 x 1 x m a m x m x m x 1 x m 1 ȳ x 1 y x m y 36 Excel Excel MMULT(), MIVERSE() Ctrl Shift, Enter

24 4 3 (8-10 ) 31 5 ( ),, z = (x x)/σ(x), t = 10z ( ) X Y 5 Y X, b = Cov(x, y)/v (x), a = ȳ b x 33 ( ) 15 X, Y, Z Z X Y, Excel MIVERSE(), MMULT()

25 5 4 (11-13 ) , / (x k ) (f k ) (f k /) () 1 f k ( = 100 ) n ( n = 7 ), x, V (x), σ(x) x = x 1f 1 + x f + + x n f n V (x) = (x 1 x) f 1 + (x x) f + + (x n x) f n = x 1 f 1 + x f + + x n f n σ(x) = V (x) x 41 7, 44 1 V (x) = x ( x) r k = f k /, x = x 1 r 1 + x r + + x n r n V (x) = (x 1 x) r 1 + (x x) r + + (x n x) r n = ( x 1 r 1 + x r + + x n r n ) x, x = = 4994 V (x) = = x k p k, x 1 x x n p 1 p p n 1, ,

26 6 bababababababababababababababab 4 (11-13 ) x = x 1 p 1 + x p + + x n p n V (x) = (x 1 x) p 1 + (x x) p + + (x n x) p n = ( ) x 1 p 1 + x p + + x n p n x σ(x) = V (x) 4 n k nc k, n(n 1)(n ) (n k + 1) nc k = k(k 1)(k ) 1 n! = k! (n k)! k! k, k! = k(k 1) 1 0! = 1, A (A ) P (A), p, A Ā (A ) q A P (A) = p P (Ā) = q (= 1 p) n A k n, k, n k, nc k p k q n k n, p B(n, p) x k : 0 1 k n p k : nc 0 q n nc 1 p q n 1 nc k p k q n k nc n p n 1 (p + q) n = n C 0 q n + n C 1 p q n 1 + n C p q n + + n C n p n, p + q = 1, nc 0 q n + n C 1 p q n n C k p k q n k + + n C n p n = 1 p k (k = 0, 1,, n) 1

27 4 7 x, x = = = V (x), V (x) = n x k p k k=0 n k n C k p k q n k k=0 n n! k k! (n k)! pk q n k k=1 n 1 n! = (k + 1) (k + 1)! (n k 1)! pk+1 q n k 1 k=0 = np = np = n 1 k=0 (n 1)! k! (n k 1)! pk q n k 1 n x k p k x k=0 n k=0 k n! k! (n k)! pk q n k (np) n 1 = np (k + 1) = np k=0 n 1 k=1 k = np (n 1)p (n 1)! k! (n k 1)! pk q n k 1 n p (n 1)! k! (n k 1)! pk q n k 1 + np n p n k=0 = n(n 1)p np n p = np(1 p) (n )! k! (n k )! pk q n k np n p bababababababababababababababab B(n, p),, x = np, V (x) = np(1 p), σ(x) = np(1 p)

28 8 4 (11-13 ) COMBI() nc s, n s, nc s = n! s! (n s)! Excel COMBI nc s = COMBI(n, s) BIOMDIST() n, p, s nc s p s (1 p) n s Excel BIOMDIST nc s p s (1 p) n s = BIOMDIST(s, n, p, FALSE), s, 41 s nc k p k (1 p) n k = BIOMDIST(s, n, p, TRUE) k=0, 1 4 p = , C (1 09) 5 = = , P, P = 100 C C C C C C = =

29 m, m, σ (m, σ ) σ, f(x), f(x) = 1 e (x m) /σ πσ X a b, P (a X b) = 1 b e (x m) /σ dx πσ a,, 0, 1 (0, 1) 0, 1, 0 x, x X 1, X,, X n, m, σ X i nm, nσ X X = X 1 + X + + X n nm nσ X,

30 30 4 (11-13 ) 41 ( ) X 1, X,, X n, m, σ ( ) lim P X1 + X + + X n nm x = 1 x e u / du n nσ π, (0, 1) p n B(n, p),, n 4, 1 4 p = , , n = 100, p = 09, np, np(1 p) np = = 90 np(1 p) = (1 09) = 9 90, 3, x = 95, = 167, ( ) 05, 95, = , 0 1, a x < b a x < b,, U(a, b) b x = x 1 a b a dx = 1 b a = a + b [ x ] b a

31 46 31, RAD() b ( V (x) = x 1 a + b a b a dx = 1 [ ] x 3 b ( a + b b a 3 = (b a) 1 a ) Excel RAD() RAD() 0 1, F9, ), (,, ) , , , 50, ( ) 7 95% ( 5%) 1% (%)

32 3 4 (11-13 ) 41 4 p = , , , ,

33 33 5 ( ) S : 1?3? (??)?4? (?15:00??16:00?) E : 1?4? (??)?4? (?15:00??16:00?)

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