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1 ( S/E) 006 7 30 0 (1 ) 01 Excel 0 7 3 1 (-4 ) 5 11 5 1 6 13 7 (5-7 ) 9 1 1 9 11 3 Simplex 1 4 (shadow price) 14 5 (reduced cost) 14 3 (8-10 ) 17 31 17 3 18 33 19 34 35 36 Excel 3 4 (11-13 ) 5 41 5 4 6 43 9 44 9 45 30 46 31 5 33

0 (1 ) 0 (1 ) 01 Excel Excel ( ) = Excel Excel =5+ 5 + 7 =5-5 3 =5* 5 10 =5/ 5 5 =5^ 5 5 ( ), 0, Excel, Excel 13E+05 13 10 5 13000 13E-05 13 10 5 0000013-13E+04 13 10 4 1300-13E-04 13 10 4 000013 ( ),, 13E+05 13000, (F) ( ),, ( ) ( ) $, =A1 =$A$1 ( ) ( ) =A$1 =$A1, F4, ( ),, (-4 ), (5-7 ), Excel,, (V), (I),

0 7 3 0 7 r, ( + ) 1% r = 001 x (x ) y, y 1 1 x = 10000, r = 01 (10%) 0 10,000 1,000 11,000 1 11,000 1,100 1,100 1,100 1,10 13,310 3 13,310 1,331 14,641 4 14,641 1,464 16,105 5 16,105 1,611 17,716 6 17,716 1,77 19,487 7 19,487 1,949 1,436 8 1,436,144 3,579 7 19,487, 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 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,, 70 01 7%, 01%, 0 3% 100 1, 1 1, 1 1, 1

5 1 (-4 ) 11 75 6677, 003784, 55 1 0 00345 100000 345 1 00051 99655 51 00038 99604 38 75 03784 6677 55 76 0416 640 67 77 04606 61530 834 110 54918 4 111 57147 1 11 59378 1 0, ( ), p = 1 0 a p, a,,, ( ) = ( ) 100000 = ( ) 100000 a, p,, p = 0 1 a ( ) = ( 100000 = ( 100000

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, 004 13,300 80, 017 16,900 (006 13,860 ) 65, 794,500 3 1 1 004 1358%, 017 1830% ( ) 65, 50% 1%

13 7 13 ( ) 100,000, 0% 100,000 1,000,000, 18% 1,000,000, 15%, 100 ( 15%) 006 1 1, 0% 1 31, 0% 15% = 5% (5 ), 100 15 00 ( 15%) 006 1 1, 15% 1 31 (30 ) 1 1 170, 1 31 00, 170, 00 15%, 5 5, 4 5 11 ( ) 1, 1 1 ( ) %,

8 1 (-4 ) 11 ( ) (0 ) 1 ( 1) 0 0, 1, 0 59 1358% ( ) 01% 60 81, 40 ( 40 ) :, 13 ( ) 0, 65 81 50%, % = 0 14 3000, 35 3%, 1,, 35 35 = 0 15 3%, 1,, 1 16 3000, 1 35, 6000 60000000 35 1, 35 = 0

9 (5-7 ) 1 1 1 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 (5-7 ) 4 6 1 (1) x 4y + 6z = 1 3 1 11 () 3x + y + z = 11 1 3 3 14 (3) x 3y 3z = 14 1 3 6 (4) (1) x y + 3z = 6 3 1 11 1 3 3 14 1 3 6 0 7 7 7 (5) () (4) 3 7y 7z = 7 0 1 6 0 (6) (3) (4) y 6z = 0 1 3 6 0 1 1 1 (7) (5) 7 y z = 1 0 1 6 0 1 0 1 4 (8) (4) + (7) x + z = 4 0 1 1 1 0 0 7 1 (9) (6) + (7) 7z = 1 1 0 1 4 0 1 1 1 0 0 1 3 (10) (9) ( 7) z = 3 1 0 0 1 (11) (8) (10) x = 1 0 1 0 (1) (7) + (10) y = 0 0 1 3, 4 6 1 (1) x 4y + 6z = 1 3 1 11 () 3x + y + z = 11 1 3 3 14 (3) x 3y 3z = 14 1 3 6 (4) (1) 0 7 7 7 (5) () (4) 3 0 1 6 0 (6) (3) (4) 1 0 1 4 (8) (4) + (7) 0 1 1 1 (7) (5) 7 0 0 7 1 (9) (6) + (7) 1 0 0 1 (11) (8) (10) x = 1 0 1 0 (1) (7) + (10) y = 0 0 1 3 (10) (9) ( 7) z = 3 1 3 1 3x 3y + 6z = 15 x + 4y + z = 10 x + y + 6z = 1

11 A B, X Y 1 X, A B, 1 Y, 6 A 1 B X Y 1 9 16 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 + 0 10 (6, 8) y = 1 3 x + 10 10 30 x,,, p (6, 8) p max p = 9 6 + 16 8 = 18 18, X 6, Y 8

1 bababababababababababababababab (5-7 ) = 1 1 + + A, B x + 6y = 6 + 6 8 = 60 x + y = 6 + 8 = 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 6 1 0 0 60 (A) 1 0 1 0 0 (B) 9 16 0 0 1 0 (P ) 1/3 1 1/6 0 0 10 5/3 0 1/6 1 0 10 11/3 0 8/3 0 1 160 0 1 1/5 1/5 0 8 (Y ) y + 1 5 a 1 5 b = 8 1 0 1/10 3/5 0 6 (X) x 1 10 a + 3 5 b = 6 0 0 3/10 11/5 1 18 ( ) 3 10 a + 11 5 b + p = 18 Simplex, 3 10 a + 11 b + p = 18 ( ) 5

3 Simplex 13 p (a 0, b 0), a, b, p a = b = 0, p (X), (Y ), max p = 18 y + 1 5 a 1 5 b = 8 (Y ) x 1 10 a + 3 5 b = 6 (X) a = b = 0, X, Y x = 6, y = 8 a = b = 0,, ( ) ),, x y a b p 6 1 0 0 60 (10) 1 0 1 0 0 (0) 9 16 0 0 1 0 1/3 1 1/6 0 0 10 (30) 5/3 0 1/6 1 0 10 (6) 11/3 0 8/3 0 1 160 0 1 1/5 1/5 0 8 1 0 1/10 3/5 0 6 0 0 3/10 11/5 1 18 1) ( 9, 16),, ( ) ), 3) ( 1 ), 1), ( 1 ) 4), 1), p

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 ( ), 3 10 60 + 11 5 0 = 18 3 10, 11 5 bababababababababababababababab 3 10, 11 5,,, (shadow price) = 1 1 + + 5 (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 + 10 10 30 x (x, y) = (0, 10) p max p = 6 0 + 0 10 = 00 0 + 6 10 = 60, 0 + 10 = 10, B 10

5 (reduced cost) 15 Simplex 1 x y a b p 6 1 0 0 60 (10) 1 0 1 0 0 (0) 6 0 0 0 1 0 1/3 1 1/6 0 0 10 5/3 0 1/6 1 0 10 /3 0 10/3 0 1 00, 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 + 1 6 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), 10 3 60 + 0 0 = 00 3 (reduced cost) X 1, 3, X 1, 3 ( )

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 15 + 9x, C y = 03y 4z X, Y, 1 1, 101 x + y 101 101, C = C x + C y ( : x = 360, y = 650, C = 1435) p = 61, R ( : x = 35, y = 635, R = 13949)

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 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, 10 50 x i t i t i = 50 + 10 z i t i = 50 + 10 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 )

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

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)

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

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)

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

4 3 (8-10 ) 31 5 ( ),, z = (x x)/σ(x), t = 10z + 50 3 ( ) 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()

5 4 (11-13 ) 41 100 7, / (x k ) (f k ) (f k /) 395 45 41 7 007 45 455 44 1 01 455 485 47 18 018 485 515 50 4 04 515 545 53 1 01 545 575 56 14 014 575 605 59 4 004 100 () 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 = 41 007 + 44 01 + + 59 004 = 4994 V (x) = 41 007 + 44 01 + + 59 004 4994 = 17764 x k p k, x 1 x x n p 1 p p n 1, 41 007, 41 007

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 1 + + n C k p k q n k + + n C n p n = 1 p k (k = 0, 1,, n) 1

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)

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 = 09 100, 95 4 100 95 4 100C 95 09 95 (1 09) 5 = 758750 09 95 01 5 = 0033865804 95 4, 95 100 P, P = 100 C 95 09 95 01 5 + 100 C 96 09 96 01 4 + 100 C 97 09 97 01 3 + 100 C 98 09 98 01 + 100 C 99 09 99 01 1 + 100 C 100 09 100 = 758750 09 95 01 5 + 3915 09 96 01 4 + 161700 09 97 01 3 + 4950 09 98 01 + 100 09 99 01 1 + 1 09 100 = 0057576886

43 9 43 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 00 01 0 03 04 05 06 07 08 09 00 0000 0040 0080 010 0160 0199 039 079 0319 0359 01 0398 0438 0478 0517 0557 0596 0636 0675 0714 0753 0 0793 083 0871 0910 0948 0987 106 1064 1103 1141 03 1179 117 155 193 1331 1368 1406 1443 1480 1517 04 1554 1591 168 1664 1700 1736 177 1808 1844 1879 05 1915 1950 1985 019 054 088 13 157 190 4 06 57 91 34 357 389 4 454 486 517 549 07 580 611 64 673 704 734 764 794 83 85 08 881 910 939 967 995 303 3051 3078 3106 3133 09 3159 3186 31 338 364 389 3315 3340 3365 3389 10 3413 3438 3461 3485 3508 3531 3554 3577 3599 361 11 3643 3665 3686 3708 379 3749 3770 3790 3810 3830 1 3849 3869 3888 3907 395 3944 396 3980 3997 4015 13 403 4049 4066 408 4099 4115 4131 4147 416 4177 14 419 407 4 436 451 465 479 49 4306 4319 15 433 4345 4357 4370 438 4394 4406 4418 449 4441 16 445 4463 4474 4484 4495 4505 4515 455 4535 4545 17 4554 4564 4573 458 4591 4599 4608 4616 465 4633 18 4641 4649 4656 4664 4671 4678 4686 4693 4699 4706 19 4713 4719 476 473 4738 4744 4750 4756 4761 4767 0 477 4778 4783 4788 4793 4798 4803 4808 481 4817 1 481 486 4830 4834 4838 484 4846 4850 4854 4857 4861 4864 4868 4871 4875 4878 4881 4884 4887 4890 3 4893 4896 4898 4901 4904 4906 4909 4911 4913 4916 4 4918 490 49 495 497 499 4931 493 4934 4936 5 4938 4940 4941 4943 4945 4946 4948 4949 4951 495 6 4953 4955 4956 4957 4959 4960 4961 496 4963 4964 7 4965 4966 4967 4968 4969 4970 4971 497 4973 4974 8 4974 4975 4976 4977 4977 4978 4979 4979 4980 4981 9 4981 498 498 4983 4984 4984 4985 4985 4986 4986 30 4987 4987 4987 4988 4988 4989 4989 4989 4990 4990 44 X 1, X,, X n, m, σ X i nm, nσ X X = X 1 + X + + X n nm nσ X,

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 = 09 100, 95 4 41 41, n = 100, p = 09, np, np(1 p) np = 100 09 = 90 np(1 p) = 100 09 (1 09) = 9 90, 3, x = 95, 95 90 3 = 167, 90 95 0455 90 ( ) 05, 95, 05 0455 = 00475 45, 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

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, ), (,, ) 0 1 40, 1 40 0 1, 46 43 7 1500, 50, ( ) 7 95% ( 5%) 1% (%) 100 33 500 164

3 4 (11-13 ) 41 4 p = 09 400, 351 4 4 7, 3 0 8 43 450, 3 14 163,

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