Part () () Γ Part , - PDF Free Download

Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9

4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35 8.. 35 8.. 39 9. 4., 4.. 4.. 46.3. 49. 5.. 5.. 55.3. 64. 7.. 7.. 7

3.3. 7 Part 4. 78 3. 78 3.. 78 3.. 78 3.3. 79 4. 8 4.. 8 4.. 84 5. 85 5.. 85 5.. 86 6. 87 6.. χ - 87 6.. t- 9 6.3. F - 93 6.4. 97 Part 5. 99 7. 99 7.. 99 7.. 3 7.3. 4 7.4. 6 8. 7 8.. 7 8.. 8 9. 9.. 9..

4 9.3. () 4. 7.. 8.. 8 Part 6.......3. 3. 6.. 7.. 3 3. 34 3.. 34 3.. 38 3.3. 44 4. 45 5. 5 5.. 5 5.. 53 Part 7. 54 6. 54 6.. 54 6.. 56 6.3. 56 7. 57 7.. 57 7.. 6 8. 6

5 8.. 6 8.. 64 8.3. 65 67 References 68

6 a () () ysawano@tmu.ac.jp () () (3) (4) () () (3) () () (3) (4) (5)

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

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 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! = 44 54 44 = 36 () 5! = = 4 (3) 4! = 48 48 = 96 () 4 4 96 = 44 (4) ( ) A A B, C, D, E 4 4 4 4! 4 4! = 9 ( ) A B, C, D, E 5 A, F 4! = 48 A 48 = 576 764.. 4 4 R,B,Y,G () (a) 4 (b) 3 (c) (d) ()

.. () (a) 4! = 4 (b) 4 4 3! = 6 4 6 = 48 (c) 4 3 = (d) 3 84 (). (a) (b) (b) 4 + 48 = 48.... 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

(8) 5.3. () C 3 = () P 3 = 7 (3) C 5 = 6 (4) C 5 = 5 (5) 4 6 8 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

.5. U = {,, 3, 4, 5, 6} 5 A / A {4, 5} A.5. {4} = A.6. 66 8.6. 8/66 3. 3... 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

3 8 3.4. ( ) (, ) ( ) (, ) ( ) (, ) ( ) (, ) Ω = {(, ), (, ), (, ), (, )} 3... 3.5. () () (3) /6 (4) / 3.6. 5 8.8 3.7. () 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

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 6 3.. 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=

5 3.. U = {,, 3, 4, 5, 6} 3 A 3 A 3.. 3 A {3} A 3 A 4. 4... 4 3 4 3 4 3 44.4. 33.3 4. ( ). 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)

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 6 3 4.3. () 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) = 4 + 5 = 5. (3) A 96 B 7 C 6 96 96 + 7 + 6 = 8 8 + 6 + 5 = 8 9. (4) {} ( ) P ({ }) = 6 = 8 648 ( ) P ({ }) = 5 6 6 = 5 36 = 9 648 ( ) P ({ }) = 5 6 5 6 6 = 5 6 = 75 648 ( ) P ({ }) = 5 6 5 6 5 6 3 = 5 648 { } P ({ }) = P ({ }) + P ({ }) + P ({ }) + P ({ }) = 398 648 (a) p P ({ } { }) P ({ }) p = P { ({ }) = = = 8 } P ({ }) P ({ }) 398 = 54 99 (b) p P ({ } { }) P ({ }) p = P { ({ }) = = = 5 } P ({ }) P ({ }) 398

7 4.4. 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 3 4 = 3 4, P ( ) = 3 3 4 = 3 4 P ( ) = 3 3 4 = 6 4, P ( ) = 3 3 4 = 6 4 () = P ( ) + P ( ) + P ( ) = 4 + 4 + 3 4 = 4.

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

9 P ({ }) = 4 = 3, P ({ }) = 4 = 3 P (D ) = 3 + 4 + + = 3 3. () D D B, C P (D B ) = 9 + 3 = 3, P (B C ) = 9 + 6 = 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 () : : :

3 4 3 3 4 4 () 3 4 3 4 3 3 (3) 3 4

3 3 3 4 4 3 4.8. Ω = {(, ), (, ), (, ), (, )} A B A = {(, ), (, )}, B = {(, ), (, )}, A B = {(, )} P (A) =, P (B) =, P (A B) = P (A)P (B) = P (A B) 4 A, B 4.9. 3 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

() 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)

3 N 4.3.. 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 ) 4.6. 3 U, U, U 3 4, 5, 6 6, 5, 4 U, U, U 3 (),, 3 U () 4, 5 U (3) 6 U 3 U

4 4.6. 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 ) = 4 3 9 = 6 45 P (A H )P (H ) + P (A H )P (H ) + P (A H 3 )P (H 3 ) = 5 + 7 + 8 = 8 + + 5 7 P (H A) = 7 53 7 = 53 = 53 7 4.7 ( ). 4 4 () () 4.7. () 4 /4 () 4 / 3/4 / = 3/8 () P ( ) = 4 P ( ) = 3 4 P ( ) = P ( ) =

5 P ( ) = P ( ) P ( ) + P ( ) P ( ) = 4 + 3 4 = 3 8 4.8. 9 X 9 5 9 8 X 8 8 8 8 45 (a), (b) 3, 4 (c) 5 (d) 6 4.8. = 5 45 3 = 9 = 8 6 6 = = 8 6 6 = 3

6 Part. 5. () b a f(x) dx y x, x 5..8.6.4.. 3 4 5 dx x y x, x..8.6.4.. 3 4 5 log x dx y log x, < x <

7 4 6 8...4.6.8. 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

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 ) =

9 5... 5.3. 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 + 5.4. () 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 = 5.4 5.5. 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

3 5.3. 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

. 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 6. 6... 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 3 75 4 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]

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] 6.5. 6.4. 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 6... 6.5 ( ). 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 ) 6.5. 6.6 ( ). f [, R] f( R x + y + z ) dx dy dz = 4π f(r)r dr {x +y +z R } 6.6 6.7 6.7. 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 N x + x + + x N = 6.3.. N N N N N 6 (6.3) A = A N = 6 6 3........ 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 3 6 6 (x + x + x 3 ) 3 x 4. N (x + x + x 3 + x 4 + + 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 Part 3. x π, y y sin x..8.6.4....5..5..5 3. y sin x y sin x = π y sin x π 7. 7... 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 7.3. 7.4. A χ A { (x A) χ A (x) = (x / A) 7.5. k =,, 3,... k k 7... 7.6.

35 (), 3, 4, 5, 6 () = 5 6, 3, 4, 5, 6 (3) [, ].5.5.5.5,.,.3,, 4,.35, e = 7.7. 7 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 8. 8... 8. (). () 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 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 8.7 ( ). 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 = + 3 + + 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 8.. () 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 / 4 + 4 = /3 /4 6x( x) dx = [ 3x x 3] /3 /4 = 7 7 5 3 = 89 864 8.. () ( ) () ( ) 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 8... 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..5 4 4.5. 8.4. 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

4 F (α).8.6.4. 4 4 9. 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 ) 3 9.. (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) =

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 3 + + a k k +,,,..., 9.a a a 3 a + a 4 + a 3 8 + + 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 9.4. 9.3 () x () a, a,..., a k {, } n P (X = a, X = a,, X k = a k ) (3) X, X,..., X k 9.4. ()

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 E[X] E(X).3.,, 4, 9, 4 (++4+9+4) = 6 5 5, 4, 6, 8, 96 ( + 4 + 6 + 8 + 96) = 59.6 5 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.5..5..5 4 4.5 (). 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 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 + + 3 3 + = 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 (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..7.8.9. P (X = k) = ( 4 5 5 X( ) P (a X b) = b a ) k 4e 4x dx () m

46 () M.8. () m = 4xe 4x dx = e 4x dx = 5 4 () M M = 5 + 4 5 4 5 + + k 4 ( ) k 4 5 = 5 5 4 ( ) 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. ( ). 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 ].7. 5 54, 35, 57, 49, 55 X.7. E[X] = (54 + 35 + 57 + 49 + 55) = 5 5 V [X] = 5 {(54 5) + (35 5) + (57 5) + (49 5 ) + (55 5) } = 63. V [X] = 5 {54 + 35 + 57 + 49 + 55 } 5 = 63.

48.8. [, ] 6 X f(x) = x( x) 88.8. 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 = 88 65.9. 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 ) = 63 5 8 5 = 36 5.5.4.3....4.6.8..9. () () 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 = 3 63 8 63 = 63. ( (4) P X ) / = (x 3 x 4 ) dx = 3 8

49.3... (). 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] = ( + + 8 + 9 + ) = 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] = 4 + 8 + 4 4 + 8 6 + 4 8 = 7 4 V [X] = ( 7 ) + ( 7 ) + ( 4 7 ) + ( 6 7 ) + ( 8 7 ) = 5 4 4 8 4 4 4 8 4 4 4 6 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]

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) 75 + 5 = 8 3.4. 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 = 3 4 4 = x 3 ( x) dx = 3 4 5 3! 5! = 3 4 3 = 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 +

5 (5) σ[x] = V [X] = T + T + 3 T + 4..... ( ). 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, 5 5 5 3 4 5 6.3 ( ).

5 48, 34,, 35, 84, 77, 87, 75, 35, 85, 45, 78,, 63,, 48, 86, 4, 5, 63, 4, 54, 6, 9, 59 9 9 3 39.3. () 4.6 4.6.4 5. 3. 8.3 4.6.48.8 4.56 3. 7.68 3..8 4.6 4.96.4 5..4. X, X,..., X n () () (3).5 (). 5 75 5 75.6. 34, 4, 3, 34, 34, 99, 73, 3, 5, 43

53 4 35 3 5 5.7.,, 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. ( + + 8 + 9 + ) = 6 5 5 (( 6) + ( 6) + (8 6) + (9 6) + ( 6) ) = 5 (( 5) + ( 4) + + 3 + 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

54.9. 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... X Y 34 33 34 3 56 4 9 4 5 39 44 6 4 8 7 6 97 8 4 76 9 88 39 48 5 X Y.. 8 6 4 4 6 8. (). 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

56.4 (). Cov[X, Y ], r[x, Y ] X Y 93 59 86 94 3 67 88 4 43 58 5 93 46 6 58 7 6 39 8 4 46 9 78 89 55 8 6 4 4 6 8.4. () E[X] = (93 + 86 + 67 + 43 + 93 + 58 + 6 + 4 + 78 + ) = 63. () E[Y ] = (59 + 94 + 88 + 58 + 46 + + 39 + 46 + 89 + 55) = 58.4 (3) E[X ] = (93 + 86 + 67 + 43 + 93 + 58 + 6 + 4 + 78 + ) = 46.3 (4) E[Y ] = (59 + 94 + 88 + 58 + 46 + + 39 + 46 + 89 + 55 ) = 4.4 (5) V [X] = E[X ] E[X] = 69.69 (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 ] = (93 59 + 86 94 + 67 88 + 43 58 + 93 46 +58 + 6 39 + 4 46 + 78 89 + 55) = 3863. () 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 ]

57.5 ( ). 8 6 4 4 6 8.6 ( ). 8 6 4 4 6 8.7 ( ). 8 6 4 4 6 8

58.8 ( ). 8 6 4 4 6 8.9 ( ). 8 6 4 4 6 8.. 3 X Y 3 5 4 4 3 6 8 4 4 5 5 5 6 6 8 7 3 8 4 6 9 7 9 3 X Y 9 5 4 3 3 6 4 4 5 4 8 6 5 7 4 8 5 8 9 9 5 8 3 X Y 9 7 3 3 4 4 5 5 8 4 6 5 4 7 5 4 8 8 9 9 6 3

59.. X Y (X, Y ) 8 6 4 4 6 8 8 6 4 4 6 8 3 8 6 4 4 6 8

6 3.96,,.89 k > b Y = k X + b k > b Y = k X + b.. Y a X Y 9 3 8 49 5 67 9 4 85 7 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

() 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 6 59 348 4 78 684 7 48 34 57 63 3969 65 45 5 3 3 9 3 7 89 6 6 36 7 5 6 6 4 86 () 3385.6 376.8 58. 56 68

6.4. r[x, Y ] 3 4 5 6 7 8 9 X 5 6 5 8 9 Y 5 6 7 6 7.4. X Y X Y XY 5 5 5 3 4 4 4 4 4 5 6 36 6 6 5 5 5 5 5 7 8 6 64 36 48 8 9 7 8 49 63 9 6 36 6 7 49 7 4.8 5.6 33.8 4.4 3. () X = 33.8 4.8 = 3.843893 () Y = 4.4 5.6 = 3.665976 33.8 4.4 3. (3) = 3.843893 3.665976 =.4788 8 6 4 4 6 8.5. 3 4 5 6 7 8 9 4 93 3 58 89 63 6 6 4 9 9 8 8 5 4 6 69 49 77 9 (): (): (3): (4): (5):

63.5. ().6. 3 4 5 6 7 8 9 99 7 63 5 54 9 44 75 58 59 89 6 56 68 57 3 44 78 6 56 (): ():(3): (4): (5):.6. (3).7. 3 4 5 6 7 8 9 89 75 58 99 6 56 68 7 89 6 99 49 7 44 78 8 94 (): (): (3): (4): (5):.7. (5).8. () () (3) (4) (5).8. () x y s cm () (3) (4) (5) x y

64.3...9 ( ). 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 : 85474 X X X 4 () 3 4 3.3. () X r X () r = 887 89588, r 3 = 8739 887, r 4 = 8633 8739, r 5 = 85474 8633 r = 4 r r 3 r 4 r 5 = 4 85474 89588 =.9897 = ( 3 3 + + ) 4 48.7.3 ( ). 45, 4, 5, 74, 49, 5, 3, 9, 79, 78 A B C A > B > C.3. A = 5.8 > B = 4.4 > C = 3..33 ( ). 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

65.34. () 56, 88, 6, 88,, 4, 89, 4, 8, 37, 49, 66, 76, 35, 6, 8, 5, 86, 4, 79 4 5/ + 56/ = 53.5 () 9 73, 98, 3, 58, 79, 96, 3, 73, 37, 73, 44, 36, 7,, 6, 9, 74, 44, 35 73 58.35 ( ). 3, 5,,, 3,, 7,,, 8, 7, 8,, 5, 7, 9, 7, 6, 3, 6, 6.35. 7 8 9.3.36 ( )..36. 3 :.37 ( ). x, x,..., x N N (x j m) j=.38. 8 9 7 9 86 6 9 7 67 3 97 39 83 47 43 64 98.38. = 54.7 = 6.8 7.3, 5..,.5.39 ( ). 8 69 7 67 69 77 73 79 6 78 57 63 76 84 6 8 8.39. 3 4 5 6 7 8 69 7 67 69 77 73 79 6 7.75.75.5 3.75.75 6.5.5 8.5.75 3.6.56 4.6 3.6 39.6 5.6 68.6 5.56 3.9 5.58,.67

66 3 4 5 6 7 8 78 57 63 76 84 6 8 8 7.65 5.375 5.65 9.65 3.375.375.65 7.375 8.375 8.89 44.4 9.64.39 9.39.89 54.39 7.4 9.98 9.64, 7.9.4 ( ). 6,, 3, 4, 5 5 A A A B C D E F 5 3 5 3 3 5 3 5.4. A B C D E F 5 3 5 3 3.7.46 3 5 3 5 3.63 () A (5 3.7)/.46 =.5356634 () A ( 3)/.63 =.474487 (3) A =.8847.4 ( ). 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) 866.43 ( 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) 3.44. 56, 58, 6, 353, 387, 47, 56, 68, 6, 78

67 7 6 5 4 3 ().5 =.75 58.5 + 6.75 = 6.5 ().75 = 8.5 68.75 + 6.75 = 68.75 (3) 68.75 6.5 = 348.5.45 ( ). 9, 95, 39, 3, 55, 9, 96, 4, 7, 3, 54, 7, 53, 75, 68, 7, 8, 4, 9, 85 3 8.45. () 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) = 46.5.46 ( ). 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 4 3 4 6 8 () () (3) (4).3. (b) (X ) (c) (X ) 3 (d) (X ) 4 3 4 5 6 7 8 9 3 4 5 X 6 5 6 7 4 6 6 9 5 4 6 (a) 4 9 4 4 6 4 9 9 4 4 5 6 7.33 (b) -8-7 -8 8-64 8 7-7 8 8 5-64 - (c) 6 8 6 6 56 6 8 8 6 6 65 56 478 () 4 () 7.33 (3) n = 5 = n (n )(n ) (4) n = 5 = ( ) 3 xi m = 5 σ 4 3 ( ) 8 7.33 = 8 7.33 =.84 i= n(n + ) (n )(n )(n 3) ( ) 4 xi m 3(n ) σ (n )(n 3) i= 5 6 = 4 3 478 3 4 4 7.334 3 = 3.73.48 ( ).

69 5, 4, 7, 5, 9, 6, 5,, 5, 8, 6,,, 5, 5, 8,, 3.48..779867,.7949.49 ( ). 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

7 z 5 7 p(7) = 5.... 9 9.33 9. ( ). 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.

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 4 3 3 [, ] x, y sin x sin y.5..6. x, y x y 4 4 X, X,..., X N,... P (X = ) = P (X = 3) = / N (X + X + + X N ).6..3...7 ( ). X, X,..., X k,... lim n X + X + + X n ne[x ] n

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+

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 k.976563.5448 7.6947E-3 -.5448.7488.698489 5.535E-5 -.698489.546875.8646853 6.7597E-9 -.8646854 3.7875.775836.3383 -.76444 4.3769535.59653.5399763 -.54346 5.6346875.985367.39894863 -.58675444 6.885.3938594.5399763 -.5538883 7.94535.494673.3383 -.494536468 8.9895783.5645394 6.7597E-9 -.56453934 9.9993438.579594749 5.535E-5 -.579594749.583883 7.6947E-3 -.583883 () n = k.898.45365.66334.57669 3.8789.439.975.5393 4.4655.33686.9335.566547 5.4785767.3377778.375548 -.397 6.3696447.8694444.856548 -.8963 7.7398833.65388888.638957 -.34993 8.34354.68756943.6748689 -.75 9.67938.3583459.3669438.76847.76975.3947685.3994344.486659.67938.3583459.3669438.76847.34354.68756943.6748689 -.75 3.7398833.65388888.638957 -.34993 4.3696447.8694444.856548 -.8963 5.4785767.3377778.375548 -.397 6.4655.33686.9335.566547 7.8789.439.975.5393 8.898.45365.66334.57669

75 (3) n = 4 k 9.4869.78643.94665.5434.77943.437935.6889.557.57.664894.69564.37.584.6688.666.93798 3.9445.3468449.3446668 -.878 4.658.66744866.65945688 -.79978 5.36584738.569.43563 -.39538 6.5763653.8767344.7958965 -.58379 7.8768.55957.543858 -.8377 8.38747.369.36633.549 9.94655.3775784.379493.938.537688.39645695.39894863.4939.94655.3775784.379493.938.38747.369.36633.549 3.8768.55957.543858 -.8377 4.5763653.8767344.7958965 -.58379 5.36584738.569.43563 -.39538 6.658.66744866.65945688 -.79978 7.9445.3468449.3446668 -.878 8.584.6688.666.93798 9.57.664894.69564.37 3.77943.437935.6889.557 3.4869.78643.94665.5434.9. 4 3 6 4 6 4 9 48 6.9. () ().. 4 95 3/4 9

76 4 35 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 5.35.565.35 3.35 4.565 5.35 (b) n = 6 n 6.565.9375.34375 3.35 4.34375 5.9375 6.565

78 Part 4. 3. 3... 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 -6-3 3 6 9 p /64 3/3 5/64 5/6 5/64 3/3 /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 3.3. 3 4 5 6 7 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 = 8 + 8 = 64. 3... 3.4 ( ). a, b (3.) k a k b n k nc k = n a (a + b) n k=

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.) 3.3.. 3.5 ( ). 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] = 5 4 4 5 3.7 (). < 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=

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 = 8 6 5 6 = 5 (c) σ[x] = V [X] = 5 () (a) E[ 47X + 6] = 47 3 + 6 = 4 + 6 = 35 (b) V [ 47X + 6] = ( 47) 5 = 555 (c) σ[ 47X + 6] = 555 = 35 (3) (a) E[X + Y + ] = 3 + 3 + = 8 (b) V [X + Y + ] = V [X] + V [Y ] = 5 (c) σ[x + Y + ] = V [X + Y + ] = 5 3.9. 3 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 ( + + 3 + 4 + 5 + 6) = 7 () V [X ] = 6 ( + + 3 + 4 + 5 + 6 ) ( ) 7 = 9 6 49 4 = 35

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 ] = 35 35 7 σ[x + X + X 3 ] = = 3.. 7 4 X 76 Y σ[x 4Y + 3] 3.. V [X] = 7 6 5 = V [Y ] = 76 6 = 9 σ[x 4Y + 3] = V [X 4Y + 3] = 4V [X] + 6V [Y ]] = 344 = 86 4. 4... 4. ( ). ( ) () 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) [ ].4.3.. 4 4

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.) =.796 4.3. 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 () 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) =.477.43 =.74 P (.5 X.4) = P ( X.5) + P ( X.4) = φ(.5) + φ(.4) =.95 +.49 =.64 () Y = X 4 Y P (38 X 43) = P ( Y 3) = φ(3) + φ() =.477 +.499 =.976, P (45 X 45) = P (.5 X.5) = φ(.5) + φ(.5) =.95 +.494 =.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) =.999 4.5. 3 47 5 () 9 = 5 + () 3 4 X 4.5. () 5 = 45 5 + P ( X.5) =.95 9 47 5 = 6 ().95 Y P (47 Y 69.5) =.95 69 C 3 C D D

84 D 4. 4. 4.6. 5 3 5 45 A N(, ) P (A >.58) =.5 4.6. X 3 = Y N(, ) X = 45 Y =.6 5 Y.58 5 + = 5 + 45 3 5 4.7. 5 8 A N(, ) P ( < A <.64) =.4 = 76 4.7. 9 5 +.64 8 4.8. 3 3 36 A N(, ) P ( < A <.53) =.9 4.8. 4.9. 9 3 3 37 A N(, ) P ( < A <.53) =.9 4.9. 4. ( ). N(,.5) X P (X 9) P (X t) =.6 t 4.. P (X < 9) =.549538, t =.3865 4... X N(, ) P (X > a) = ( ) exp t dt π a P (X > a) 4. ( ). a > P (X > a) < ) exp ( a πa a

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 45 47 9 5 4 5. 5... {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= 3 5.. 5.. 3 [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 5... 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 5.7 ( ). X, X,..., X N N(, σ ) N N N N Z Z X. = 6 6 6 X. Z N X....... N N N(N ) N(N ) N(N ) N(N ) Z, Z,..., Z N N(, σ ). 5.7 5.8. 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 6. 6.. 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 9 6.3. [, ) Y m χ - f(x) = C x m e x χ(, ) (x)

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. = 6 6 6 X. Z N X....... N N N(N ) N(N ) N(N ) N(N ) X n

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 - 6.5. X, X, X 3 3 3 3 A = 3 3 6 i= i= Y 3 3 3 Y = X X Y 3 X 3 3 3 6 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 3 3 3 3 ( ) = X + X + X X + X + X 3 3 3 9 χ - = Y + Y + Y 3 Y = Y + Y,,,... X P (X = k) = λk k! e λ (k =,,,...) k )

9 6.6. Po(3) 3 4 5 6 7.498.493.4.4.68.8.54.6 6.7 6.7 ( ). 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) =.975 6.8. 39 i= λ λi e i! = λ = 57.5 λ = 8.6 λ f 8 (t) dt = P (X > λ) > P (X > 57.5) =.975 6.9 ( ). Po(9) X X =,,, 3, 4, 5, 6, 7, 8

9 6.9..34.688.499897 3.49949 4.3373755 5.676879 6.9939 7.764 8.375564 9.375564.85876 6.. 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

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 3.5..5 4 4

93 6.4 ( ). 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. = 6 6 6 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 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 3 + + X n ) > α } m (Y + Y 3 + + 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. n n n n Z Z. = 6 6 6 Z n....... n n(n ) n(n ) n(n ) n(n ) W W. W m m m m m = 6 6 6....... 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. /7 3 4 5 6 7.334.57.35.88.6.6.6.3 6.9 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. 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 -.68.5 9 9 9C i >.5 6.. n = 9, k = 9 m =, m = 4 6.9 9 9 9C i = f,4 (x) dx > f,4 (x) dx =.5 i= 6. ( ). (, 4) F -.68.5 (4, ) F - m.975 m i=.68 6.. 6. m = /.68

97 6.3. 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 ] = 4 + 9 + 4 + = 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) 5 4 6.4 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) 6.4.. 6.4 (). 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. 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 Part 5. 7. 7... 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) 7.. 7. 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=

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) ) = 3.36 9 (3) 9 5.95(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) ) = 5.95 7.5 ( ). 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,...)

(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,...)

() {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

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

4 () X, X,..., X n () Y n () lim n nθ σ = lim P ( Y n m > θ) = n Y n 7.3.. 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=

5 7.6. 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 )

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 7.4.. 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

7 8. 8... [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 95 8.. 89.6 436 95 8.. Y = 436 N = σ = 89.6 ( Y.96 σ, Y +.96 σ ) = (436.96 6, 436 +.96 6) = (44.4, 448.4) N N 8.. 95 N X, X,..., X N ( ) X + X + + X N P I =.95 N I 95 95 95