Part () () Γ Part ,

Size: px
Start display at page:

Download "Part () () Γ Part ,"

Transcription

1 Contents a Part

2 Part () () Γ Part ,

3 Part χ t F Part

4 () Part Part

5 References 68

6 6 a () () () () (3) (4) () () (3) () () (3) (4) (5)

7 7 (6) (7) (8) () () (3) 7

8 8... () () (3) exp(a) e a (4) R.... () () (3) (4) y = π exp ( x ).3.. () (a) (b) () (a) A B A B A B (b) A B A B (3) (a) (b) (4) (a) (b) (5) (a) (b) (6) (a) (b) (7) (a) (b) (c) (8) (a) N(, ) (b) N(m, σ ) m σ σ Y = X m σ N(m, σ ) X N(, ) Y

9 9 Part..... k n n k n! np k = = n(n )(n ) (n k + ) (n k)! n! n! = n(n )(n )! =.. 5 A, B, C, D, E F, G 7 () () (3) (4) A, F.. () 7! = 54 6! = = 36 () 5! = = 4 (3) 4! = = 96 () = 44 (4) ( ) A A B, C, D, E ! 4 4! = 9 ( ) A B, C, D, E 5 A, F 4! = 48 A 48 = R,B,Y,G () (a) 4 (b) 3 (c) (d) ()

10 .. () (a) 4! = 4 (b) 4 4 3! = = 48 (c) 4 3 = (d) 3 84 (). (a) (b) (b) = k n n k n! n(n )(n ) (n k + ) nc k = = k!(n k)! k! ( n n C k = k). (a + a + + a N ) n = (a + b) n = nc k a k b n k k= l,l,...,l N, l +l + +l N =n n! l!l! l N! a l a l a N l N l,l,...,l N, l +l + +l N =n l + l + + l N = n l, l,..., l n a + a + + a N = M. a + a + + a N = M a, a,..., a N N H M = M+N C N.3. () 3 () 3 (3) (4) 5 (5) x = m y = n m, n (, ) (3, 5) (6) X + Y + Z = 8 X, Y, Z (7) 5

11 (8) 5.3. () C 3 = () P 3 = 7 (3) C 5 = 6 (4) C 5 = 5 (5) C 3 = 56 (6) C 8 = 9 (7) 5 H = 4 C = (8) X + Y + Z + W + V = X, Y, Z, W, V x + y + z + w + v = 5 9 C 4 = 6.4. {,, 3, 4, 5} {,, 3, 4, 5, 6, 7, 8, 9, } σ {,, 3, 4, 5} {,, 3, 4, 5, 6, 7, 8, 9, } σ(), σ(), σ(3), σ(4), σ(5) 5 () σ(),..., σ(5) () σ(),..., σ(5) σ (3) σ() < σ() < σ(3) < σ(4) < σ(5) σ (4) σ() + σ() + σ(3) + σ(4) + σ(5) =.4. x, x, x 3, x 4, x 5 () (x, x, x 3, x 4, x 5 ) () (x, x, x 3, x 4, x 5 ) (3) x < x < x 3 < x 4 < x 5 (x, x, x 3, x 4, x 5 ) (4) x + x + x 3 + x 4 + x 5 = (x, x, x 3, x 4, x 5 ) () 5 = () P 5 = 34 (3) C 5 = 5 (4) A + B + C + D + E = α + β + γ + δ + ε = 5 9 C 5 = 6 A, B, C, D, E A E 5 σ() + σ() + σ(3) + σ(4) = α + β + γ + δ = 6 ε 9 C 3 = 84 σ() + σ() + σ(3) + σ(4) = σ 84

12 .5. U = {,, 3, 4, 5, 6} 5 A / A {4, 5} A.5. {4} = A / N n N n/n 3. (). () Ω () (3) (4) (5) Ω (6) ω (ω Ω) (7) (8) ( ) 3.. () {,, 3, 4, 5, 6} (),, 3, 4, 5, 6 (3) {} {} {6} (4) 64 (5) {, } {3, 4} 3.3. {a, a,..., a k } ( ) a ( ) a ( ) a 3 ( ) {a, a,..., a k } = {a, b, c} { c A {a, b, c} b A a A { c A {a, b} c A {a, c} b A { c A {a} c A {b, c} b A a A { c A {b} c A {c} b A c A

13 ( ) (, ) ( ) (, ) ( ) (, ) ( ) (, ) Ω = {(, ), (, ), (, ), (, )} () () (3) /6 (4) / () A B A B () A B A B (3) A B A, B (4) A B A, B (5) (6) A A A c (7) A B P (A)P (B) = P (A B) (8) A B A B = 3.8. () A B A / B (a) A B (b) A B A B A / B (c) A B (d) A B () A B, A B (a) A B A B A B (b) A B (c) A B (d) A B (3) A B, A B, A B, A B (a) A B (b) A B (c) A B (d) A B

14 4 (4) A B C A B, C (5) U = {,, 3, 4, 5, 6},, 3, 4, 5, 6 (a) 4 4 U (b) {4} {4} U (c) 3, 4 3, 4 U 3, 4 U 3 U 4 U (d) {3, 4} 3, 4 {3, 4} U 3.9 ( ). A A P (A) [, ] () P (A) () P (Ω) = (3) A, B P (A B) = P ((A) + P (B) n ) A, A,..., A n P A i = P (A i ) 3.. i= () 3 6 P ({3}) = 6 (), 3 P ({, }) = 3 (3) P ({3}) = (4) A {,, 3, 4, 5, 6} A A 6 A i= Ω = {ω = (ω, ω,..., ω j,...) : ω j {, } (j =,,...)} P ({ω : ω j = j }) = P {ω : ω = ω = = ω j = < ω j = } j= = P ({ω : ω = ω = = ω j = < ω j = }) = j= j = j= ( ) {A i } i= = P A i = P (A i ) i= σ- i=

15 5 3.. U = {,, 3, 4, 5, 6} 3 A 3 A A {3} A 3 A ( ). A, B P (A) > A B P (A B) P (B A) = P A (B) = P (A) A, B () A B () A B 4.. () () 4.3. () /3 () / 4.3. () P (A) = 8, P (B) = 7, P (A B) = P A(B)

16 6 () U A, B P (A) = 4, P (B) = A, B 5 P (A B) (3) A B 6 C.5 A, B, C 48 4 A (4) A A () P A (B) = P (A B) P (A) = 4 5 () A, B P (A B) = P (A)P (B) = P (A B) = P (A) + P (B) P (A B) P (A B) = = 5. (3) A 96 B 7 C = = 8 9. (4) {} ( ) P ({ }) = 6 = ( ) P ({ }) = = 5 36 = ( ) P ({ }) = = 5 6 = ( ) P ({ }) = = { } P ({ }) = P ({ }) + P ({ }) + P ({ }) + P ({ }) = (a) p P ({ } { }) P ({ }) p = P { ({ }) = = = 8 } P ({ }) P ({ }) 398 = (b) p P ({ } { }) P ({ }) p = P { ({ }) = = = 5 } P ({ }) P ({ }) 398

17 A, B, C, D A B, C, D B, C, D, 3, 4 C, D A, B, C D B, C, D () A () A C (3) A (4) A B 4.4. B, C, D B = B C = C D = D () A B (C D) A P (B (C D)) = P (B)P (C D) = P (B)( P (C D)) = P (B)( P (C)P (D)) = 4 () A C B C 6 P B (C D) (B C) = (3) A 3 4 P (B C) P (B (C D)) = 3 (4) A B B P (B) P (B (C D)) = 3 B, C, D P ( ) = 3 4 = 4, P ( ) = 3 4 = 4 P ( ) = 3 4 = 4, P ( ) = 3 4 = 4 P ( ) = = 3 4, P ( ) = = 3 4 P ( ) = = 6 4, P ( ) = = 6 4 () = P ( ) + P ( ) + P ( ) = = 4.

18 8 () C = P ( ) + P ( ) = = 6. P A (C ) = P ( C A ) P (A ) = 6 4 = 3. (3) = = 3 4. (4) B = P ( ) + P ( ) + P ( ) + P ( ) =. P A (C ) = P ( B A ) P (A ) = 3 4 = A, B, C, D A B, C, D B, C, D A D 4 B C { } P ({ }) = = 3 3 A B, C, D { P ({ } }) = = 9 3 () D () D B, C (B, C) B, C B, C 4.5. () P ({ }) = = 3 3, P ({ }) = = 9 3 P ({ }) = = 4 3, P ({ }) = = 4 3 P ({ }) = 4 = 3, P ({ }) = 3 4 = 6 3

19 9 P ({ }) = 4 = 3, P ({ }) = 4 = 3 P (D ) = = 3 3. () D D B, C P (D B ) = = 3, P (B C ) = = 5 3 3, P (D ) = 3 P D (B ) =, P D (C ) = 5 = 5 7 C 4.6. A, B, B P (A) > P A (B B ) = P A (B ) P A (B B ) P (B B ) = P (B ) P (B B ) 4.6. P A (B B ) = P (A (B B )) P (A) = P ((A B ) (A B )) P (A) = P ((A B ) P ((A B ) (A B )) P (A) = P A (B ) P A (B B ) 4... A, B P (A)P (B) = P (A B) P (A) > P A (B) = P (B) 4.7 ( ). A B, A c B, A B c, A c B c () : : :

20 () (3) 3 4

21 Ω = {(, ), (, ), (, ), (, )} A B A = {(, ), (, )}, B = {(, ), (, )}, A B = {(, )} P (A) =, P (B) =, P (A B) = P (A)P (B) = P (A B) 4 A, B b g Ω = {(b, b, b), (b, g, b), (b, b, g), (b, g, g), (g, b, b), (g, b, g), (g, g, b), (g, g, g)} A B A = {(b, g, b), (b, b, g), (b, g, g), (g, b, b), (g, b, g), (g, g, b)} B = {(b, b, b), (b, g, b), (b, b, g), (b, g, g)}, A B = {(b, b, g), (b, g, b), (g, b, b)} P (A) = 3 4, P (B) =, P (A B) = 3 8 P (A)P (B) = P (A B) A, B 4.. a, b n r n r A a B b

22 () a b P (A) = r n, P (B) = r n, P A(B) = r n = P (B) A, B () a b P (A) = r n, P (B) = r n, P A(B) = r n P (B) A, B 4.. (5) () 4 () (3) () 67 67, (), (3) 6 4, (4) 3 A = {, 4, 5 }, B = {3, 6 } (), (), (3), (4) A = { 3 }, B = { } A B = (), (), (3), (4) 4.. () = 67 4 () () A B = () (3) 3 (5) 4.. X A = {X =, 3, 6} B = {X =, 5} (): (): (3): (4): (5): 4.. (5) 4.3. X A = {X =, 3, 6} B = {X =, 5} (): (): (3): (4): (5): 4.3. (3)

23 3 N H, H,..., H N Ω = H i i =,,..., N P (H i ) P (A H i ) P (H i A) N N H i H H H N i= N 4.4. H, H,..., H N Ω = H i N A Ω P (A) = P (A H i )P (H i ) i= ( ) ( N N N ). A Ω Ω = H i P (A) = P (A Ω) = P A H i = P A H i i= ( 3.9) N N P (A) = P (A H i ) = P (A H i )P (H i ) i= i= i= i= i= i= i= H i i =,,..., N P (H i ) P (A H i ) P (H i A) 4.5 ( ). H, H,..., H N Ω = N H i A Ω P (A) > i= P (H i A) = P (A H i )P (H i ) N j= P (A H j)p (H j ) (i =,,..., N). 4.4 P (H i A) = P (A H i) P (A) = P (A H i )P (H i ) N j= P (A H j)p (H j ) U, U, U 3 4, 5, 6 6, 5, 4 U, U, U 3 (),, 3 U () 4, 5 U (3) 6 U 3 U

24 H, H, H 3 U, U, U 3 A P (H ) =, P (H ) = 3, P (H 3) = 6, P (A H ) = 6 45, P (A H ) = 45, P (A H 3) = 5 45 P (A H ) = = 6 45 P (A H )P (H ) + P (A H )P (H ) + P (A H 3 )P (H 3 ) = = P (H A) = = 53 = ( ). 4 4 () () 4.7. () 4 /4 () 4 / 3/4 / = 3/8 () P ( ) = 4 P ( ) = 3 4 P ( ) = P ( ) =

25 5 P ( ) = P ( ) P ( ) + P ( ) P ( ) = = X X (a), (b) 3, 4 (c) 5 (d) = = 9 = = = = 3

26 6 Part. 5. () b a f(x) dx y x, x dx x y x, x log x dx y log x, < x <

27 x 5.. (). 5. (). () f : [a, ) R b > a f [a, b] (5.) a f(x) dx = lim R R a f(x) dx a f(x) dx () f : [a, c) R a b < c f [a, b] (5.) c a f(x) dx = lim R c R a f(x) dx c a f(x) dx (3) f : [a, c) (c, b] R a A < c, c < B b A, B f [a, A] [B, b] (5.3) b a f(x) dx = lim A c A a f(x) dx + lim B c b B f(x) dx (4) 5.. b a f(x) dx

28 8 () () x a dx [log x] R x a [ dx = a + xa+ log R x a dx = R a+ a + x a dx = ] R R lim R (a = ) (a ) (a = ) (a ) = a + x a dx (a ) (a < ) x a dx x a dx = lim x a dx ε ε [log x] x a ε (a = ) [ ] dx = a + xa+ (a ) log ε x a dx = ε a+ a + (3) (4) x a dx = lim ε (a = ) (a ) = a + cos x dx ε ε x a dx + lim R R R x a dx = (a ) (a > ) cos x dx = sin R R dx (5) x dx x = lim dx ε ε ε x + lim dx ε x ( (6) x + ) dx x + dx (7) = π x (8) log x dx = lim ε = lim ε ε log x dx + lim ε lim ε log ε = ε log x dx = lim ε ε log x dx + lim ε ε log x dx = lim ε [x log x x] ε ε log( x) dx ε log x dx = lim ε ( ε log ε + ε ) =

29 a < b m, n N b a (x a) m (b x) n dx = m!n! (b a)m+n+ (m + n + )! b. (x a) m (b x) n dx = n a m () n 3 π sin n x dx = () n π sin n x dx = π π cos n x dx = n n cos n x dx = n n b a (x a) m+ (b x) n dx n 3 n 3 n 3 n π. t = π x n (sin n+ x cos x) = (n + ) sin n x cos x sin n+ x = (n + ) sin n x (n + ) sin n+ x (5.4) (n + ) π π sin n x dx = (n + ) sin n+ x dx () () π dx = π π sin x dx = n π sin n x dx π sin n x dx = π. (5.4) n π sin n x dx π π sin n x dx = (n ) sin n x dx π sin n x dx = = π

30 Γ. Γ(t) = 5.6. u t e u du exp( x ) dx = π. 3 n x n exp( x ) ( x 4n x n ) 4n exp( x ) e 3 n x x n ) 4n exp( x ) ( x 4n = exp( x ) { exp e 3 n x ( ))} x + 4n log ( x 4n t 4 t + log( t) t, exp( t) t exp( x ) dx ) 4n exp( x ) ( x 4n { )} exp( x ) exp ( x4 4n { ( exp( x ) exp )} 4n /3 4n /3 exp( x ) n exp( x ) dx = lim n n n ( x n x = sin θ n 5.5 (5.4) π lim 4n n ( x ) 4n dx = 4n 4n ( x sin 8n + θ dθ = lim n 4n ) 4n 4n ) 4n dx exp( x ) dx + n 3 /3 n dx = lim n ( x ) 4n dx n π ( x ) 4n dx = 4n sin 8n + θ dθ π π sin 8n θ dθ sin 8n + θ dθ = π nc k n! n! 5.7 ( ). lim n n! n n+ e n = π

31 . n! = 3 t n e t dt t = n(s + ) n! n n+ e = n n s n ( s ) n ds n n! = n n+ e n ((s + )e s ) n dt ((s + )e s ) n dt max(, (s + )( s)) (s + )e s e s / ((s + )e s ) n dt n n 5.5 π n! n n+ e = n ((s + )e s ) n dt π n e ns / ds = π f(x, y) dx dy d c b a [a,b] [c,d] ( ) b f(x, y) dx dy x y a ( ) d f(x, y) dy dx y x ( 4 ) 6.. x y 3 dx dy = x y 3 dy dx = 6.. [,] [3,4] c x dx = 75 () [, ] [3, 4] x, 3 y 4 x 3, y 4 () x y 3 dx dy x y 3 dx dy [,] [3,4] [,] [3,4] (3) dx dy dx R F (x, y) x + y R f(x, y) f(x, y) dx dy = χ D (x, y)f (x, y) dx dy x +y R [ R,R] [ R,R]

32 3 6.3 ( ). f D = {(x, y) R : x + y R } (6.) f(x, y) dx dy = f(r cos θ, r sin θ)r dr dθ D [,R] [,π] g [, R] g( R x + y ) dx dy = π rg(r) dr D X = ax + by, Y = cx + dy D D (6.) f(x, Y ) dx dy = f(ax + by, cx + dy) ad bc dx dy D D ( ). f B(R) = {(x, y, z) R 3 : x + y + z R } f(x, y, z) dx dy dz = {x +y +z R } ( π π ( ) ) R f(r sin θ cos φ, r sin θ sin φ, r cos θ)r sin θdr dθ dφ f( x + y + z ) ( ). f [, R] f( R x + y + z ) dx dy dz = 4π f(r)r dr {x +y +z R } f [, R] f( x + x + + x N ) dx dy dz {x +x + +x N R } = N x + x + + x N = R f(r)r N dr

33 33 N x + x + + x N = N N N N N 6 (6.3) A = A N = N N(N ) N(N ) N N N(N ) y y. y N = A N x x.. x N x + x + x 3 + x 4 + x N N N N N N x x = (x + x ) x (x + x + x 3 ) 3 x 4. N (x + x + x 3 + x x N ) + x N N(N ) N(N ) (x, x,..., x N ) (y, y,..., y N ) x + x + + x N = y + y + + y N 6.8. = f(x, x, x 3,..., x n ) dx dx dx 3 dx n f(y, y, y 3,..., y n ) dy dy dy 3 dy n

34 34 Part 3. x π, y y sin x y sin x y sin x = π y sin x π Ω M P (Ω, M, P ) (Ω, M, P ) Ω ( ) 7. ( ). () {x, x,..., x n,...} () (a, b) 7.. Ω = {(a, b) : a, b 6, a, b } X : Ω R X(a, b) = a X A χ A { (x A) χ A (x) = (x / A) 7.5. k =,, 3,... k k

35 35 (), 3, 4, 5, 6 () = 5 6, 3, 4, 5, 6 (3) [, ] ,.,.3,, 4,.35, e = x, y x y 5 x, y x y 7.7. () 47 7 > 47 7 () x y x = y (3) (/3) 4 (4) x y x = y (). () X : Ω {, ±, ±,...} n p n = P (X = n) {p n } {n=, ±,...} () X : Ω R a b a, b P (a X b) {P (a X b)} a b P (a X) a (3) K X : Ω K K A P (X A)

36 36 8. ( ). µ R ( ) X µ P (X (a, b)) = µ((a, b)), a < b a, b µ = P X 8.3. X Ω = {,, 3, 4, 5, 6} Ω = {, } p X 8.4. X : Ω X () ω Ω = {(, ), (, ), (, ), (, )}, X = {a, b} X((, )) = X((, )) = X((, )) = a, X((, )) = b ω (,) (,) (,) (,) P ({ω}) /4 /4 /4 /4 () p X p X X ω a b p X ({ω}) 3/4 /4 X, X,..., X n P (X > a ) = P (X > a ) = = P (X n > a n ) R P (a X b) 8.5 ( ). R ν ν((a, b)) = b a f(x) dx, < a < b < f ν f ν ν F (x) = x f(y) dy 8.6. ν f f(y) dy =

37 ( ). A χ A, A { (x A ) χ A (x) = A (x) = (x X \ A = A c ) 8.8 (). U(a, b) f(x) = b a χ (a,b)(x) 8.9. U(5, 8) X P (6 < X < 7) 8.9. f(x) = 3 χ (5,8)(x) 7 8 P (6 < X < 7) = 6 3 χ (5,8)(x) dx = 5 3 dx = 3 8..,,, 3, 3,, 4, 4, 3 4,, 5, 5, 3 5, 4 5,... a, a,... lim N N {n =,,..., N : a < a n < b} = b a a, b a, b a, a,..., a N m(m + 3) N = m + = N m l( [, m]) a, b a, a,..., a N l a < a n < b a n [lb] [la] N {n =,,..., N : a < a n < b} = N lb la < [lb] [la] < lb la + N = m N m(m + 3) N {n =,,..., N : a < a n < b} N m lim N m ([lb] [la]) l= m l(b a) < m N l= N {n =,,..., N : a < a n < b} = b a 8.. X f(x) = αx( x)χ (,) (x) x R () f α () P (X > /) (3) P (/4 < X < /3)

38 () P ( < X < ) = f(x) dx = = α = 6 () P (X > /) = / (3) P (/4 < X < /3) = f(x) dx = αx( x) dx = α 6 6x( x) dx = [ 3x x 3] = 3 / = /3 /4 6x( x) dx = [ 3x x 3] /3 /4 = = () ( ) () ( ) 8.. () x x x E = ( x) dx + x dx = () A, B x, y E = 4 A, B = max( x, y) A B = max( x, y) A B = max(x, y) ( = = = = y ( y y dy + A, B = max(x, y) ) max( x, y) + max( x, y) + max(x, y) + max(x, y) dx dy max(x, y) dx dy max(x, y) dx dy + ) y dx dy + x y ( ) x dx dy y ( y ) dy = 3 max(x, y) dx dy

39 X 8.5 X F (α), α R F (α) = P (X > α) F (α) 8.3. X F (α) () lim F (α) = α () lim F (α) = α. () lim F (α) = = α () lim F (α) = = α 8.4. X F (α) p, q F (α) = p tan α + q p, q P ( < X < ) tan tan ( π/, π/) tan x lim α F (α) = lim F (α) = π α p + q =, π p + q = p = π, q = P ( < X < ) = P (X > ) P (X ) = ( π π 4 + ) = 4

40 4 F (α) P (X A) = P ({X A}) P (X = a) = P ({X = a}) 9. ( ). () X, Y A, B P (X A, Y B) = P (X A)P (Y B) () X, X,..., X n B, B,..., B n P (X B, X B,, X n B n ) = P (X B )P (X B ) P (X n B n ) (X, Y, Z) a, b P (X = a, Y = a, Z = b) = P (X = a, Y = b, Z = a) = P (X = b, Y = a, Z = a) = P (X = b, Y = b, Z = b) = 4 P (X = Y = a) = 4, P (X = a, Y = b) = 4, P (X = b, Y = a) = 4, P (X = Y = b) = 4 P (X = a) =, P (X = b) =, P (Y = a) =, P (Y = b) =

41 4 P (X = a, Y = a) = P (X = a)p (Y = a) = 4, P (X = a, Y = b) = P (X = a)p (Y = b) = 4, P (X = b, Y = a) = P (X = b)p (Y = a) = 4, P (X = b, Y = b) = P (X = b)p (Y = b) = 4 X, Y Y, Z Z, X X, Y, Z = P (X = a, Y = a, Z = a) P (X = a)p (Y = a)p (Z = a) = 8 X X X, X,..., X n a, a,..., a n X P (X = a, X = a,, X n = a n ) = P (X = a )P (X = a ) P (X n = a n ) R X, X,..., X n a, a,..., a n R P (X > a, X > a,, X n > a n ) = P (X > a )P (X > a ) P (X n > a n ) 9.3. a + a + a a k k +,,,..., 9.a a a 3 a + a 4 + a a k +, k.a a a 3.a a a 3 () () [, ) A A [, ) () [, ) x k x =.x x x k () = k (x k {, }) j=. () x k =. () k =,,... x [, ) X k (x) = x k X k, X, X,..., X k,... X k = k X k = k () x () a, a,..., a k {, } n P (X = a, X = a,, X k = a k ) (3) X, X,..., X k 9.4. ()

42 4 () x X = a, X = a,, X k = a k x =.a a a k + α k (α (, )) [, ) x k P (X = a, X = a,, X k = a k ) = k (3) () P (X k = a k ) = k P (X = a, X = a,, X k = a k ) = P (X = a )P (X = a ) P (X k = a k ).,..... A B 3 A B () A, B 6 () A B : A B 8 4 (3) A 3/4 B /4 A 9 B 3 () (3). ( ). E[X] () X a, a,..., a N P (X = a k ) = p k E[X] := N a k p k () X a, a,... P (X = a k ) = p k E[X] := a k p k (3) (a, b) X f(x) E[X] = b a k= k= xf(x) dx a = b =

43 43 E[X] E(X).3.,, 4, 9, 4 ( ) = 6 5 5, 4, 6, 8, 96 ( ) = E[X] = E[X ].4. P (X (a, b)) = b X x π( + (x + ) dx = lim ) R,R.3 a π( + (x + ) ) dx R R x π( + (x + ) ) dx (). a, b X, Y () E[aX + b] = ae[x] + b () E[X + Y ] = E[X] + E[Y ] (3) X, Y E[XY ] = E[X]E[Y ]. X a, a,..., a N Y b, b,..., b M 3 N N N () E[aX+b] = (aa j +b)p (X = a j ) = a a j P (X = a j )+b P (X = a j ) = ae[x]+b j= j= N N () E[X] = a j P (X = a j ), E[Y ] = b k P (Y = b k ) N N E[X] = a j P (X = a j ) = j= j= a j M k= k= j= P (X = a j, Y = b k ) = N j= j= k= M a j P (X = a j, Y = b k )

44 44 E[Y ] = j= k= N j= k= M b k P (X = a j, Y = b k ) N M E[X + Y ] = (a j + b k )P (X = a j, Y = b k ) = E[X] + E[Y ] (3) P (X = a j )P (Y = b k ) = P (X = a j, Y = b k ) N N N N E[X]E[Y ] = a j b k P (X = a j )P (Y = b k ) = a j b k P (X = a j, Y = b k ) = E[XY ] j= k= j= k= A χ A.6. () x < + x + x + + x n + () x < + x + 3x + + n x n + (3).6. () x () + x + x + + x n + = x + x + 3x + + n x n + = ( x) (3) n n = n n = n=.7. r H h V v X v V () H u (H u) du () h H P ( h h ) (3) P ( h h ) = h f(s) ds f(p) p f(p), p H

45 45 (4) E[h].7. () H u(h u) du = H4 () P ( h h ) = h = H3 (H h ) 3 (3) () f(p) = d dp P ( h p) = d H 3 (H p) 3 dp (4) E[h] = H x + y = h f(h ) dh = H 4 H 3 H 3 = 3(H h) H 3, p H.8. x + y = P A = (, ) () P = (cos θ, sin θ) AP θ π () θ [, π] AP.8. () AP = ( cos θ) + sin θ = cos θ = 4 sin θ = sin θ = sin θ θ π () E[AP ] = π sin θ π dθ = 4 π P (X = k) = ( X( ) P (a X b) = b a ) k 4e 4x dx () m

46 46 () M.8. () m = 4xe 4x dx = e 4x dx = 5 4 () M M = k 4 ( ) k 4 5 = ( ) 75...,, 3, 4, 5 8, 9, 3, 3, 3,, 4, 4, 7 3. ( ). X V [X] := E[(X E[X]) ] = E[X ] E[X] V [X] X V [X] V (X) E[(X E[X]) ] = E[X ] E[X].5 E[E[X]X] = E[X]E[X] = E[X] E[E[X] ] = E[X].. () X a, a,..., a N P (X = a k ) = p k ( N N ) V [X] = a k p k a k p k k= () X a, a,... P (X = a k ) = p k ( ) V [X] = a k p k a k p k k= (3) (a, b) X f(x) ( b ) b V [X] = x f(x) dx xf(x) dx a = b = V [X] a k= k= a

47 47. ( ). a, b X V [ax + b] = a V [X]. (ax + b ae[x] b) = a (X E[X]).5.3 ( ). X, Y V [X+Y ] = V [X]+V [Y ]. V [X+Y ] = E[(X+Y ) ] E[X+Y ] E[X+Y ] = E[X]+E[Y ] V [X + Y ] = E[X + XY + Y ] E[X] E[X]E[Y ] E[Y ] X, Y.5 E[XY ] = E[X]E[Y ] V [X + Y ] = E[X ] E[X] + E[Y ] E[Y ] + E[XY ] E[X]E[Y ] = V [X] + V [Y ].4. V [X + Y ] = V [X] + V [Y ] V [X Y ] = V [X] V [Y ]..3 V [X Y ] = V [X + ( Y )] = V [X] + V [ Y ] = V [X] + ( ) V [Y ] = V [X] + V [Y ].3.5. X, Y X = X = X = Y = /4 Y = /4 /4 Y = /4 E[XY ] = E[X] = E[Y ] = P (X =, Y = ) = P (X = )P (Y = ) X, Y.6 ( ). E[XY ] E[X ]E[Y ] X = ty Y = tx t. X = φ(t) = E[ tx Y ] = t E[X ] te[x]e[y ] + E[Y ] t D D = 4E[X] E[Y ] 4E[X ]E[Y ] D 4E[X] E[Y ] 4E[X ]E[Y ] E[XY ] E[ X ]E[ Y ] , 35, 57, 49, 55 X.7. E[X] = ( ) = 5 5 V [X] = 5 {(54 5) + (35 5) + (57 5) + (49 5 ) + (55 5) } = 63. V [X] = 5 { } 5 = 63.

48 48.8. [, ] 6 X f(x) = x( x) V [X] = 88 x( x) = 36 (x 6) V [X] = 88 36(x 6) dx 88 (x 6) x( x) dx (x 6) 4 dx = a > X f(x) { f(x) = ax 3 a x ( x)χ [,] (x) = 3 ( x) ( x ) ( ) () a () E[X] (3) V [X] ( (4) P X ) ( 3 ) = = () () f(x) dx = a = a =. x f(x) dx = a 3 = 3. (3) E[X ] = x f(x) dx = a 4 = V [X] = 4 9 = = 63. ( (4) P X ) / = (x 3 x 4 ) dx = 3 8

49 (). X V [X] σ[x] σ[x] σ(x).. X P (X = ) = P (X = ) = P (X = 8) = P (X = 9) = P (X = ) = 5. E[X] V [X] σ[x].. E[X] = ( ) = 6 5 V [X] = 5 (( 6) + ( 6) + (8 6) + (9 6) + ( 6) ) = 4 σ[x] = V [X] = 4.. X () X, X,,. P (X = ) = 4, P (X = ) = 8, P (X = 4) = 4, P (X = 6) = 8, P (X = 8) = 4. () 3 X. X,,... () E[X] = = 7 4 V [X] = ( 7 ) + ( 7 ) + ( 4 7 ) + ( 6 7 ) + ( 8 7 ) = σ[x] = 5 σ[x] = 4 () P (X = ) = 8, P (X = ) = 3 8, P (X = ) = 3 8, P (X = 3) = 8 () E[X] = 3, V [X] = 3 4, σ[x] = 3.3 (). 5 5, 39, 3, 99, 97, 5, 99, 48, 98, 57, 5 96, 8, 96, 97, 5, 93, 4, 38, 3, 3, 9,, 99, 8 (),, 3,..., 3 () X E[X] σ[x] σ[x]

50 5 (3) X A 5 + A E[X] σ[x] σ[x] () 7.3. () 44, 484, 59, 576, 65, 676, 79, 784, 84, 9 () E[X] = 8, σ[x] = 3 (3) = P (a X b) = 3 4 b a x( x) dx, a b [, ] X X.4. E[X] = 3 4 V [X] = 3 4 σ[x] = 5 x ( x) dx = = x 3 ( x) dx = ! 5! = = 5.5. T f : R R { f(t) = aχ [,] (t)t T + a t T + ( t ) = ( ) () f(t) X a () a P ( X, 5) (3) E[X] (4) V [X] (5) σ[x].5. () f(t) dt = a T + a = T + () a P ( X.5) = (3) E[X] = (4) V [X] = t (T + )t T + dt = T + T + 3 t (T + )t T + dt ( T + T + 3 ) = T + T + (T + 4)(T + 3) f(t) dt = T +

51 5 (5) σ[x] = V [X] = T + T + 3 T ( ). n n x, x,..., x n () x, x,..., x n () (3) (4) x = x j n (5) S = n j= (x j x) (6) (a) n n/ (b) n (n )/ (n + )/ j=.. 5,, 4, 3, 5,,, 4, 3, 5, 4, 5, 5, 5, 5, 3, 5, 5,,,,,,, 4, 3, 5, 4, 5, 5, 5, 5, 4, 5, 3, 5, 5, 3, 4,,, 3, 3,,, 3, 3, 4, 3,, 5, 5, 4, 4, 4,, 4, 4, 4, 4,, 4, ( ).

52 5 48, 34,, 35, 84, 77, 87, 75, 35, 85, 45, 78,, 63,, 48, 86, 4, 5, 63, 4, 54, 6, 9, () X, X,..., X n () () (3).5 () , 4, 3, 34, 34, 99, 73, 3, 5, 43

53 ,, 3, 3 4 X X X.7. P (X = 3) = /6, P (X = 4) = /3, P (X = 5) = /3, P (X = 6) = /6.8. 5,, 8, 9,.8. ( ) = (( 6) + ( 6) + (8 6) + (9 6) + ( 6) ) = 5 (( 5) + ( 4) ) = 4.9. N n µ σ N a, a,..., a N n X, X,..., X n () X = X + X + + X n () S = X + X + + X n X n n i<j n i, j a ij i<j n n a ij = i= j=i+ (a + a + + a n ) = a j + j= a ij i<j n a i a j

54 54.9. Ω = {(a i, a i,..., a in ) : a i, a i,..., a in } () Ω E[X] = n Ω i,i,...,i n,,..., n (a i + a i + + a in ) a {i, i,..., i n } i, i,..., i n n n N C n a n N P n Ω = N P n E[X] = N n N P n a j = n N P n j= (N n)! N! (N )! (N n)! N a j = N j= N j= a j () [ X + X ] + + X n E = n N N a j = σ + µ j= E[X X ] = i<j N i<j N i= a i a j P (X = a i, X = a j ) = i= N(N ) i<j N a i a j ( N ) N N a i a j = a i a i = N µ a i = N µ Nµ Nσ E[X X ] = µ N σ X, X X i, X j (i j) n i<j n i= E[X i X j ] = n n µ n n(n ) σ E[X ] = n n n µ nn(n ) σ + n µ + n σ = µ + n σ n n(n ) σ E[S ] = n n σ + n N(n ) n(n ) σ = n(n ) σ

55 55... X Y X Y (). X, Y r[x, Y ] = Cov[X, Y ] σ[x]σ[y ] Cov[X, Y ] = E[(X E[X])(Y E[Y ])] σ[x]σ[y ] V [X] = E[(X E[X]) ] = E[X ] E[X]. (). X, Y () Cov[X, Y ] = E[XY ] E[X]E[Y ] () V [X + Y ] = V [X] + V [Y ] + Cov[X, Y ] (3) X, Y Cov[X, Y ] = V [X + Y ] = V [X] + V [Y ].6.3 ( ). r[x, Y ] = Y = ax +b a > b r[x, Y ] = Y = ax + b a < b

56 56.4 (). Cov[X, Y ], r[x, Y ] X Y () E[X] = ( ) = 63. () E[Y ] = ( ) = 58.4 (3) E[X ] = ( ) = 46.3 (4) E[Y ] = ( ) = 4.4 (5) V [X] = E[X ] E[X] = (6) V [Y ] = E[Y ] E[Y ] = 6.84 (7) σ[x] = V [X] = 4.87 (8) σ[y ] = V [Y ] = 4.74 (9) E[XY ] = ( ) = () Cov[X, Y ] = E[XY ] E[X]E[Y ] = 78.6 E[XY ] E[X]E[Y ] () r[x, Y ] = =.9 σ[x]σ[y ]

57 57.5 ( ) ( ) ( )

58 58.8 ( ) ( ) X Y X Y X Y

59 59.. X Y (X, Y )

60 6 3.96,,.89 k > b Y = k X + b k > b Y = k X + b.. Y a X Y b 9 a.. Y = 6X + 97 a = 43 b = 55.. D = {(x, y) R : x, y, x + y } f(x, y) = αχ D (x, y)( x y) α () α () E[X], E[Y ] (3) E[XY ] (4) E[X ], E[Y ] (5) V [X], V [Y ] (6) Cov[X, Y ] (7) r[x, Y ] χ D (x, y)( x y).. () ( x y) dx dy = α = 6 6 D

61 () E[X] = 6 D x( x y) dx dy = 6 6 ( x E[X] = 6 6 y( x y) dx dy = D 4 (3) E[X ] = 6 x ( x y) dx dy = 6 D x( x) dx = 4 ( x ) x( x y) dy dx y ) x ( x y) dy E[Y ] = dx y x E[X ] = 6 x ( x) dx = E[Y ] = 6 y ( x y) dx dy = D ( x ) (4) E[XY ] = 6 xy( x y) dx dy = 6 xy( x y) dy dx y D x E[XY ] = (5) V [X] = V [Y ] = E[X ] E[X] = 6 = 3 8 x( x) 3 dx = (6) Cov[X, Y ] = E[XY ] E[X]E[Y ] = 6 = 8 (7) r[x, Y ] = Cov[X, Y ] = σ[x]σ[y ] 3.3 ( ). X 3, 64, 68, 7, 4, 59, 5,, 87, 6.3. () X ()

62 6.4. r[x, Y ] X Y X Y X Y XY () X = = () Y = = (3) = = (): (): (3): (4): (5):

63 63.5. () (): ():(3): (4): (5):.6. (3) (): (): (3): (4): (5):.7. (5).8. () () (3) (4) (5).8. () x y s cm () (3) (4) (5) x y

64 ( ). x, x,..., x N () N (x + x + + x N ) () N x x x N { ( (3) + + )} N x x N.3. () ().3. () : 89588, : 887, 3 : 8739, 4 : 8633, 5 : X X X 4 () () X r X () r = , r 3 = , r 4 = , r 5 = r = 4 r r 3 r 4 r 5 = =.9897 = ( ) ( ). 45, 4, 5, 74, 49, 5, 3, 9, 79, 78 A B C A > B > C.3. A = 5.8 > B = 4.4 > C = ( ). x, x,..., x N x x x N () (a) N x (N+)/ (b) N (x N/ + x N/+ )/ () y = x j (j =,,..., N) j y

65 () 56, 88, 6, 88,, 4, 89, 4, 8, 37, 49, 66, 76, 35, 6, 8, 5, 86, 4, / + 56/ = 53.5 () 9 73, 98, 3, 58, 79, 96, 3, 73, 37, 73, 44, 36, 7,, 6, 9, 74, 44, ( ). 3, 5,,, 3,, 7,,, 8, 7, 8,, 5, 7, 9, 7, 6, 3, 6, ( ) :.37 ( ). x, x,..., x N N (x j m) j= = 54.7 = , 5..,.5.39 ( ) ,.67

66 , ( ). 6,, 3, 4, 5 5 A A A B C D E F A B C D E F () A (5 3.7)/.46 = () A ( 3)/.63 = (3) A = ( ). x, x,..., x N x x x N () x N () x (3) x N x.4. 84, 96, 553, 859, 33, 96, 94, 545, 787 () 96 () 94 (3) ( 3 ). x, x,..., x N x x x N () < p < p A (N + )p/ q r A = ( r)x q + rx q+ () 5 (3) 3 75 (4) , 58, 6, 353, 387, 47, 56, 68, 6, 78

67 ().5 = = 6.5 ().75 = = (3) = ( ). 9, 95, 39, 3, 55, 9, 96, 4, 7, 3, 54, 7, 53, 75, 68, 7, 8, 4, 9, () 5 = 3 () 6 = 39 (3) = 55 (4) = 68 (5) 5 = 75 (6) 6 = 8 (7) 7 = 84 () 5 = 33 () = 6.5 (3) 75 3 = 79.5 (4) 8 = 84. (5) = ( ). x, x,..., x N σ > () () n (n )(n ) n(n + ) (n )(n )(n 3) ( ) 3 xi m σ i= ( ) 4 xi m σ i= 3(n ) (n )(n 3).47.,,, 6,, 5, 6, 7,, 4, 6, 6, 9,, 5

68 () () (3) (4).3. (b) (X ) (c) (X ) 3 (d) (X ) X (a) (b) (c) () 4 () 7.33 (3) n = 5 = n (n )(n ) (4) n = 5 = ( ) 3 xi m = 5 σ 4 3 ( ) = =.84 i= n(n + ) (n )(n )(n 3) ( ) 4 xi m 3(n ) σ (n )(n 3) i= 5 6 = = ( ).

69 69 5, 4, 7, 5, 9, 6, 5,, 5, 8, 6,,, 5, 5, 8,, , ( ). Z + = {,,,...} X p x = P ({x}) g(z) = p x z x X x= { λ k.5. λ g(z) = x= λ x x! e λ z x = e λ+λxz k! e λ } k Z.5. p(n) n + p(n)z n = ( z k ) p(5).5. k= j k j n= k= ( z k ) = ( + z j + + z kj + ) j =,,... j= k + k + + n k n + = n z k z k z 3k3 z n kn + p(n)z n z 6 n= ( + z + z + z 3 + z 4 + z 5 )( + z + z 4 ) = + z + z + z 3 + z 4 + z 5 + z + z 3 + z 4 + z 5 + z 4 + z 5 = + z + z + z 3 + 3z 4 + 3z 5 ( + z 3 )( + z 4 )( + z 5 ) = ( + z 3 + z 4 + z 5 ) z 6 ( + z j + + z kj + ) j= = 5 ( + z j + + z kj + ) j= = ( + z + z + z 3 + z 4 + z 5 )( + z + z 4 )( + z 3 )( + z 4 )( + z 5 ) = ( + z + z + z 3 + z 4 + z 5 )( + z + z 4 )( + z 3 )( + z 4 )( + z 5 ) = ( + z + z + z 3 + 3z 4 + 3z 5 )( + z 3 + z 4 + z 5 ) = + z + z + z 3 + 3z 4 + 3z 5 + z 3 + z 4 + z 5 + z 4 + z 5 + z 5 = + z + z + z 3 + 5z 4 + 7z 5

70 7 z 5 7 p(7) = ( ). X p > P ( X > λ) λ p E[ X p ] p = P ( X E[X] > λ) V [X] λ. P ( X > λ) = E[χ { X >λ} ] λ p χ { X >λ} X p... ( ). ( ( ) ). X, X,..., X k,... lim n n (X + X + + X n ) = E[X ] 7..3 ( ( ) ). X, X,..., X k,... X M M [ lim E n n (X ] + X + + X n ) E[X ] =. Y = X E[X ], Y = X E[X ],... Y, Y,..., Y n X, X,..., X n E[X ] = E[X ] = = E[X n ] = [ E n (X ] + X + + X n ) = n E[X + X + + X n ] + n E[X i X j ] i<j n = n E[X + X + + X n ] + n = n E[X ] i<j n E[X i ]E[X j ].4.

71 7 () X, X,..., X n X X + X + + X n lim = E[X ] n n () 45 (3) (4).4. () () (3) (4) N /N!.5. [, ] x, y x y [, ] x, y sin x sin y x, y x y 4 4 X, X,..., X N,... P (X = ) = P (X = 3) = / N (X + X + + X N ) ( ). X, X,..., X k,... lim n X + X + + X n ne[x ] n

72 7 V [X ] N(, V [X ]) f [ ( )] X + X + + X n ne[x ] f = n lim E n ( ) t f(t) exp dt πv [X ] V [X ] {Y n } n= µ f lim E[f(Y n)] = n f(t)dµ(t) [a, b] χ [a,b] f(t) χ [a,b] (t) g(t) f, g f g I g(t)dµ(t) f(t)dµ(t) dµ(t) µ(y I) g(t)dµ(t) I χ [a,b] (t)dµ(t) f(t)dµ(t) Y (a, b) lim P (Y n I) = µ(i) n k X k n (X + X + + X n ) n [ ] E n (X + X + + X n ) = E [X + X + + X n ] = n E[X ] = ne[x ] n n V [ ] n (X + X + + X n ) = n V [X + X + + X n ] = n n V [X ] = V [X ] n X v, v / ( ) P (X + X + + X n ) (a, b) b ) exp ( x n πv a v dx a n a n b n lim = n b n n! πn n+ e n. P (X + X + + X n = k) = X, X,..., X n n + k = n nc n+k n! nc n+k n n (n) n+ π(n + k) n+k+ (n k) n k+ = n n+ π(n + k) n+k+ (n k) n k+

73 x = v k n 73 nc n+k n nn+ π (n + ) n n n v x ( x n v ) n+ n n v x x v = ( + ) n n v x ( x ) n+ n v x x πn nv nv = ( v ) n ( n πv nv x + ) n v x ( x ) + n v x x nv nv ( lim n ( x = exp = exp nv x v ( x v ) exp ) ) n ( + ) n v x ( x ) + n v x x nv nv ( x v ) ) exp ( x v v n nc n+k n (a, b) ( ) v exp x n πv v P ( ) (X + X + + X n ) (a, b) n πv b a ) exp ( x v dx.8. P (X = ) = P (X = ) = / {X j } j= X + X + + X n n N(, ) [a, b] a, b ) b a P b a ( a < X + X + + X n < b n b a π exp ) ( x dx exp π n X + X + + X n n, n +, n + 4,...,,..., n 4, n, n ) ( a / n () n =

74 74 k E E E E E E () n = k

75 75 (3) n = 4 k () () /4 9

76 X, X,..., X N,... P (X = ) = P (X = ) = / N (X + X + + X N ).... n = 5, 6, 7 n C k k n+k.. () (a) n = 5 n (b) n = 6 n

77 (c) n = 7 n

78 78 Part n 3. (). X,,,..., n P (X = k) = n C k p k q n k X B(n, p) p q = p n p q = p ( n p(x; n, p) = p x) x q n x, x =,,,..., n 3.. P P + 6 P X 3.. X p /64 3/3 5/64 5/6 5/64 3/3 / ABCDE A, B, C, D, E A 7 X X = A, B, C, D, E q X q X, X = A, B, C, D, E A E D C B A E D /8 7/8 /8 35/8 35/8 /8 7/3 /8 p A = 7 C 7 = 7 64, p B = p E = 35 8, p C = p D = = ( ). a, b (3.) k a k b n k nc k = n a (a + b) n k=

79 79 (3.) k(k ) a k b n k nc k = n(n ) a (a + b) n k=. (a + b) n = nc k a k b n k a k= n (a + b) n = k n C k a k b n k k= a (3.) (3.) ( ). X B(n, p) E[X] = n p. p n k p k ( p) n k nc k X E[X] = k p k ( p) n k nc k (3.) E[X] = k= k p k ( p) n k nc k = n p(p + p) n = n p k= 3.6. X B(n, p) n =, p = E[X] = (). < p < q = p X B(n, p) V [X] = n p q σ[x] = n p q. P (X = k) = n C k p k q n k V [X] = k p k ( p) n k nc k n p = k= k(k ) p k ( p) n k nc k + k= k p k ( p) n k nc k n p (3.) (3.) V [X] = n(n )p + np n p = np( p) = npq 3.8. X 8 8 Y () E[X], V [X], σ[x] k=

80 8 () E[ 47X + 6], V [ 47X + 6], σ[ 47X + 6] (3) E[X + Y + ], V [X + Y + ], σ[x + Y + ] 3.8. (a) p n X E[X] = n p ( 3.5), V [X] = n p( p) = n p q ( 3.7), σ[x] = n p q ( 3.7) (b) E[a X +b] = a E[X]+b (.5), V [a X +b] = a V [X] (.), σ[a X +b] = a σ[x]. (c) X, Y E[X + Y ] = E[X] + E[Y ] (.5), V [X + Y ] = V [X] + V [Y ] (.3) () (a) E[X] = n p = 8 6 = 3 (b) V [X] = n p q = = 5 (c) σ[x] = V [X] = 5 () (a) E[ 47X + 6] = = = 35 (b) V [ 47X + 6] = ( 47) 5 = 555 (c) σ[ 47X + 6] = 555 = 35 (3) (a) E[X + Y + ] = = 8 (b) V [X + Y + ] = V [X] + V [Y ] = 5 (c) σ[x + Y + ] = V [X + Y + ] = X, X, X 3 () E[X ] () V [X ] (3) σ[x ] (4) V [7X + 58] (5) σ[x + X + X 3 ] 3.9. (a) a, b E[a X + b] = a E[X] + b (.5), V [a X + b] = a V [X] (.3) (b) X, Y, Z E[X + Y + Z] = E[X] + E[Y ] + E[Z] (.5), V [X + Y + Z] = V [X] + V [Y ] + V [Z] (.3) () E[X ] = 6 ( ) = 7 () V [X ] = 6 ( ) ( ) 7 = = 35

81 8 (3) σ[x ] = 5 V [X ] = 6 (4) V [7X + 58] = 49V [X ] = 75 (5) V [X + X + X 3 ] = V [X ] + V [X ] + V [X 3 ] = 4V [X ] + V [X ] + V [X 3 ] = σ[x + X + X 3 ] = = X 76 Y σ[x 4Y + 3] 3.. V [X] = = V [Y ] = 76 6 = 9 σ[x 4Y + 3] = V [X 4Y + 3] = 4V [X] + 6V [Y ]] = 344 = ( ). ( ) () µ R, σ > ϕ(x : µ, σ ) = exp (x µ) πσ σ µ σ () X µ σ X N(µ, σ ) (3) µ =, σ = f(x) = ) exp ( x π N(, ) (4) < ε <.5 N(, ) X P (X M) = ε M u(ε) [ ]

82 8 4.. X N(3, 4) X ) (x 3) exp ( 8π 8 Y = (X 3) E[X] = 3, V [X] = 4 E[Y ] = (E[X] 3) =, V [Y ] = 4 V [X] = Y N(, ) P ( < Y < ) =.398 P (3 < X < 3.) =.398 P (.8 < X < 3.) = X, Y,, 4 φ(a), a > () P (X 6) () P (6 Y ) φ(a) = P ( X a) (a > ) 4.3. () P (X 6) = P ( X 6) + P (X ) = φ(6) + () Z = Y Z P (6 Y ) = P ( Z ) = φ() + φ() 4.4. X φ(a) = P ( X a) φ(.) =.4, φ(.) =.793, φ(.3) =.8, φ(.4) =.55, φ(.5) =.95, φ(.6) =.6, φ(.7) =.58, φ(.8) =.88, φ(.9) =.36, φ(.) =.34, φ(.) =.364, φ(.) =.384, φ(.3) =.43, φ(.4) =.49, φ(.5) =.433, φ(.6) =.445, φ(.7) =.455, φ(.8) =.464, φ(.9) =.47, φ(.) =.477, φ(.) =.48, φ(.) =.486, φ(.3) =.489, φ(.4) =.49, φ(.5) =.494, φ(3.) =.499 () P ( X <.7), P (.3 X.), P (.5 X.4) () X N(4, ) P (38 X 43) P (45 X 45) (3) 85 6 (4) 7 5 X P (X 3) 4.4.

83 83 () P (X a), P (X > a) a > a P ( X <.7) = P ( X.7) = φ(.7) =.58 P (.3 X.) = P ( X.) P ( X.3) = =.74 P (.5 X.4) = P ( X.5) + P ( X.4) = φ(.5) + φ(.4) = =.64 () Y = X 4 Y P (38 X 43) = P ( Y 3) = φ(3) + φ() = =.976, P (45 X 45) = P (.5 X.5) = φ(.5) + φ(.5) = =.689 (3) Y = X 85 P (X 6) = P (Y.5) P (Y.5) =.6 = 6 (4) Y = X P (X 3) = P (Y 3) = () 9 = 5 + () 3 4 X 4.5. () 5 = P ( X.5) = = 6 ().95 Y P (47 Y 69.5) = C 3 C D D

84 84 D A N(, ) P (A >.58) = X 3 = Y N(, ) X = 45 Y =.6 5 Y = A N(, ) P ( < A <.64) =.4 = A N(, ) P ( < A <.53) = A N(, ) P ( < A <.53) = ( ). N(,.5) X P (X 9) P (X t) =.6 t 4.. P (X < 9) = , t = X N(, ) P (X > a) = ( ) exp t dt π a P (X > a) 4. ( ). a > P (X > a) < ) exp ( a πa a

85 85. P (X > a) = π a ( ) exp t dt < πa a ( ) t exp t dt = ) exp ( a πa ( ) ( ) 4.. exp t t dt < exp t dt = ( exp ) = < π π π πe {a i } N i= {b i} N i= a i b i (i =,,..., N) N [a, b ] [a, b ] [a N, b N ] = [a j, b j ] = {(x, x,..., x N ) : a i x i b i } j= [a, b] [c, d] [e, f], a, b, c, d, e, f R, a < b, c < d, e < f a x b, c y d, e z f 5.3 ( ). A R N ν(a) = ν f ν A f(x) dx f 5.4. X, X,..., X N f, f,..., f N (X, X,..., X N ) f f f N P (a X b, a X b,, a N X N b N ) = = b a f (x ) dx b b a a N a bn b a f (x ) dx bn a N f N (x N ) dx N f (x )f (x ) f N (x N ) dx dx dx N. P (a X b, a X b,, a N X N b N ) = P (a X b )P (a X b ) P (a N X N b N ) P (a i X i b i ) = bi a i f i (x i ) dx i (i =,,..., N)

86 X, X,..., X N a X + a X + + a N X N 5.5 (). σ, σ > X, X N(m, σ ), N(m, σ ) X + X N(m + m, σ + σ ) 5.6. N(m, σ ) + N(m, σ ) +. X m, X m m = m = ) P (X + X > a) = exp ( x πσ σ σ y dx dy σ x+y>a X = x σ, Y = y σ (6.) Z = P (X + X > a) = exp ( X + Y ) dx dy π σ X+σ Y >a σ σ + σ X + σ σ + σ Y, W = σ σ + σ X σ σ + σ Y P (X + X > a) = π = π = π Z Z>a σ +σ Z>a σ +σ a σ +σ P (X + X > a) = π σ + σ exp ( Z + W ) dz dw ( exp ( Z + W ) ( exp ( Z + W ) Z σ + σ, W W σ + σ a ) dw dz ) dw dz ( exp ( Z + W ) (σ + σ ) (6.) 5.6 W ( Z ) P (X + X > a) = exp dz π(σ + σ ) (σ + σ ) Z>a X + X ) dw dz

87 ( ). X, X,..., X N N(, σ ) N N N N Z Z X. = X. Z N X N N N(N ) N(N ) N(N ) N(N ) Z, Z,..., Z N N(, σ ) X, X,..., X N N(, σ ) a X +a X + +a N X N N(, a + a + + a N ). R = a + a + + a N V = R a R X + a R X + + a N R X N N(, ) ( a R, a ) R, an V R χ -. χ - 6. (χ - ). X, X,..., X n Y = X + X + + X n Y χ - χ m(α) = P (Y α) 6.. X, X,..., X 9 9 χ - Y = X + X + + X [, ) Y m χ - f(x) = C x m e x χ(, ) (x)

88 88 C ( m ) C = Γ = e t t m dt f(x) dx = C C (6.) (6.3). α > P (Y α) (6.) P (Y α) = ( π) n n i= yi nα exp π ( ) P (Y α) = C r n exp ( r r α, r ) (6.) = C r n exp ( r dr r = R r α, r (6.3) P (Y α) = C α ( i= R n exp( R) dr ) y i dy ) dr dθ χ - X, X,..., X N X = X + X + + X N N a a n..... a n a nn Y X Y a a n. =..... X. a n a nn Y n A N (6.3) N N N N Z Z X. = X. Z N X N N N(N ) N(N ) N(N ) N(N ) X n

89 89 N (X i X) = N N i= N (X i X i X + X ) = N i= N N (X i X) = N i= N (X i X) = N N i= N X i X + X = N i= N X i X i= N Z i N Z = N i= N i= Z i N X i X 6.4 ( ). X, X,..., X N X i N(m, S ) Y = S (X i X) Y N χ X, X, X A = i= i= Y Y = X X Y 3 X Y, Y, Y 3 Y = 3 (X + X + X 3 ), Y = (X X ), Y 3 = 6 (X + X X 3 ) ( Y = X + X + X X + X + X ( ) = X + X + X X + X + X χ - = Y + Y + Y 3 Y = Y + Y,,,... X P (X = k) = λk k! e λ (k =,,,...) k )

90 Po(3) ( ). n χ - f n (x) λ > k λ λi e i! = f k+ (t) dt i= λ. f k+ (t) k C k f k+ (t) = C k t k e t/ χ (, ) (t) C k ( C k = t k e dt) t/ = ( k+ t k e dt) t = k+ k! λ > λ f k+ (t) dt = C k t k e t/ dt = k+ C k t k e t dt λ t k e t dt = ( t k kt k k(k )t k k!)e t + C λ λ f k+ (t) dt = k+ C k (λ k + kλ k + k(k )λ k + + k!)e λ ( ) λ k = k! + λk (k )! + + e λ 6.8. λ = m + n, m, n λ 39 i= λ λi e i! >.975 λ X 8 χ - P (X > 57.5) = i= λ λi e i! = λ = 57.5 λ = 8.6 λ f 8 (t) dt = P (X > λ) > P (X > 57.5) = ( ). Po(9) X X =,,, 3, 4, 5, 6, 7, 8

91 t-. t- 6. (t- ). Y, X, X,..., X N N(, ) Z = Y N X j N N t- < ε <.5 P (Z M) = ε M t n (ε) 6.. X, X, X 3, X 4 N(, ) () Z = Z 3 = () Z 4 = X j= X + X 3 t- Z = X X + X 4 t- X 4 X 4 X + X 3 (X + X + X 3 ) 6 (X + X + X 3 ) 3 t- 6. (t- ). N t- f(x) = C ( + x N C ) N+ ( C = + x N ) N+ dx (x R). t- 6.α R P (Z > α) P (Z > α) = exp ( y + x + x ) + + x N dy dx dx... dx N (π) N+ y> α x +x + +xn N x, x,..., x N 6.7 P (T > α) = C r N exp ( y + r ) dy dr y > α N r

92 9 C P (T > ) = / C y, r 6.3 P (T > α) = C π ( )(R cos θ) N R exp ( R tan α N ( ( R π = C π = C ( ) tan α N ( ) cos N θ tan α N R R N exp R N cos N θ exp ( R N exp ) ( R dr α C P (T > α) = C tan θ = t P (T > α) = C α/ N π ( ) cos N θ dθ tan α N (t + ) N+ dt = C α ( R ) dθ dr ) ) ) dr dθ ) dr dθ (t + N) N+ dt 6.3. N = 5 t- x = tan θ 3π 6 = dx ( + x ) 3 dx ( + x /3) 3 = 3 3π 8 f(x) = 8 ) 3 ( 3 + x 5 t- 3π

93 ( ). X, X,..., X N N(m, σ ) T = N X m X = S N (X + X + + X N ), S = N (X i X) N N t- i=. T T = σ N (X + X + + X N mn) N (X j X) σ N (6.3) A N N N N N Z Z X m. = X m. Z N X N m N N(N ) N(N ) N(N ) N(N ) T = σ Z N N j= Z j σ Z /σ, Z /σ,..., Z N /σ T N j= 6.3. F -. F - X ρ X ρ X ρ 6.5 (F - ). Y χ (m) Y χ (n) Z = n m Y Y (m, n) F - (m, n) F - X, X,..., X m Y, Y,..., Y n Z = m X j Y j m n j= 6.6 (F - ). F (j, k) F - F {Cx j/ (jx + k) j+k (x ) f(x) = f j,k (x) = (x < ) = Cχ (, )(x)x j/ (jx + k) j+k j=

94 94 C f(x) dx =. P (F > α) = C exp ( X + X + + X n + Y + Y ) + + Y m dx dy D dx dy = dx dx... dx n dy dy... dy m { D = n (X + X X n ) > α } m (Y + Y Y m ) (m + n ) C P (F > ) = C X, X,..., X n Y, Y,..., Y m P (F > α) = C r n r m exp ( r ) + r dr dr r >( n m α)/ r tan θ = n m α θ ( π/, π/) r, r tan θ = x π P (F > α) = C cos n θ sin m θ dθ θ ( ) n ( ) x m P (F > α) = C ( n x + x + x + dx = C xm m α)/ ( n m α)/ (x + ) m+n y = x dx P (F > α) = C n m α m y (y + ) m+n dy = C α y m (my + n) m+n dy 6.7 ( ). m, n X, X,..., X n, Y, Y,..., Y m X, X,..., X n N(m x, σ ) Y, Y,..., Y m N(m y, σ ) X = n X i, i= Y = m m Y i, i= SX = (X i X), SY = n m i= m (Y i Y ) i= (n, m ) F- F = ns X n ms Y m S X ( ) S Y

95 95. n n n n Z Z. = Z n n n(n ) n(n ) n(n ) n(n ) W W. W m m m m m = m m(m ) m(m ) m(m ) m(m ) σ X m x X m x. X n m x σ Y m y Y m y. Y m m y Z, Z,..., Z n, W, W,..., W m F = Z j m W j n m j= F - F (n, m ) j= k 6.8. / k 6.9 ( ). k < n m = (k + ), m = (n k) 6.6 (m, m ) F - f m,m p (, ), q = p k nc i p i q n i = i= nc i p i q n i = i=k+ m p m q m p m q f m,m (t) dt f m,m (t) dt

96 96. n! (n k )!k! p n! (n k )!k! = = n! (n k)!k! t k ( t) n k dt p p t k ( t) n k dt t k (( t) n k ) dt n! (n k)!k! pk ( p) n k n! + (n k)!(k )! p t k ( t) n k dt n! k t k ( t) n k dt = nc i p i q n i (n k )!k! p i= m t x = m t + m n! ( ) k+ ( ) n k m t m I = dt k!(n k )! m t + m m t + m m p m q f m,m m, m C k C f m,m (t) dt = nc i p i q n i m p m q i= p = C = 6.. X (m, n) F - Y (n, m) F - t > P (X > t) + P (Y > t ) =. XY = P (X > t) + P (Y > t ) = P (X > t) + P (X > t ) = P (X > t) + P (X < t) = 6. ( ). (, 4) F C i > n = 9, k = 9 m =, m = C i = f,4 (x) dx > f,4 (x) dx =.5 i= 6. ( ). (, 4) F (4, ) F - m.975 m i= m = /.68

97 X,..., X 4 4 Z, Z,..., Z A = 5 (X + X + + X 5 ) B = 8 (X + X + + X 8 ) () Z = 9X +3X 7X 3 +, () Z = X +3X +X 3 +X 4, (3) Z 3 = X +X + +X 8, (4) Z 4 = (X A) + (X A) + + (X 5 A), (5) Z 5 = X Z3 /8, 5 (6) Z 6 = X + X + + X 8 8B, (7) Z 7 = A, (8) Z 8 = X + X + X 3 + X 4, Z 4 X6 + X 9 3Z 4 (9) Z 9 = 4(X + X 3 + X 4 ), () Z = 5A + X 3 + (A) (B) 6.3. () X, X, X 3 E[Z ] = 9E[X ] + 3E[X ] 7E[X 3 ] + =, V [Z ] = 8V [X ] + 9V [X ] + 49V [X 3 ] = 39 Z N(, 39) () E[Z ] =, V [Z ] = = 8 Z N(, 8) (3) χ - Z 3 χ (8) (4) χ -Z 4 χ (4) (5) t-z 5 t(8) (6) Z 6 = (X B) +(X B) + +(X 8 B) Z 6 χ (7) (7) Z 7 = X + X + + X 5 (X A) + (X A) + + (X 5 A) Z 7 t(4) X6 + +X 9 (8) Z 8 = X + X + + X 4 (9) Z 9 = (X A) + (X A) + + (X 5 A) Z 8 t() X + X 3 + X 4 3 F - 4 Z 9 F (4, 3) () E[Z ] =, V [Z ] = 6 Z N(, 6) () N(, 39) Z () Z 9 = 4 χ (4) 3 χ (3) (3) Z = N(, 6) (). T, T,... T = {T i T i } i= T Ex(λ) T f(x) = λ exp( λ x)χ (, ) (x) t > N t = min{k : T k t} t = < t < t < < t k {N tj N tj } k j=

98 98. a, a,..., a k P (N tj N tj = a j, j =,,..., k) = P (N t = a, N t = a + a,..., N tk = a + a + + a k ) = P (T a t < T a+, T a+a t < T a+a +,..., T a+a + +a k t k < T a+a + +a k +) U k = T k T k, k =,,... U + U + + U a +a + +a j t j < U + U + + U a +a + +a j + P = u +u + +u a +a + +a j t j t j <u +u + +u a +a + +a j + j=,,...,k u +u + +u a +a + +a j t j t j<u +u + +u a +a + +a j + j=,,...,k u +u + +u a +a + +a k t k λ a a a k du du... du a +a + +a j + exp(λ (u + u + + u a+a + +a k +) u a+a + +a k + t k < u + u + + u a+a + +a k + λ a a a k du du... du a +a P = + +a k exp(λ t k )

99 99 Part X, X,..., X n X, X,..., X n X, X + X + + X n, X n n X, X,..., X n 7.. A = 7, B = 76, C = 77, D = 8, E = 69, F = 65, G = 78, H = 8, I = 75, J = 74 cm () 7(cm) () 74(cm) (3) (cm) () () 7, 74 (), (), (3) A = 7, B = 76, C = 77, D = 8, E = 69, F = 65, G = 78, H = 8, I = 75, J = 74 cm X j X i i= E X j X i 9 = V [X ] i= 9 i= j= j= X j X i 7.3 (). n X, X,..., X n X j X i n n i= j= j=

100 7.4. A = 7, B = 76, C = 77, D = 8, E = 69, F = 65, G = 78, H = 8, I = 75, J = 74 cm () 74.8(cm) (A + B + C + D + E + F + G + H + I + J) = 74.8 () 3.36(cm ) ( (A 74.8) + (B 74.8) + (C 74.8) + (D 74.8) + (E 74.8) + (F 74.8) +(G 74.8) + (H 74.8) + (I 74.8) + (J 74.8) ) = (3) (cm ) ( (A 74.8) + (B 74.8) + (C 74.8) + (D 74.8) + (E 74.8) + (F 74.8) +(G 74.8) + (H 74.8) + (I 74.8) + (J 74.8) ) = ( ). n X, X,..., X n T (X, X,..., X n ) 7.6. n B(n, p) X p X n 7.7 ( ). T = T n = T (X, X,..., X n ) θ k > lim P ( T θ < k) = lim P ( T n θ < k) = n n 7.8. X, X,..., X N,... X ( P X = ) ( = p, P X = 3 ) = p, P (X = ) = p 4 4 p (, /) () E[X ] () E[X ] Y N = N (X + X + + X N ) (N =, 3,...)

101 (3) E[Y N ] (4) E[Y N ] (5) L = lim N NV (Y N ) (6) (a) a > L a = lim N P ( Y N E[X ] > a) (b) L a Y N p 7.8. () p () 5 8 p (3) p 5 (4) 8N p + N N (5) L = 5 8 p p p (6) (a) P ( Z > λ) E[ Z ] (λ > ) λp Z = Y N E[Y N ] λ = a p = P ( Y N E[Y N ] > a) λ E[ Y N E[Y N ] ] = λ V [Y N ] (7.) P ( Y N E[X ] > a) = P ( Y N E[Y N ] > a) a V [Y N] lim NV [Y N] = 5 N 8 p p (7.) lim N NV [Y N ] = lim N N ( 5 8 p p E[X ] = p (7.) (7.) (7.3) lim N P ( Y N p > a) = ) = (b) (7.3) a > Y N (), () 7.9 ( ). T = T (X, X,..., X n ) θ E[T ] = θ 7.. p (, /) X, X,..., X N,... X P (X = ) = p, P (X = ) = p, P (X = ) = p Y N = N (X + X + + X N ) (N =, 3,...)

102 () {c j } N j= (c + c + + c N ) (a) (b) () E[X ] (3) E[X ] (4) E[Y N ] (5) E[Y N ] (6) {a N } N= {b N} N= Z N = a N Y N + b N Y N (N =, 3,...) Z N p 7.. () (a) N (b) (N N)/ () E[X ] = 3p (3) E[X ] = 5p (4) E[Y ] = 3p (5) E[Y ] = 5 N p + 9N 9 N p (6) 3a N p + 5 N b Np + 9N 9 N b N p = p a N = 5 7N 7, b N = N 9N 9 7. (). X, X,..., X n f θ (x) L(θ) = f θ (x )f θ (x ) f θ (x n ) θ ˆθ ˆθ = T (x, x,..., x n ) T (X, X,..., X n ) θ T 7.. X, X,..., X N,... (, ) X (7.4) P (a < X < b) = b a p θ e θx dx (b > a > ) p θ θ π (7.5) e x dx = () (7.5) (7.4) p θ () (X, X ) a < b, c < d a, b, c, d P (a < X < b, c < X < d) = f(x, x ) dx dx [a,b] [c,d] f(x, y) f(x, y) f(x, y) x, y > (3) (X, X,..., X N ) f(x, x,..., x N ) (4) x = x + x + + x N θ

103 3 7.. π θ () e θx dx = θ p θ = π () f(x, y) = p θ exp( θ(x + y )) (3) f(x, x,..., x N ) = p N θ exp( θ(x + x + + x N )) (4) x = x + x + + x N d { } θ N exp( θ x ) = N dθ θ N exp( θ x ) x θ N exp( θ x ) θ = N x θ = N (X + X + + X N ) 7... T = T n = T (X, X,..., X n ) θ k > 7.3. lim P ( T θ < k) = lim P ( T n θ < k) = n n () X, X,..., X n m = E[X ], σ = V [X ] Y n = n (X + X + + X n ) θ > P ( Y n E[X ] > θ) nθ σ () n 7.3. () (.) Z P ( Z > θ) θ E[ Z ] Z = Y n m P ( Y n m > θ) θ E[ Y n m ] E[ Y n m ] = n V [X + X + + X n ] = n σ ( )P ( Y n m > θ) nθ σ

104 4 () X, X,..., X n () Y n () lim n nθ σ = lim P ( Y n m > θ) = n Y n X {P θ } θ Θ θ Θ g(θ) X T g E θ [T ] = g(θ) θ Θ 7.4. θ θ n j X j = X j = θ θ (x, x,..., x n ) R = x j T = R n T (x, x,..., x n ) T (3.) (3.) [ ] R E θ [T j ] = E θ n = n n C j θ j ( θ) n j = θ θ( θ) + n j= T θ T n n T = R(R ) n(n ) (3.) (3.) ( ) E θ [T ] = θ + θ( θ) n θ n = θ X = {(x, x,..., x n ) : x j =, }, P θ = {B(, θ) n } θ (,) G(α, ν) f(x) = Γ(ν) αν x ν e αx χ (, ) (x) Γ(ν) Γ(ν) = t ν e t dt Γ(ν + ) = νγ(ν) 7.5. ν > θ > X γ- G(θ, ν) ˆθ = X/ν ] [ ] X E [ˆθ = E = ν ν Γ(ν) θ ν x ν e x/θ θγ(ν + ) dx = = θ νγ(ν) j=

105 X Po(θ) ˆθ = X ˆθ θ θ r θ x (7.6) x! e θ z x = exp(θ(z )) x= (7.6) r z = θ x x! e θ x(x )(x ) (x r + ) = θ r x= ˆθ r = X(X )(X ) (X r + ) θ r 7.7. X B(n, θ), < θ < g(θ) = θ T (x) ( n T (x)θ x) x ( θ) n x = θ, < θ < x= θ ( n T (x)θ x) x+ ( θ) n x =, < θ < θ x= = lim θ x= ( n x) T (x)θ x+ ( θ) n x = 7.8. P P m R P X = X j n j= E P [X] = P, P P X P 7.9. P P V R P X = X j n j= S = n (X j X) ( ) j= E P [S ] = n n σ (P ) S σ (P ) n U = (X j X) ( ) n j= E P [U ] = σ (P ) U σ (P )

106 6 7.. θ > [, θ] U(, θ) n (X, X,..., X n ) X n = X + X + + X N θ Nn 7.. E[X] = E[X ] + E[X ] + + E[X n ] n = θ/ n n = θ X, X,..., X n θ p θ (x) = P (X i = x θ) x, x,..., x n L(θ x, x,..., x n ) = p θ (x )p θ (x ) p θ (x n ) log L(θ x, x,..., x n ) = log(p θ (x )p θ (x ) p θ (x n )) = 7.. θ > f(x; θ) = θe θx χ [, ) (x) (X, X,..., X N ) log p θ (x j ) () I, I,..., I N (, ) P (X I, X I,, X N I N ) () X [, ) g(θ) = θ N e θx θ (3) θ ˆθ f(x; θ) j= 7.. () P (X I, X I,, X N I N ) = θ N e θ(x +x + +x N ) dx dx... dx n I I I N () g(θ) = θ N e θx g (θ) = Nθ N e θx Xθ N e θx g θ = N X (3) θ N e θ(x+x+ +x N ) () X = x + x + + x N N N ˆθ = ˆθ = x + x + + x N X + X + + X N

107 [a, b] θ X, X,..., X n ˆθ (X, X,..., X n ) ˆθ (X, X,..., X n ) ˆθ (X, X,..., X n ) ˆθ (X, X,..., X n ) [ˆθ (X, X,..., X n ), ˆθ (X, X,..., X n ) P (ˆθ (X, X,..., X n ) θ ˆθ (X, X,..., X n )) P m ± ε α P (θ < ˆθ (X, X,..., X n ) = α, P (θ > ˆθ (X, X,..., X n ) = α ˆθ (X, X,..., X n ) ˆθ (X, X,..., X n ) X, X,..., X N N N = E[X ] = m V [X ] = σ, σ > Y N = X + X + + X N N Y N m σ /N P ( m.58 σ N < Y N < m +.58 σ N ) =.99 N N Y Y N = Y ( Y.58 σ, Y +.58 σ ) N N m 99 ( Y.96 σ, Y +.96 σ ) N N m Y = 436 N = σ = 89.6 ( Y.96 σ, Y +.96 σ ) = ( , ) = (44.4, 448.4) N N N X, X,..., X N ( ) X + X + + X N P I =.95 N I

Part. 4. () 4.. () 4.. 3 5. 5 5.. 5 5.. 6 5.3. 7 Part 3. 8 6. 8 6.. 8 6.. 8 7. 8 7.. 8 7.. 3 8. 3 9., 34 9.. 34 9.. 37 9.3. 39. 4.. 4.. 43. 46.. 46..

Part. 4. () 4.. () 4.. 3 5. 5 5.. 5 5.. 6 5.3. 7 Part 3. 8 6. 8 6.. 8 6.. 8 7. 8 7.. 8 7.. 3 8. 3 9., 34 9.. 34 9.. 37 9.3. 39. 4.. 4.. 43. 46.. 46.. Cotets 6 6 : 6 6 6 6 6 6 7 7 7 Part. 8. 8.. 8.. 9..... 3. 3 3.. 3 3.. 7 3.3. 8 Part. 4. () 4.. () 4.. 3 5. 5 5.. 5 5.. 6 5.3. 7 Part 3. 8 6. 8 6.. 8 6.. 8 7. 8 7.. 8 7.. 3 8. 3 9., 34 9.. 34 9.. 37 9.3.

More information

..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A

..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A .. Laplace ). A... i),. ω i i ). {ω,..., ω } Ω,. ii) Ω. Ω. A ) r, A P A) P A) r... ).. Ω {,, 3, 4, 5, 6}. i i 6). A {, 4, 6} P A) P A) 3 6. ).. i, j i, j) ) Ω {i, j) i 6, j 6}., 36. A. A {i, j) i j }.

More information

A B P (A B) = P (A)P (B) (3) A B A B P (B A) A B A B P (A B) = P (B A)P (A) (4) P (B A) = P (A B) P (A) (5) P (A B) P (B A) P (A B) A B P

A B P (A B) = P (A)P (B) (3) A B A B P (B A) A B A B P (A B) = P (B A)P (A) (4) P (B A) = P (A B) P (A) (5) P (A B) P (B A) P (A B) A B P 1 1.1 (population) (sample) (event) (trial) Ω () 1 1 Ω 1.2 P 1. A A P (A) 0 1 0 P (A) 1 (1) 2. P 1 P 0 1 6 1 1 6 0 3. A B P (A B) = P (A) + P (B) (2) A B A B A 1 B 2 A B 1 2 1 2 1 1 2 2 3 1.3 A B P (A

More information

6.1 (P (P (P (P (P (P (, P (, P.

6.1 (P (P (P (P (P (P (, P (, P. (011 30 7 0 ( ( 3 ( 010 1 (P.3 1 1.1 (P.4.................. 1 1. (P.4............... 1 (P.15.1 (P.16................. (P.0............3 (P.18 3.4 (P.3............... 4 3 (P.9 4 3.1 (P.30........... 4 3.

More information

6.1 (P (P (P (P (P (P (, P (, P.101

6.1 (P (P (P (P (P (P (, P (, P.101 (008 0 3 7 ( ( ( 00 1 (P.3 1 1.1 (P.3.................. 1 1. (P.4............... 1 (P.15.1 (P.15................. (P.18............3 (P.17......... 3.4 (P................ 4 3 (P.7 4 3.1 ( P.7...........

More information

ii 3.,. 4. F. ( ), ,,. 8.,. 1. (75% ) (25% ) =7 24, =7 25, =7 26 (. ). 1.,, ( ). 3.,...,.,.,.,.,. ( ) (1 2 )., ( ), 0., 1., 0,.

ii 3.,. 4. F. ( ), ,,. 8.,. 1. (75% ) (25% ) =7 24, =7 25, =7 26 (. ). 1.,, ( ). 3.,...,.,.,.,.,. ( ) (1 2 )., ( ), 0., 1., 0,. (1 C205) 4 10 (2 C206) 4 11 (2 B202) 4 12 25(2013) http://www.math.is.tohoku.ac.jp/~obata,.,,,..,,. 1. 2. 3. 4. 5. 6. 7. 8. 1., 2007 ( ).,. 2. P. G., 1995. 3. J. C., 1988. 1... 2.,,. ii 3.,. 4. F. ( ),..

More information

統計学のポイント整理

統計学のポイント整理 .. September 17, 2012 1 / 55 n! = n (n 1) (n 2) 1 0! = 1 10! = 10 9 8 1 = 3628800 n k np k np k = n! (n k)! (1) 5 3 5 P 3 = 5! = 5 4 3 = 60 (5 3)! n k n C k nc k = npk k! = n! k!(n k)! (2) 5 3 5C 3 = 5!

More information

n ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................

More information

.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z (g f(x = g(

.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z (g f(x = g( 06 5.. ( y = x x y 5 y 5 = (x y = x + ( y = x + y = x y.. ( Y = C + I = 50 + 0.5Y + 50 r r = 00 0.5Y ( L = M Y r = 00 r = 0.5Y 50 (3 00 0.5Y = 0.5Y 50 Y = 50, r = 5 .3. (x, x = (, u = = 4 (, x x = 4 x,

More information

ii 3.,. 4. F. (), ,,. 8.,. 1. (75%) (25%) =7 20, =7 21 (. ). 1.,, (). 3.,. 1. ().,.,.,.,.,. () (12 )., (), 0. 2., 1., 0,.

ii 3.,. 4. F. (), ,,. 8.,. 1. (75%) (25%) =7 20, =7 21 (. ). 1.,, (). 3.,. 1. ().,.,.,.,.,. () (12 )., (), 0. 2., 1., 0,. 24(2012) (1 C106) 4 11 (2 C206) 4 12 http://www.math.is.tohoku.ac.jp/~obata,.,,,.. 1. 2. 3. 4. 5. 6. 7.,,. 1., 2007 (). 2. P. G. Hoel, 1995. 3... 1... 2.,,. ii 3.,. 4. F. (),.. 5... 6.. 7.,,. 8.,. 1. (75%)

More information

x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x

x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x [ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),

More information

II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2

II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2 II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh

More information

( )/2 hara/lectures/lectures-j.html 2, {H} {T } S = {H, T } {(H, H), (H, T )} {(H, T ), (T, T )} {(H, H), (T, T )} {1

( )/2   hara/lectures/lectures-j.html 2, {H} {T } S = {H, T } {(H, H), (H, T )} {(H, T ), (T, T )} {(H, H), (T, T )} {1 ( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 1 2011 ( )/2 2 2011 4 1 2 1.1 1 2 1 2 3 4 5 1.1.1 sample space S S = {H, T } H T T H S = {(H, H), (H, T ), (T, H), (T, T )} (T, H) S

More information

201711grade1ouyou.pdf

201711grade1ouyou.pdf 2017 11 26 1 2 52 3 12 13 22 23 32 33 42 3 5 3 4 90 5 6 A 1 2 Web Web 3 4 1 2... 5 6 7 7 44 8 9 1 2 3 1 p p >2 2 A 1 2 0.6 0.4 0.52... (a) 0.6 0.4...... B 1 2 0.8-0.2 0.52..... (b) 0.6 0.52.... 1 A B 2

More information

ii 3.,. 4. F. (), ,,. 8.,. 1. (75% ) (25% ) =9 7, =9 8 (. ). 1.,, (). 3.,. 1. ( ).,.,.,.,.,. ( ) (1 2 )., ( ), 0. 2., 1., 0,.

ii 3.,. 4. F. (), ,,. 8.,. 1. (75% ) (25% ) =9 7, =9 8 (. ). 1.,, (). 3.,. 1. ( ).,.,.,.,.,. ( ) (1 2 )., ( ), 0. 2., 1., 0,. 23(2011) (1 C104) 5 11 (2 C206) 5 12 http://www.math.is.tohoku.ac.jp/~obata,.,,,.. 1. 2. 3. 4. 5. 6. 7.,,. 1., 2007 ( ). 2. P. G. Hoel, 1995. 3... 1... 2.,,. ii 3.,. 4. F. (),.. 5.. 6.. 7.,,. 8.,. 1. (75%

More information

renshumondai-kaito.dvi

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

More information

(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y

(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y [ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)

More information

18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C

18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C 8 ( ) 8 5 4 I II III A B C( ),,, 5 I II A B ( ),, I II A B (8 ) 6 8 I II III A B C(8 ) n ( + x) n () n C + n C + + n C n = 7 n () 7 9 C : y = x x A(, 6) () A C () C P AP Q () () () 4 A(,, ) B(,, ) C(,,

More information

st.dvi

st.dvi 9 3 5................................... 5............................. 5....................................... 5.................................. 7.........................................................................

More information

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi) 0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()

More information

211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,

More information

(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0

(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0 1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45

More information

1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2

1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2 2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6

More information

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11

More information

Untitled

Untitled II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j

More information

meiji_resume_1.PDF

meiji_resume_1.PDF 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

More information

X G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2

More information

熊本県数学問題正解

熊本県数学問題正解 00 y O x Typed by L A TEX ε ( ) (00 ) 5 4 4 ( ) http://www.ocn.ne.jp/ oboetene/plan/. ( ) (009 ) ( ).. http://www.ocn.ne.jp/ oboetene/plan/eng.html 8 i i..................................... ( )0... (

More information

(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t

(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t 6 6.1 6.1 (1 Z ( X = e Z, Y = Im Z ( Z = X + iy, i = 1 (2 Z E[ e Z ] < E[ Im Z ] < Z E[Z] = E[e Z] + ie[im Z] 6.2 Z E[Z] E[ Z ] : E[ Z ] < e Z Z, Im Z Z E[Z] α = E[Z], Z = Z Z 1 {Z } E[Z] = α = α [ α ]

More information

() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n (

() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n ( 3 n nc k+ k + 3 () n C r n C n r nc r C r + C r ( r n ) () n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (4) n C n n C + n C + n C + + n C n (5) k k n C k n C k (6) n C + nc

More information

I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google

I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google I4 - : April, 4 Version :. Kwhir, Tomoki TA (Kondo, Hirotk) Google http://www.mth.ngoy-u.c.jp/~kwhir/courses/4s-biseki.html pdf 4 4 4 4 8 e 5 5 9 etc. 5 6 6 6 9 n etc. 6 6 6 3 6 3 7 7 etc 7 4 7 7 8 5 59

More information

II 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K

II 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K II. () 7 F 7 = { 0,, 2, 3, 4, 5, 6 }., F 7 a, b F 7, a b, F 7,. (a) a, b,,. (b) 7., 4 5 = 20 = 2 7 + 6, 4 5 = 6 F 7., F 7,., 0 a F 7, ab = F 7 b F 7. (2) 7, 6 F 6 = { 0,, 2, 3, 4, 5 },,., F 6., 0 0 a F

More information

入試の軌跡

入試の軌跡 4 y O x 4 Typed by L A TEX ε ) ) ) 6 4 ) 4 75 ) http://kumamoto.s.xrea.com/plan/.. PDF) Ctrl +L) Ctrl +) Ctrl + Ctrl + ) ) Alt + ) Alt + ) ESC. http://kumamoto.s.xrea.com/nyusi/kumadai kiseki ri i.pdf

More information

n ( (

n ( ( 1 2 27 6 1 1 m-mat@mathscihiroshima-uacjp 2 http://wwwmathscihiroshima-uacjp/~m-mat/teach/teachhtml 2 1 3 11 3 111 3 112 4 113 n 4 114 5 115 5 12 7 121 7 122 9 123 11 124 11 125 12 126 2 2 13 127 15 128

More information

.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =,

.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =, [ ] IC. r, θ r, θ π, y y = 3 3 = r cos θ r sin θ D D = {, y ; y }, y D r, θ ep y yddy D D 9 s96. d y dt + 3dy + y = cos t dt t = y = e π + e π +. t = π y =.9 s6.3 d y d + dy d + y = y =, dy d = 3 a, b

More information

tokei01.dvi

tokei01.dvi 2. :,,,. :.... Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 4 3. (probability),, 1. : : n, α A, A a/n. :, p, p Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN

More information

i

i i 3 4 4 7 5 6 3 ( ).. () 3 () (3) (4) /. 3. 4/3 7. /e 8. a > a, a = /, > a >. () a >, a =, > a > () a > b, a = b, a < b. c c n a n + b n + c n 3c n..... () /3 () + (3) / (4) /4 (5) m > n, a b >, m > n,

More information

II 2 II

II 2 II II 2 II 2005 yugami@cc.utsunomiya-u.ac.jp 2005 4 1 1 2 5 2.1.................................... 5 2.2................................. 6 2.3............................. 6 2.4.................................

More information

4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.

4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5. A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c

More information

2011de.dvi

2011de.dvi 211 ( 4 2 1. 3 1.1............................... 3 1.2 1- -......................... 13 1.3 2-1 -................... 19 1.4 3- -......................... 29 2. 37 2.1................................ 37

More information

2000年度『数学展望 I』講義録

2000年度『数学展望 I』講義録 2000 I I IV I II 2000 I I IV I-IV. i ii 3.10 (http://www.math.nagoya-u.ac.jp/ kanai/) 2000 A....1 B....4 C....10 D....13 E....17 Brouwer A....21 B....26 C....33 D....39 E. Sperner...45 F....48 A....53

More information

A

A A 2563 15 4 21 1 3 1.1................................................ 3 1.2............................................. 3 2 3 2.1......................................... 3 2.2............................................

More information

I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )

I A A441 : April 15, 2013 Version : 1.1 I   Kawahira, Tomoki TA (Shigehiro, Yoshida ) I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17

More information

18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α

18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α 18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α 2 ), ϕ(t) = B 1 cos(ω 1 t + α 1 ) + B 2 cos(ω 2 t

More information

x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y)

x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy

More information

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,. 9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,

More information

( ) sin 1 x, cos 1 x, tan 1 x sin x, cos x, tan x, arcsin x, arccos x, arctan x. π 2 sin 1 x π 2, 0 cos 1 x π, π 2 < tan 1 x < π 2 1 (1) (

( ) sin 1 x, cos 1 x, tan 1 x sin x, cos x, tan x, arcsin x, arccos x, arctan x. π 2 sin 1 x π 2, 0 cos 1 x π, π 2 < tan 1 x < π 2 1 (1) ( 6 20 ( ) sin, cos, tan sin, cos, tan, arcsin, arccos, arctan. π 2 sin π 2, 0 cos π, π 2 < tan < π 2 () ( 2 2 lim 2 ( 2 ) ) 2 = 3 sin (2) lim 5 0 = 2 2 0 0 2 2 3 3 4 5 5 2 5 6 3 5 7 4 5 8 4 9 3 4 a 3 b

More information

微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.

微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます.   このサンプルページの内容は, 初版 1 刷発行時のものです. 微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)

More information

W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)

W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2) 3 215 4 27 1 1 u u(x, t) u tt a 2 u xx, a > (1) D : {(x, t) : x, t } u (, t), u (, t), t (2) u(x, ) f(x), u(x, ) t 2, x (3) u(x, t) X(x)T (t) u (1) 1 T (t) a 2 T (t) X (x) X(x) α (2) T (t) αa 2 T (t) (4)

More information

21 2 26 i 1 1 1.1............................ 1 1.2............................ 3 2 9 2.1................... 9 2.2.......... 9 2.3................... 11 2.4....................... 12 3 15 3.1..........

More information

all.dvi

all.dvi 5,, Euclid.,..,... Euclid,.,.,, e i (i =,, ). 6 x a x e e e x.:,,. a,,. a a = a e + a e + a e = {e, e, e } a (.) = a i e i = a i e i (.) i= {a,a,a } T ( T ),.,,,,. (.),.,...,,. a 0 0 a = a 0 + a + a 0

More information

2016 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 16 2 1 () X O 3 (O1) X O, O (O2) O O (O3) O O O X (X, O) O X X (O1), (O2), (O3) (O2) (O3) n (O2) U 1,..., U n O U k O k=1 (O3) U λ O( λ Λ) λ Λ U λ O 0 X 0 (O2) n =

More information

1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =

1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ = 1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A

More information

変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy,

変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy, 変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy, z + dz) Q! (x + d x + u + du, y + dy + v + dv, z +

More information

³ÎΨÏÀ

³ÎΨÏÀ 2017 12 12 Makoto Nakashima 2017 12 12 1 / 22 2.1. C, D π- C, D. A 1, A 2 C A 1 A 2 C A 3, A 4 D A 1 A 2 D Makoto Nakashima 2017 12 12 2 / 22 . (,, L p - ). Makoto Nakashima 2017 12 12 3 / 22 . (,, L p

More information

i

i 009 I 1 8 5 i 0 1 0.1..................................... 1 0.................................................. 1 0.3................................. 0.4........................................... 3

More information

newmain.dvi

newmain.dvi 数論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published

More information

Z: Q: R: C: sin 6 5 ζ a, b

Z: Q: R: C: sin 6 5 ζ a, b Z: Q: R: C: 3 3 7 4 sin 6 5 ζ 9 6 6............................... 6............................... 6.3......................... 4 7 6 8 8 9 3 33 a, b a bc c b a a b 5 3 5 3 5 5 3 a a a a p > p p p, 3,

More information

I L01( Wed) : Time-stamp: Wed 07:38 JST hig e, ( ) L01 I(2017) 1 / 19

I L01( Wed) : Time-stamp: Wed 07:38 JST hig e,   ( ) L01 I(2017) 1 / 19 I L01(2017-09-20 Wed) : Time-stamp: 2017-09-20 Wed 07:38 JST hig e, http://hig3.net ( ) L01 I(2017) 1 / 19 ? 1? 2? ( ) L01 I(2017) 2 / 19 ?,,.,., 1..,. 1,2,.,.,. ( ) L01 I(2017) 3 / 19 ? I. M (3 ) II,

More information

(1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10)

(1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10) 2017 12 9 4 1 30 4 10 3 1 30 3 30 2 1 30 2 50 1 1 30 2 10 (1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10) (1) i 23 c 23 0 1 2 3 4 5 6 7 8 9 a b d e f g h i (2) 23 23 (3) 23 ( 23 ) 23 x 1 x 2 23 x

More information

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

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 49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 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 = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r

More information

20 4 20 i 1 1 1.1............................ 1 1.2............................ 4 2 11 2.1................... 11 2.2......................... 11 2.3....................... 19 3 25 3.1.............................

More information

006 11 8 0 3 1 5 1.1..................... 5 1......................... 6 1.3.................... 6 1.4.................. 8 1.5................... 8 1.6................... 10 1.6.1......................

More information

IA hara@math.kyushu-u.ac.jp Last updated: January,......................................................................................................................................................................................

More information

1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C

1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C 0 9 (1990 1999 ) 10 (2000 ) 1900 1994 1995 1999 2 SAT ACT 1 1990 IMO 1990/1/15 1:00-4:00 1 N 1990 9 N N 1, N 1 N 2, N 2 N 3 N 3 2 x 2 + 25x + 52 = 3 x 2 + 25x + 80 3 2, 3 0 4 A, B, C 3,, A B, C 2,,,, 7,

More information

20 6 4 1 4 1.1 1.................................... 4 1.1.1.................................... 4 1.1.2 1................................ 5 1.2................................... 7 1.2.1....................................

More information

simx simxdx, cosxdx, sixdx 6.3 px m m + pxfxdx = pxf x p xf xdx = pxf x p xf x + p xf xdx 7.4 a m.5 fx simxdx 8 fx fx simxdx = πb m 9 a fxdx = πa a =

simx simxdx, cosxdx, sixdx 6.3 px m m + pxfxdx = pxf x p xf xdx = pxf x p xf x + p xf xdx 7.4 a m.5 fx simxdx 8 fx fx simxdx = πb m 9 a fxdx = πa a = II 6 ishimori@phys.titech.ac.jp 6.. 5.4.. f Rx = f Lx = fx fx + lim = lim x x + x x f c = f x + x < c < x x x + lim x x fx fx x x = lim x x f c = f x x < c < x cosmx cosxdx = {cosm x + cosm + x} dx = [

More information

20 9 19 1 3 11 1 3 111 3 112 1 4 12 6 121 6 122 7 13 7 131 8 132 10 133 10 134 12 14 13 141 13 142 13 143 15 144 16 145 17 15 19 151 1 19 152 20 2 21 21 21 211 21 212 1 23 213 1 23 214 25 215 31 22 33

More information

1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1

1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1 1 I 1.1 ± e = = - =1.602 10 19 C C MKA [m], [Kg] [s] [A] 1C 1A 1 MKA 1C 1C +q q +q q 1 1.1 r 1,2 q 1, q 2 r 12 2 q 1, q 2 2 F 12 = k q 1q 2 r 12 2 (1.1) k 2 k 2 ( r 1 r 2 ) ( r 2 r 1 ) q 1 q 2 (q 1 q 2

More information

) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4

) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4 1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev

More information

1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1

1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω ω α 3 3 2 2V 3 33+.6T m T 5 34m Hz. 34 3.4m 2 36km 5Hz. 36km m 34 m 5 34 + m 5 33 5 =.66m 34m 34 x =.66 55Hz, 35 5 =.7 485.7Hz 2 V 5Hz.5V.5V V

More information

(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fou

(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fou (Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fourier) (Fourier Bessel).. V ρ(x, y, z) V = 4πGρ G :.

More information

II R n k +1 v 0,, v k k v 1 v 0,, v k v v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ k σ dimσ = k 1.3. k

II R n k +1 v 0,, v k k v 1 v 0,, v k v v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ k σ dimσ = k 1.3. k II 231017 1 1.1. R n k +1 v 0,, v k k v 1 v 0,, v k v 0 1.2. v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ kσ dimσ = k 1.3. k σ {v 0,...,v k } {v i0,...,v il } l σ τ < τ τ σ 1.4.

More information

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

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 No.2 1 2 2 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 δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j

More information

( ) 2002 1 1 1 1.1....................................... 1 1.1.1................................. 1 1.1.2................................. 1 1.1.3................... 3 1.1.4......................................

More information

chap1.dvi

chap1.dvi 1 1 007 1 e iθ = cos θ + isin θ 1) θ = π e iπ + 1 = 0 1 ) 3 11 f 0 r 1 1 ) k f k = 1 + r) k f 0 f k k = 01) f k+1 = 1 + r)f k ) f k+1 f k = rf k 3) 1 ) ) ) 1+r/)f 0 1 1 + r/) f 0 = 1 + r + r /4)f 0 1 f

More information

1.2 y + P (x)y + Q(x)y = 0 (1) y 1 (x), y 2 (x) y 1 (x), y 2 (x) (1) y(x) c 1, c 2 y(x) = c 1 y 1 (x) + c 2 y 2 (x) 3 y 1 (x) y 1 (x) e R P (x)dx y 2

1.2 y + P (x)y + Q(x)y = 0 (1) y 1 (x), y 2 (x) y 1 (x), y 2 (x) (1) y(x) c 1, c 2 y(x) = c 1 y 1 (x) + c 2 y 2 (x) 3 y 1 (x) y 1 (x) e R P (x)dx y 2 1 1.1 R(x) = 0 y + P (x)y + Q(x)y = R(x)...(1) y + P (x)y + Q(x)y = 0...(2) 1 2 u(x) v(x) c 1 u(x)+ c 2 v(x) = 0 c 1 = c 2 = 0 c 1 = c 2 = 0 2 0 2 u(x) v(x) u(x) u (x) W (u, v)(x) = v(x) v (x) 0 1 1.2

More information

untitled

untitled 2 : n =1, 2,, 10000 0.5125 0.51 0.5075 0.505 0.5025 0.5 0.4975 0.495 0 2000 4000 6000 8000 10000 2 weak law of large numbers 1. X 1,X 2,,X n 2. µ = E(X i ),i=1, 2,,n 3. σi 2 = V (X i ) σ 2,i=1, 2,,n ɛ>0

More information

Gmech08.dvi

Gmech08.dvi 145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2

More information

n Y 1 (x),..., Y n (x) 1 W (Y 1 (x),..., Y n (x)) 0 W (Y 1 (x),..., Y n (x)) = Y 1 (x)... Y n (x) Y 1(x)... Y n(x) (x)... Y n (n 1) (x) Y (n 1)

n Y 1 (x),..., Y n (x) 1 W (Y 1 (x),..., Y n (x)) 0 W (Y 1 (x),..., Y n (x)) = Y 1 (x)... Y n (x) Y 1(x)... Y n(x) (x)... Y n (n 1) (x) Y (n 1) D d dx 1 1.1 n d n y a 0 dx n + a d n 1 y 1 dx n 1 +... + a dy n 1 dx + a ny = f(x)...(1) dk y dx k = y (k) a 0 y (n) + a 1 y (n 1) +... + a n 1 y + a n y = f(x)...(2) (2) (2) f(x) 0 a 0 y (n) + a 1 y

More information

さくらの個別指導 ( さくら教育研究所 ) A 2 P Q 3 R S T R S T P Q ( ) ( ) m n m n m n n n

さくらの個別指導 ( さくら教育研究所 ) A 2 P Q 3 R S T R S T P Q ( ) ( ) m n m n m n n n 1 1.1 1.1.1 A 2 P Q 3 R S T R S T P 80 50 60 Q 90 40 70 80 50 60 90 40 70 8 5 6 1 1 2 9 4 7 2 1 2 3 1 2 m n m n m n n n n 1.1 8 5 6 9 4 7 2 6 0 8 2 3 2 2 2 1 2 1 1.1 2 4 7 1 1 3 7 5 2 3 5 0 3 4 1 6 9 1

More information

December 28, 2018

December 28, 2018 e-mail : kigami@i.kyoto-u.ac.jp December 28, 28 Contents 2............................. 3.2......................... 7.3..................... 9.4................ 4.5............. 2.6.... 22 2 36 2..........................

More information

i 18 2H 2 + O 2 2H 2 + ( ) 3K

i 18 2H 2 + O 2 2H 2 + ( ) 3K i 18 2H 2 + O 2 2H 2 + ( ) 3K ii 1 1 1.1.................................. 1 1.2........................................ 3 1.3......................................... 3 1.4....................................

More information

[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s

[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s [ ]. lim e 3 IC ) s49). y = e + ) ) y = / + ).3 d 4 ) e sin d 3) sin d ) s49) s493).4 z = y z z y s494).5 + y = 4 =.6 s495) dy = 3e ) d dy d = y s496).7 lim ) lim e s49).8 y = e sin ) y = sin e 3) y =

More information

II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re

II ( ) (7/31) II (  [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re II 29 7 29-7-27 ( ) (7/31) II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I Euler Navier

More information

x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s

x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s ... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z

More information

Microsoft Word - 表紙.docx

Microsoft Word - 表紙.docx 黒住英司 [ 著 ] サピエンティア 計量経済学 訂正および練習問題解答 (206/2/2 版 ) 訂正 練習問題解答 3 .69, 3.8 4 (X i X)U i i i (X i μ x )U i ( X μx ) U i. i E [ ] (X i μ x )U i i E[(X i μ x )]E[U i ]0. i V [ ] (X i μ x )U i i 2 i j E [(X i

More information

II (10 4 ) 1. p (x, y) (a, b) ε(x, y; a, b) 0 f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a 2 x a 0 A = f x (

II (10 4 ) 1. p (x, y) (a, b) ε(x, y; a, b) 0 f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a 2 x a 0 A = f x ( II (1 4 ) 1. p.13 1 (x, y) (a, b) ε(x, y; a, b) f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a x a A = f x (a, b) y x 3 3y 3 (x, y) (, ) f (x, y) = x + y (x, y) = (, )

More information

all.dvi

all.dvi 38 5 Cauchy.,,,,., σ.,, 3,,. 5.1 Cauchy (a) (b) (a) (b) 5.1: 5.1. Cauchy 39 F Q Newton F F F Q F Q 5.2: n n ds df n ( 5.1). df n n df(n) df n, t n. t n = df n (5.1) ds 40 5 Cauchy t l n mds df n 5.3: t

More information

30

30 3 ............................................2 2...........................................2....................................2.2...................................2.3..............................

More information

1

1 1 1 7 1.1.................................. 11 2 13 2.1............................ 13 2.2............................ 17 2.3.................................. 19 3 21 3.1.............................

More information

I, II 1, A = A 4 : 6 = max{ A, } A A 10 10%

I, II 1, A = A 4 : 6 = max{ A, } A A 10 10% 1 2006.4.17. A 3-312 tel: 092-726-4774, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ 1. 2. A 1. 1. 2. 3. 4. 5. 2. ɛ-δ 1. ɛ-n

More information

4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx

4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx 4 4 5 4 I II III A B C, 5 7 I II A B,, 8, 9 I II A B O A,, Bb, b, Cc, c, c b c b b c c c OA BC P BC OP BC P AP BC n f n x xn e x! e n! n f n x f n x f n x f k x k 4 e > f n x dx k k! fx sin x cos x tan

More information

( 30 ) 30 4 5 1 4 1.1............................................... 4 1.............................................. 4 1..1.................................. 4 1.......................................

More information

IA 2013 : :10722 : 2 : :2 :761 :1 (23-27) : : ( / ) (1 /, ) / e.g. (Taylar ) e x = 1 + x + x xn n! +... sin x = x x3 6 + x5 x2n+1 + (

IA 2013 : :10722 : 2 : :2 :761 :1 (23-27) : : ( / ) (1 /, ) / e.g. (Taylar ) e x = 1 + x + x xn n! +... sin x = x x3 6 + x5 x2n+1 + ( IA 2013 : :10722 : 2 : :2 :761 :1 23-27) : : 1 1.1 / ) 1 /, ) / e.g. Taylar ) e x = 1 + x + x2 2 +... + xn n! +... sin x = x x3 6 + x5 x2n+1 + 1)n 5! 2n + 1)! 2 2.1 = 1 e.g. 0 = 0.00..., π = 3.14..., 1

More information

keisoku01.dvi

keisoku01.dvi 2.,, Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 4 Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 5 Mon, 2006, 401, SAGA, JAPAN Dept.

More information

1. 1 BASIC PC BASIC BASIC BASIC Fortran WS PC (1.3) 1 + x 1 x = x = (1.1) 1 + x = (1.2) 1 + x 1 = (1.

1. 1 BASIC PC BASIC BASIC BASIC Fortran WS PC (1.3) 1 + x 1 x = x = (1.1) 1 + x = (1.2) 1 + x 1 = (1. Section Title Pages Id 1 3 7239 2 4 7239 3 10 7239 4 8 7244 5 13 7276 6 14 7338 7 8 7338 8 7 7445 9 11 7580 10 10 7590 11 8 7580 12 6 7395 13 z 11 7746 14 13 7753 15 7 7859 16 8 7942 17 8 Id URL http://km.int.oyo.co.jp/showdocumentdetailspage.jsp?documentid=

More information

ii

ii ii iii 1 1 1.1..................................... 1 1.2................................... 3 1.3........................... 4 2 9 2.1.................................. 9 2.2...............................

More information

2 2 MATHEMATICS.PDF 200-2-0 3 2 (p n ), ( ) 7 3 4 6 5 20 6 GL 2 (Z) SL 2 (Z) 27 7 29 8 SL 2 (Z) 35 9 2 40 0 2 46 48 2 2 5 3 2 2 58 4 2 6 5 2 65 6 2 67 7 2 69 2 , a 0 + a + a 2 +... b b 2 b 3 () + b n a

More information