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. 39. 4.. 4.. 43. 46.. 46.. 47.3. 47 Part 4. 5. 5

3.. 5.. 5.3. 5 3. 53 3.. 53 3.. 55 4. 55 4.. 55 4.. 56 5. 57 5.. χ - 57 5.. t- 59 5.3. F - 6 Part 5. 64 6. 64 6.. 64 6.. 65 6.3. 66 6.4. 68 7. 69 7.. 69 7.. 7 8. 7 8.. 7 8.. 73 8.3. 74 9. 78 9.. 78 9.. 79

4 Part 6. 8. 8.. 8.. 8.3. 8. 85.. 85.. 86. 86.. 87.. 9.3. 93 3. 93 4. 94 4.. 94 4.. 97 Part 7. 98 5. 98 5.. 98 5.. 5.3. 6. 6.. 6.. 5 7. 5 7.. 5 7.. 7 7.3. 9

5 Refereces

6 : () () ysawao@tmu.ac.jp () () (3) (4) () () (3) () () (3) (4)

7 (5) () () (3) 7 () () (3) (4) (5) (6) (7) (8) (9) () () () (3) (4) (5) () () (3) exp(a) e a (4) R

8 Part..... k k! P k = = ( )( ) ( k + ) ( k)!!! = ( )( )! =.. 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) ()..

9 () (a) 4! = 4 (b) 4 4 3! = 6 4 6 = 48 (c) 4 3 = (d) 3 84 (). (a) (b) (b) 4 + 48 = 48.... k k! ( )( ) ( k + ) C k = = k!( k)! k! ( ) C k = k. (a + b) = (a + a + + a N ) = C k a k b k k= l,l,,l N, l +l + +l N =! l!l! l N! a l a l a N l N l,l,,l N, l +l + +l N = l + l + + l N = l, l,, l a + a + + a N = M. a + a + + a N = M a, a,, a N.3. N H M = M+N C N () 3 () 3 (3) (4) 5 (5) x = m y = m, (, ) (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 = A, B, C, D, E α + β + γ + δ + ε = 5 9 C 5 = 6.... N N /N. ().

() Ω () (3) (4) (5) Ω (6) ω (ω Ω) (7) (8) ( ).. () {,, 3, 4, 5, 6} (),, 3, 4, 5, 6 (3) {} {} {6} (4) 64 (5) {, } {3, 4}.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 8.4. ( ) (, ) ( ) (, ) ( ) (, ) ( ) (, ) Ω = {(, ), (, ), (, ), (, )}....5. () () (3) /6

(4) /.6. 5 8.8.7. () 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 =.8 ( ). A A P (A) [, ] () P (A) () P (Ω) = (3) A, B P (A B) = P (A) ( + P (B) ) A, A,, A P A i = P (A i ).9. i= () 3 6 P ({3}) = 6 (), 3 P ({, }) = 3 (3) P ({3}) = (4) A {,, 3, 4, 5, 6} A A 6 A 6.. Ω = {ω = (ω, ω,, ω j, ) : ω j {, } (j =,, )} i= 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 ) 3 i= σ- i= 3. 3... 4 3 4 3 4 3 44.4. 33.3 3. ( ). A, B P (A) > A B P (A B) P (B A) = P A (B) = P (A) 3.. () P (A) = 8, P (B) = 7, P (A B) = P A(B) () 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 3.. () P A (B) = P (A B) P (A) () A, B P (A B) = P (A)P (B) = = 4 5 P (A B) = P (A) + P (B) P (A B) P (A B) = 4 + 5 = 5.

4 (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 { } (a) = P { ({ }) } P ({ } { }) = P ({ }) P ({ }) = P ({ }) = 8 398 = 54 99 (b) = P { ({ }) } P ({ } { }) = P ({ }) P ({ }) = P ({ }) = 5 398 3.3. 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 3.3. B, C, D B = B C = C D = D

5 () 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. () 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.

6 3.4. 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 3.4. () 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 P ({ }) = 4 = 3, P ({ }) = 4 = 3 P (D ) = 3 + 4 + + = 3. 3 () D D B, C P (D B ) = 9 + 3 C = 3, P (B C ) = 9 + 6 = 5 3 3, P (D ) = 3 P D (B ) =, P D (C ) = 5 = 5 7 3.5. A, B, B P (A) > P A (B B ) = P A (B ) P A (B B ) P (B B ) = P (B ) P (B B )

7 3.5. 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 ) 3... A, B P (A)P (B) = P (A B) P (A) > P A (B) = P (B) [ ] 3.6. Ω = {(, ), (, ), (, ), (, )} A B A = {(, ), (, )}, B = {(, ), (, )}, A B = {(, )} P (A) =, P (B) =, P (A B) = P (A)P (B) = P (A B) 4 A, B 3.7. 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 3.8. a, b r r A a B b () a b P (A) = r, P (B) = r, P A(B) = r = P (B) A, B () a b P (A) = r, P (B) = r, P A(B) = r P (B) A, B

8 3.9. (5) () 4 () 67 67, (), (3) 6 4, (4) 3 () (3) A = {, 4, 5 }, B = {3, 6 } (), (), (3), (4) A = { 3 }, B = { } A B = 3.9. (), (), (3), (4) () = 67 4 () () A B = () (3) 3 (5) 3.3.. H, H,, H N Ω = N H i i =,,, N P (H i ) P (A H i ) P (H i A) N N H i H H H N i= 3.. H, H,, H N Ω = A Ω P (A) = i= N P (A H i )P (H i ) i= i= i= H i N H i ( ) ( N N N ). A Ω Ω = H i P (A) = P (A Ω) = P A H i = P A H i (.8) N N P (A) = P (A H i ) = P (A H i )P (H i ) i= i =,,, N P (H i ) P (A H i ) P (H i A) i= i= i= i=

9 3. ( ). 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). 3. 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 ) 3.. 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 3.. 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 )P (H ) + P (A H )P (H ) + P (A H 3 )P (H 3 ) = 5 + 7 + 8 + + 5 = 8 7 P (H A) = 7 53 7 = 53 = 53 7 3.3. 4 4 () () 3.3.

() 4 /4 () 4 / 3/4 / = 3/8 () P ( ) = 4 P ( ) = 3 4 P ( ) = P ( ) = P ( ) = P ( ) P ( ) + P ( ) P ( ) = 4 + 3 4 = 3 8

Part. 4. () 4.. (). 4. (). (4.) (4.) (4.3) () f : [a, ) R b > a f [a, b] 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] 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] b a f(x) dx = lim A c A a f(x) dx + lim B c b B f(x) dx (4) 4.. b a f(x) dx () x a dx x a dx = lim R R x a dx [log x] R x a (a = ) [ ] dx = R a + xa+ (a ) log R x a dx = R a+ a + (a = ) (a ) = a + (a ) (a < )

(4.4) (4.5) () x a dx x a dx = lim ε ε x a dx [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 cos x dx (a = ) (a ) = a + R ε ε x a dx + lim R cos x dx = si R R R dx (5) x dx x = lim dx ε ε ε x + lim dx ε x x a dx = (a ) (a > ) ( (6) x + ) dx x + dx (7) = π x (8) log x dx = lim ε = lim ε ε log x dx + lim lim log( x) dx ε ε log x dx + lim lim ε ε ε log x dx lim ε log ε = ε log x dx = lim log x dx = lim [x log x ε ε ε x] ε = lim ( ε log ε + ε ) = ε

3 4... 4.3. a < b m, N b a (x a) m (b x) dx = m!! (b a)m++ (m + + )! b. (x a) m (b x) dx = a m + 4.4. () 3 π si x dx = () π si x dx = π π cos x dx = cos x dx = b a (x a) m+ (b x) dx 3 3 3 π. t = π x (si + x cos x) = ( + ) si x cos x si + x = ( + ) si x ( + ) si + x (4.6) ( + ) π π si x dx = ( + ) si + x dx () () π dx = π 4.4 4.5. π. (4.6) π 4.6. si x dx si x dx π π exp( x ) dx = π si x dx = π π si x dx = ( ) si x dx π π si x dx = = π si x dx =

4. 3 x exp( x ) ( x 4 x ) 4 exp( x ) e 3 x x ) 4 exp( x ) ( x 4 = exp( x ) { exp e 3 x ( ))} x + 4 log ( x 4 t 4 t + log( t) t, exp( t) t exp( x ) dx ) 4 exp( x ) ( x 4 { )} exp( x ) exp ( x4 4 { ( exp( x ) exp )} 4 /3 4 /3 exp( x ) exp( x ) dx = lim ( x x = si θ 4.5 (4.6) π lim 4 ( x ) 4 dx = 4 4 ( x si 8 + θ dθ = lim 4 ) 4 4 ) 4 dx exp( x ) dx + 3 /3 dx = lim ( x ) 4 dx π ( x ) 4 dx = 4 si 8 + θ dθ π π si 8 θ dθ si 8 + θ dθ = π C k!! 4.7 ( ). lim! = π + e.! = t e t dt t = (s + )! = + e ((s + )e s ) dt

5! + e = s ((s + )e s ) dt max(, (s + )( s)) (s + )e s e s / ( s ) ds ((s + )e s ) dt π! + e = ((s + )e s ) dt π e s / ds 5. 5... 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 ) 5.. x y 3 dx dy = x y 3 dy dx = 5.. [,] [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]

6 5.3 ( ). f D = {(x, y) R : x + y R } (5.) f(x, y) dx dy = f(r cos θ, r si θ)r dr dθ D [,R] [,π] 5.5. 5.4. g [, R] g( R x + y ) dx dy = π rg(r) dr D X = ax + by, Y = cx + dy D D f(x, Y ) dx dy = f(ax + by, cx + dy) ad bc dx dy D D 5... 5.5 ( ). 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 si θ cos φ, r si θ si φ, r cos θ)r si θdr dθ dφ f( x + y + z ) 5.5. 5.6 ( ). f [, R] f( R x + y + z ) dx dy dz = 4π f(r)r dr {x +y +z R } 5.6 5.7 5.7. 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 N x + x + + x N =

7 5.3.. N N N N N 6 (5.) A = A N = 6 6 3.......... N N(N ) N(N ) N N N(N ) x + x + x 3 + x 4 + x N N N N N N x x y y. = (x + x ) x 3 6 6 (x + x + x 3 ) 3 x 4 y N. 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 5.8. = f(x, x, x 3,, x ) dx dx dx 3 dx f(y, y, y 3,, y ) dy dy dy 3 dy

8 Part 3. 6. 6... Ω M P (Ω, M, P ) (Ω, M, P ) Ω ( ) 6. ( ). () {x, x,, x, } () (a, b) 6.. Ω = {(a, b) : a, b 6, a, b } X : Ω R X(a, b) = a X 6.3. 6.4. A χ A { (x A) χ A (x) = (x / A) 6... 6.5. (), 3, 4, 5, 6 () = 5 6, 3, 4, 5, 6 (3) [, ].5.5.5 7. 7... 7. (). () X : Ω {, ±, ±, } p = P (X = ) {p } {=, ±, } () 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)

9 7. ( ). µ R ( ) X µ P (X (a, b)) = µ((a, b)), a, b µ = P X a < b 7.3. X Ω = {,, 3, 4, 5, 6} Ω = {, } p X 7.4. 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 R P (a X b) 7.5 ( ). R ν ν((a, b)) = b a f(x) dx, < a < b < f ν f ν ν F (x) = x f(y) dy 7.6. ν f f(y) dy =

3 7.7 ( ). A χ A, A { (x A ) χ A (x) = A (x) = (x X \ A = A c ) 7.8 (). U(a, b) f(x) = b a χ (a,b)(x) 7.9. U(5, 8) X P (6 < X < 7) 7.9. f(x) = 3 χ (5,8)(x) P (6 < X < 7) = 7 6 8 3 χ (5,8)(x) dx = 5 3 dx = 3 7.. X f(x) = αx( x)χ (,) (x) x R () f α () P (X > /) (3) P (/4 < X < /3) 7.. () P ( < X < ) = = α = 6 () P (X > /) = / (3) P (/4 < X < /3) = f(x) dx = 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 7.. () () 7..

3 () 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 7... X 7.5 X F (α), α R F (α) = P (X > α) F (α) 7.. X F (α) () lim F (α) = α () lim F (α) = α. () lim F (α) = = α () lim F (α) = = α 7.3. X F (α) p, q F (α) = p ta α + q p, q P ( < X < ) ta ta ( π/, π/)

3 7.3. lim F (α) = lim F (α) = π α α p + q =, π p + q = p = π, q = P ( < X < ) = P (X > ) P (X ) = ( π π 4 + ) = 4 8. P (X A) = P ({X A}) P (X = a) = P ({X = a}) 8. ( ). () X, Y A, B P (X A, Y B) = P (X A)P (Y B) () X, X,, X B, B,, B P (X B, X B,, X B ) = P (X B )P (X B ) P (X B ) 3 8.. (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) = 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

33 X X X, X,, X a, a,, a X P (X = a, X = a,, X = a ) = P (X = a )P (X = a ) P (X = a ) 8.3. 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 =.x x x k () = j=. () x k =. () x k k (x k {, }) k =,, x [, ) X k (x) = x k X k, X, X,, X k, X k = k X k = k 8.4. 8.3 () x () a, a,, a k {, } P (X = a, X = a,, X k = a k ) (3) X, X,, X k 8.4. () () 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 )

34 9., 9... 9. ( ). 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 = E[X] E(X) 9..,, 4, 9, 4 ( + + 4 + 9 + 4) = 6 5 5, 4, 6, 8, 96 ( + 4 + 6 + 8 + 96) = 59.6 5 E[X] = E[X ] 9.3. P (X (a, b)) = b X x π( + (x + ) dx = lim ) R,R a π( + (x + ) ) dx R R x π( + (x + ) ) dx 9.4 (). 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

j= 35 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 N N () E[X] = a j P (X = a j ), E[Y ] = b k P (Y = b k ) j= j= N N E[X] = a j P (X = a j ) = j= a j E[Y ] = j= k= M k= N k= j= k= j= P (X = a j, Y = b k ) = N M b k P (X = a j, Y = b k ) j= j= k= M a j 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 9.5. () x < + x + x + + x + () x < + x + 3x + + x + (3) 9.5. () x () + x + x + + x + = x + x + 3x + + x + = ( x) (3) + + 3 3 + = = =

36 9.6. r H h V v X v V () H u (H u) du () h H P ( h h ) (3) P ( h h ) = H (4) E[h] h f(s) ds f(h) h f(h), h 9.6. H () u(h u) du = H4 () P ( h h ) = H3 (H h ) 3 H 3 (3) () f(h) = (4) E[h] = H h f(h ) dh = H 4 3(H h) H 3, h H 9.7. x + y = P A = (, ) () P = (cos θ, si θ) AP θ π () θ [, π] AP 9.7. () AP = ( cos θ) + si θ = cos θ = 4 si θ = si θ = si θ θ π () E[AP ] = π si θ π dθ = 4 π. 9.6 9.7 9.8. P (X = k) = ( 4 5 5 X( ) P (a X b) = b a ) k 4e 4x dx

37 () m () M 9.7. () 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 9...,, 3, 4, 5 8, 9, 3, 3, 3,, 4, 4, 7 3 9.9 ( ). 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] 9.4 E[E[X]X] = E[X]E[X] = E[X] E[E[X] ] = E[X] 9.. () 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

38 9. ( ). a, b X V [ax + b] = a V [X]. (ax + b ae[x] b) = a (X E[X]) 9.4 9. ( ). X, Y V [X +Y ] = V [X]+V [Y ]. V [X + Y ] = E[(X + Y ) ] E[X + Y ] X, Y V [X + Y ] = E[X ] E[X] + E[Y ] E[Y ] + E[XY ] E[X]E[Y ] = V [X] + V [Y ] 9. 9.3. 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 9.4 ( ). E[XY ] E[ X ]E[ Y ] X = ty Y = tx t. X = φ(t) = E[ tx Y ] t 9.5. 5 54, 35, 57, 49, 55 X 9.5. 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. 9.6. 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 )

39 9.6. () () 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 9.3.. 9.7 (). X V [X] σ[x] σ[x] σ(x) 9.8. X P (X = ) = P (X = ) = P (X = 8) = P (X = 9) = P (X = ) = 5. E[X] V [X] σ[x] 9.8. E[X] = ( + + 8 + 9 + ) = 6 5 V [X] = 5 (( 6) + ( 6) + (8 6) + (9 6) + ( 6) ) = 4 σ[x] = V [X] = 4 9.9. X () X, X,,. P (X = ) = 4, P (X = ) = 8, P (X = 4) = 4, P (X = 6) = 8, P (X = 8) = 4. () 3 X. X,,. 9.9. () 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

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 9. (). 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] (3) X A 5 + A E[X] σ[x] σ[x] () 7 9.. () 44, 484, 59, 576, 65, 676, 79, 784, 84, 9 () E[X] = 8, σ[x] = 3 (3) 75 + 5 = 8 3 9.. P (a X b) = 3 4 b a x( x) dx, a b [, ] X X 9.. 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 9.. T f : R R { f(t) = aχ [,] (t)t T + a t T + ( t ) = ( ) () f(t) X a

4 () a P ( X, 5) (3) E[X] (4) V [X] (5) σ[x] 9.. () f(t) dt = a T + a = T + () a P ( X.5) = (3) E[X] = (4) V [X] = (5) σ[x] = V [X] = T + 3 t (T + )t T + dt = T + T + 3 ( ) T + t (T + )t T + dt = T + 3 T + T + 4 T + T + (T + 4)(T + 3) f(t) dt = T +..... ( ). x, x,, x () x, x,, x () (3) (4) x = x j (5) S = j= (x j x) (6) (a) / (b) ( )/ ( + )/ j=.. X, X,, X () () (3)

4.3 (). 5 75 5 75.4.,, 3, 3 4 X X X.4. P (X = 3) = /6, P (X = 4) = /3, P (X = 5) = /3, P (X = 6) = /6.5. 5,, 8, 9,.5. ( + + 8 + 9 + ) = 6 5 5 (( 5) + ( 4) + + 3 + 4 ) = 4.6. N µ σ N a, a,, a N X, X,, X () X = X + X + + X () S = X + X + + X X i<j i, j a ij i<j a ij = i= j=i+ (a + a + + a ) = a j + j= a ij i<j a i a j.6. Ω = {(a i, a i,, a i ) : a i, a i,, a i } () Ω E[X] = Ω i,i,,i,,, (a i + a i + + a i )

43 a {i, i,, i } i, i,, i N C a N P Ω = N P E[X] = N N P a j = N P j= (N )! N! () [ X + X ] + + X E = N E[X X ] = i<j N i<j N i= (N )! (N )! N a j = N j= N a j = σ + µ j= a i a j P (X = a i, X = a j ) = i= N(N ) i<j N ( 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) i<j i= E[X i X j ] = µ (N ) σ N j= a i a j E[X ] = µ N(N ) σ + µ + σ = µ + σ (N ) σ E[S ] = σ + N( ) (N ) σ = (N ) σ a j... 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

44 X Y.7 (). 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].8 (). 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 ].9 ( ). r[x, Y ] = Y = ax + b a > b r[x, Y ] = Y = ax + b a < b. (). 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.. () 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.

45 () 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 ].. 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.. X Y (X, Y ) 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

46.. Y = 6X + 97 a = 43 b = 55.3. 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 ].3. () ( x y) dx dy = α = 6 D 6 ( x ) () E[X] = 6 x( x y) dx dy = 6 x( x y) dy dx = 6 D x( x) dx = 4 E[Y ] = 6 y( x y) dx dy = D 4 ( x ) (3) E[X ] = 6 x ( x y) dx dy = 6 x ( x y) dy dx y D x E[X ] = 6 E[Y ] = 6 y ( x y) dx dy = D (4) E[XY ] = 6 xy( x y) dx dy = 6 D x E[XY ] = ( x (5) V [X] = V [Y ] = E[X ] E[X] = 6 = 3 8 x ( x) dx = ) xy( x y) dy dx y 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.... 9 9.33 9. ( ). X p > P ( X > λ) λ p E[ X p ]

47. P ( X > λ) = E[χ { X >λ} ] λ p χ { X >λ} X p... ( ). ( ( ) ). X, X,, X k, lim (X + X + + X ) = E[X ] 6..3 ( ( ) ). X, X,, X k, X M M [ lim E (X ] + X + + X ) E[X ] =. Y = X E[X ], Y = X E[X ], Y, Y,, Y X, X,, X E[X ] = [ E (X ] + X + + X ) = E[X + X + + X ] + E[X i X j ] i<j = E[X + X + + X ] + = E[X ] i<j E[X i ]E[X j ].3...4 ( ). X, X,, X k, lim E X + X + + X E[X ] lim N(, V [X ]) [ ( )] X + X + + X E[X ] f = ( ) t f(t) exp dt πv [X ] V [X ] {Y } = µ f lim E[f(Y )] = f(t)dµ(t)

48 [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) µ(y I) Y (a, b) lim P (Y I) = µ(i) k X k (X + X + + X ) [ ] E (X + X + + X ) = E [X + X + + X ] = E[X ] = E[X ] V [ ] (X + X + + X ) = V [X + X + + X ] = V [X ] = V [X ] X v, v / ( ) P (X + X + + X ) (a, b) b ) exp ( x πv a v dx a a b lim = b! π + e. P (X + X + + X = k) = X, X,, X + k = C +k! C +k () + π( + k) +k+ ( k) k+ x = v k = + π( + k) +k+ ( k) k+ C +k + π ( + ) v x ( x v ) + v x x v = ( + ) v x ( x ) + v x x π v v = ( v ) ( πv v x + ) v x ( x ) + v x x v v

49 lim ( ( x = exp v = exp v x ( x v ) exp ) ) ( + ) v x ( x ) + v x x v v ( x v C +k ) ) exp ( x v ( ) v exp x πv v v (a, b) ( ) P (X + X + + X ) (a, b) πv b a ) exp ( x v dx

5 Part 4...... (). X,,,..., P (X = k) = C k p k q k X B(, p) p q = p p q = p ( ) p(x;, p) = p x q x, x =,,,, x.. P P + 6 P X.. X -6-3 3 6 9 p /64 3/3 5/64 5/6 5/64 3/3 /64.3. 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. p A = 7 C 7 = 7 64, p B = p E = 35 8, p C = p D = 64.....4 ( ). a, b (.) k a k b k C k = a (a + b) k= (.) k(k ) a k b k C k = ( ) a (a + b) k=

. (a + b) = 5 C k a k b k a k= (a + b) = k C k a k b k k= a (.) (.).3...5 ( ). X B(, p) E[X] = p. p k p k ( p) k C k X E[X] = k p k ( p) k C k (.) E[X] = k= k p k ( p) k C k = p(p + p) = p k=.6. X B(, p) =, p = E = 5 4 4 5.7 (). < p < q = p X B(, p) V [X] = p q σ[x] = p q. P (X = k) = C k p k q k V [X] = k p k ( p) k C k p = k= k(k ) p k ( p) k C k + k= k p k ( p) k C k p (.) (.) V [X] = ( )p + p p = p( p) = pq.8. X 8 8 Y () E[X], V [X], σ[x] () E[ 47X + 6], V [ 47X + 6], σ[ 47X + 6] (3) E[X + Y + ], V [X + Y + ], σ[x + Y + ] k=.8.

5 (a) p X E[X] = p (.5), V [X] = p( p) = p q (.7), σ[x] = p q (.7) (b) E[a X + b] = a E[X] + b ( 9.4), V [a X + b] = a V [X] ( 9.), σ[a X + b] = a σ[x]. (c) X, Y E[X + Y ] = E[X] + E[Y ] ( 9.4), V [X + Y ] = V [X] + V [Y ] ( 9.) () (a) E[X] = p = 8 6 = 3 (b) V [X] = 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 + ] = 5.9. 3 X, X, X 3 () E[X ] () V [X ] (3) σ[x ] (4) V [7X + 58] (5) σ[x + X + X 3 ].9. (a) a, b E[a X + b] = a E[X] + b ( 9.4), V [a X + b] = a V [X] ( 9.) (b) X, Y, Z E[X + Y + Z] = E[X] + E[Y ] + E[Z] ( 9.4), V [X + Y + Z] = V [X] + V [Y ] + V [Z] ( 9.) () E[X ] = 6 ( + + 3 + 4 + 5 + 6) = 7 () V [X ] = 6 ( + + 3 + 4 + 5 + 6 ) (3) σ[x ] = 5 V [X ] = 6 (4) V [7X + 58] = 49V [X ] = 75 ( ) 7 = 9 6 49 4 = 35

53 (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. 3... 3. ( ). ( ) () 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(, ) 3.. X, Y,, 4 φ(a), a > () P (X 6) () P (6 Y ) φ(a) = P ( X a) (a > ) 3.. () φ(6) () Z = Y Z P (6 Y ) = P ( Z ) = φ() + φ() 3.3. 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)

54 () X N(4, ) P (38 X 43) P (45 X 45) (3) 85 6 (4) 7 5 X P (X 3) 3.3. () P (X a), P (X > a) a > a 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 3.4. 3 47 5 () 9 = 5 + () 3 4 X 3.4. P ( X.5) =.95

() 5 + 9 47 5 = 6 55 ().95 Y P (47 Y 69.5) =.95 69 C 3 C D D D 3. 3... X N(, ) P (X > a) = ( ) exp t dt π a P (X > a) 3.5 ( ). a > P (X > a) < ) exp ( a πa a. P (X > a) = π a ( ) exp t dt < πa a ( ) t exp t dt = ) exp ( a πa 4. 4... {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 4.. 4.. 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 4.3 ( ). A R N ν(a) = ν f ν A f(x) dx f

56 4.4. 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) 4... X, X,, X N a X + a X + + a N X N 4.5 (). σ, σ > X, X N(m, σ ), N(m, σ ) X + X N(m + m, σ + σ ). X m, X m m = m = ) P (X + X > a) = exp ( x πσ σ σ y dx dy σ X = x σ, Y = y σ x+y>a P (X + X > a) = exp ( X + Y ) dx dy π σ X+σ Y >a σ Z = σ + σ X + σ σ + σ Y, W = σ σ + σ X σ σ + σ Y P (X + X > a) = π Z>a σ +σ P (X + X > a) = π σ + σ Z>a exp ( Z + W ) dz dw exp ( Z + W ) dz dw (σ + σ ) W ( Z ) P (X + X > a) = exp dz π(σ + σ ) Z>a (σ + σ ) X + X

57 4.6 ( ). 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.. χ -. 5. 5. (χ - ). X, X,, X Y = X + X + + X Y χ - χ m(α) = P (Y α) 5.. [, ) Y m χ - f(x) = C x m e x χ(, ) (x) C ( m ) C = Γ = e t t m dt f(x) dx = C C. α > P (Y α) P (Y α) = ( π) P (Y α) = C i= X i α exp r α r = R P (Y α) = C r exp α ( i= ) ( r dr ) y i dy R exp( R) dr

58 A N (5.) N N N N N Z Z X. = 6 6 6 X 3. Z N X........ N N N(N ) N(N ) N(N ) N(N ) N(N ) N N N (X i X) = N i= N (X i X) = N i= N X i X i= N Z i N Z = N i= 5.3 ( ). X, X,, X N X i N(m, S ) Y = S (X i X) Y N χ,,, X i= P (X = k) = λk k! e λ (k =,,, ) k 5.4 ( ). χ - f (x) λ > k λ λi e i! = f k+ (t) dt i=. f k+ (t) k C k λ f k+ (t) = C k t k e t/ χ (, ) (t) C k ( C k = t k e dt) t/ = k+ k! λ > λ f k+ (t) dt = C k t k e t/ dt λ N i= Z i = k+ C k t k e t dt λ

59 λ t k e t dt = ( t k kt k k(k )t k k!)e t + C ( ) λ f k+ (t) dt = k+ C k (λ k +kλ k +k(k )λ k + +k!)e λ k = k! + λk (k )! + + 5.5. 39 i= λ λi e >.975 λ i! X 8 χ -P (X > 57.5) =.975 e λ 5.5. 39 i= λ λi e i! = λ f 8 (t) dt λ = 57.5 λ = 8.575 5.. t-. 5.6 (t- ). Y, X, X,, X N N(, ) Z = Y N X j N N t- 5.7 (t- ). N t- f(x) = C ( + x N C ) N+ ( C = + x N j= ) N+ dx (x R). t 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 P (T > α) = C r N exp ( y + r ) dy dr y > α N r C P (T > ) = / C y, r P (T > α) = C π R P (T > α) = C ( )(R cos θ) N R exp ta α N π ( ) cos N θ dθ ta α N ) ( R dθ dr

6 ta θ = t P (T > α) = C α/ N (t + ) N+ dt = C α (t + N) N+ dt 5.8 ( ). 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-. T T = σ N (X + X + + X N mn) N (X j X) σ N (5.) 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 5.3. F -. F - 5.9 (F - ). Y χ (m) Y χ () Z = m Y Y (m, ) F - (m, ) F - X, X,, X m Y, Y,, Y Z = m X j Y j m j= j= j= i=

6 5. (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 C f(x) dx =. P (F > α) = C exp ( X + X + + X + Y + Y ) + + Y m dx dy D dx dy = dx dx dx dy dy dy m { D = (X + X 3 + + X ) > α } m (Y + Y 3 + + Y m ) (m + ) C P (F > ) = C X, X,, X Y, Y,, Y m P (F > α) = C r r m exp ( r ) + r dr dr r >( m α)/ r ta θ = m α θ ( π/, π/) r, r π P (F > α) = C cos θ si m θ dθ θ ta θ = x = y / ( P (F > α) = C x + = C = C = C ( m α)/ ( m m α α ) xm α)/ (x + ) m+ m y (y + ) m+ y m (my + ) m+ dy dy ( x dx x + ) m x + dx 5. ( ). m, X, X,, X, Y, Y,, Y m X, X,, X N(m x, σ ) Y, Y,, Y m N(m y, σ ) X = X i, Y = m Y i, SX = (X i X), SY = m (Y i Y ) m m i= i= i= F = S X ms Y m (, m ) F- i=

6 S X ( ) S Y. Z Z. = 6 6 6 Z....... ( ) ( ) ( ) ( ) σ X m x X m x. X m x m m m m W W Y m y. = 6 6 6 Y m y σ. W m Y....... m m y m m(m ) m(m ) m(m ) m(m ) Z, Z,, Z, W, W,, W m F = Z j m W j m j= F - F (, m ) j= k 5. ( ). k < m = (k + ), m = ( k) 5. (m, m ) F - f m,m p (, ), q = p k C i p i q i = i= m p m q f m,m (t) dt C i p i q i = i=k+ m p m q f m,m (t) dt

.! ( k )!k! p! ( k )!k! = =! ( k)!k! 63 t k ( t) k dt p p t k ( t) k dt t k (( t) k ) dt! ( k)!k! pk ( p) k! + ( k)!(k )! p t k ( t) k dt! k t k ( t) k dt = C i p i q i ( k )!k! p i= m t x = m t + m! ( ) k+ ( ) k m t m I = dt k!( k )! m t + m m t + m m p m q f m,m m, m C k C f m,m (t) dt = C i p i q i m p m q i= p = C = 5.3 ( ). 39 9 39C i >.5 i= 5.3. 59 9 59C i = i= f,4 (x) dx >.68 f,4 (x) dx =.5

64 Part 5. 6. 6... X, X,, X X, X,, X X, X + X + + X, X X, X,, X 6.. 7, 76, 77, 8, 69, 65, 78, 8, 75, 74 (cm) () 7(cm) () 74(cm) (3) (cm) () () 7, 74 (), (), (3) 6.. 6. 7, 76, 77, 8, 69, 65, 78, 8, 75, 74 (cm) X j X i i= E X j X i 9 = V [X ] i= 9 i= j= j= X j X i 6.3 (). X, X,, X X j X i i= j= j=

65 6.4. 7, 76, 77, 8, 69, 65, 78, 8, 75, 74 () 74.8(cm) () 3.36(cm ) (3) 9 5.95(cm ) 6.5 ( ). X, X,, X T (X, X,, X ) 6.6. B(, p) X p X 6.7 ( ). T = T = T (X, X,, X ) θ k > lim P ( T θ < k) = (), () 6.8 ( ). T = T (X, X,, X ) θ E[T ] = θ 6.9 (). X, X,, X f θ (x) L(θ) = f θ (x )f θ (x ) f θ (x ) θ ˆθ ˆθ = T (x, x,, x ) T (X, X,, X ) θ T 6... T = T = T (X, X,, X ) θ k > 6.. lim P ( T θ < k) = lim P ( T θ < k) =

66 () X, X,, X m = E[X ], σ = V [X ] Y = (X + X + + X ) θ > P ( Y E[X ] > θ) θ σ () 6.. () (.) Z P ( Z > θ) θ E[ Z ] Z = Y m P ( Y m > θ) θ E[ Y m ] E[ Y m ] = V [X + X + + X ] = σ ( )P ( Y m > θ) θ σ () X, X,, X () Y () lim θ σ = lim P ( Y m > θ) = Y 6.3.. X {P θ } θ Θ θ Θ g(θ) X T g E θ [T ] = g(θ) θ Θ 6.. θ θ j X j = X j = θ θ (x, x,, x ) R = x j T = R T (x, x,, x ) T (.) (.) [ ] R E θ [T j ] = E θ = C j θ j ( θ) j = θ θ( θ) + j= T θ j= T T = R(R ) ( )

67 (.) (.) ( ) E θ [T ] = θ + θ( θ) θ = θ X = {(x, x,, x ) : x j =, }, P θ = {B(, θ) } θ (,) G(α, ν) f(x) = Γ(ν) αν x ν e αx χ (, ) (x) Γ(ν) Γ(ν) = Γ(ν + ) = Γ(ν) t ν e t dt 6.. 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) 6.3. X Po(θ) ˆθ = X ˆθ θ θ r θ x x! e θ z x = exp(θ(z )) x= r z = θ x x! e θ x(x )(x ) (x r + ) = θ r x= ˆθ r = X(X )(X ) (X r + ) θ r 6.4. X B(, θ), < θ < g(θ) = θ T (x) ( ) T (x)θ x ( θ) x = x θ, < θ < x= θ ( ) T (x)θ x+ ( θ) x =, < θ < x θ x= = lim θ x= ( x) T (x)θ x+ ( θ) x =

68 6.5. P P m R P X = X j j= E P [X] = P, P P X P 6.6. P P V R P X = X j j= S = (X j X) ( ) j= E P [S ] = σ (P ) S σ (P ) U = (X j X) ( ) j= E P [U ] = σ (P ) U σ (P ) 6.7. θ > [, θ] U(, θ) (X, X,, X ) X = X + X + + X N θ N 6.7. E[X] = E[X ] + E[X ] + + E[X ] = θ/ 6.4.. X, X,, X θ p θ (x) = P (X i = x θ) x, x,, x L(θ x, x,, x ) = p θ (x )p θ (x ) p θ (x ) log L(θ x, x,, x ) = log(p θ (x )p θ (x ) p θ (x )) = log p θ (x j ) 6.8. θ > f(x; θ) = θe θx χ [, ) (x) (X, X,, X N ) () I, I,, I N (, ) P (X I, X I,, X N I N ) () X [, ) g(θ) = θ N e θx θ (3) θ ˆθ = θ j=

69 6.8. () P (X I, X I,, X N I N ) = θ N e θ(x+x+ +x N ) dx dx dx 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. 7... [a, b] θ X, X,, X ˆθ (X, X,, X ) ˆθ (X, X,, X ) ˆθ (X, X,, X ) ˆθ (X, X,, X ) [ˆθ (X, X,, X ), ˆθ (X, X,, X ) P (ˆθ (X, X,, X ) θ ˆθ (X, X,, X )) P m ± ε α P (θ < ˆθ (X, X,, X ) = α, P (θ > ˆθ (X, X,, X ) = α ˆθ (X, X,, X ) ˆθ (X, X,, X ) 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

7 7.. 89.6 436 95??. Y = 436 N = σ = 89.6 ( Y.96 σ, Y +.96 σ ) = (436.96 6, 436 +.96 6) = (44.4, 448.4) N N 7... p p N j X j = X j = Y N = X + X + + X N Y N N p p( p)/n ( ) p( p) p( p) P p.58 < Y N < p +.58 =.99 N N 7.. () 35 () 65 { (i ) X i = (i ) { (i ) X i = (i ) X i P (X i = ) = p [ ] p p N X = X + X + + X N N

7 N(p, p( p)/n) 95 X (p.96 p( p)/n, p +.96 p( p)/n) 95 99 X (p.58 p( p)/n, p +.58 p( p)/n) 95 35/ 65/ p( p) p.96 < 35 p( p) < p +.96, p( p) p.96 < 65 p( p) < p +.96.96 p( p)/n.98 N N p 35/ 65/ 35 p.96 965 < 35 35 < p +.96 965, 65 p.96 35 < 65 65 < p +.96 35 p 7.3. R P (.96 R.4) 7.3. X, X,, X..8 R = X + X + + X V [R] = V [X ] R = R E[R] σ[r] R. =..8 = 6 P (.8 R.) = P ( R ) =.686 7.4. 4 96 95 7.4. 96 4 X, X,, X 4 p = 96, E[X] = 96, σ[x] = 4 p( p) = 4 4 96 ±.96 4 X 49 ± 3.56. 4

7 8. N N x, x,, x N N (x +x + +x N ) X, X,, X N X = (X + X + + X ) X E[X] = E[X ], V [X] = V [X ] 8.. () ph () (3) 48, 67, 8, 9, 55, 53, 67, 5, 73, 74 8... X, X,, X N(m, σ ) σ X + X + + X m N(, σ /) 8.. σ 8 38 () m P ( X m < a) =.99 a () 99 8.. () σ[x m] = σ(x ) =.38 a =.56 a =.38 () [7., 9.]

73 8... X, X,, X N(m, σ ) σ Z = ( ) X + X + + X m, X = X + X + + X, V = (X j X) Z/V t- 8.3 ().. A = 99.8, B = 99.6, C = 99.6, D =., E = 99.6, F =., G = 99.7, H = 99.9, I = 99.8, J = 99.8 () A, B,, J m () A, B,, J σ (3) X, X,, X N(, σ ) X = (X + X + + X ), S = (X j X) 9 Y = X S (4) (5) 95 = 3.6 9 V P (V >.6) =.5 j= j= 8.3. () m = (A + B + C + D + E + F + G + H + I + J) = 99.8 () () σ = ((A 99.8) + (B 99.8) + (C 99.8) + (D 99.8) + (E 99.8) +(F 99.8) + (G 99.8) + (H 99.8) + (I 99.8) + (J 99.8) ) =.34 (3) 9 t- (4) Y 9 t- P (Y >.6) =.5 Y 99.8.34 >.6 (5) 99.8 [99.67, 99.93].6 < X 99.8.34 <.6

74 8.4 ()..58.59,.57,, 63,.65,.58,.64,.65,.57 5 () 8 () 8 (3) X, X,, X 8 N(m, σ ) X = 8 (X + X + + X 8 ), S = 8 8 (X i X) X [, ) S Y = 7 X m S Y Y (4) (5) 7 t- Z i= P (Z >.36) =.5, P (Z >.89) =.5 7 =.645, 5 = 35 8.4. () (.59 +.57 +, 63 +.65 +.58 +.64 +.65 +.57) 8 =.6 () ((.59.6) + (.57.6) + (, 63.6) + (.65.6) + (.58.6) +(.64.6) + (.65.6) + (.57.6) ) 8 =.5.5 =.35 (3) 7 t- (4).6.58 7 <.84.35 (5).6.58 7 >.89.35 8.3.. A, A,, A a A, A,, A a A x x x x A x x x x........ A a x a x a x a x a

75 x j = x ij, x i = a i= a j= x ij, x = a a i= j= x ij µ α, α,, α a ε ij N(, σ e) x ij = µ + α i + ε ij 8.5 (). µ α, α,, α a ε ij N(, σ e) x = a µ i = α i µ a i= j= x ij = µ + α i + ε ij x ij, x i = ( ) () x N µ, σ e ( a ) () x i N µ i, σ e ( (3) x i x N α i, (a ) )σ e a x ij, x j = a j= i= x ij. () x x = µ + a a () x i x i = µ + α i + (3) x i x x i x = α i + i= j= j= ε ij ε ij = µ i + j= ε ij a a l= j= ε ij j= ε lj = α i + a a j= ε ij a l i j= ( ) ( ) a σe + (a ) σ (a ) e = a a a + a a σ e = a a σ e (3) ε lj x ij = x + (x i x ) + (x ij i )

76 a S T = S A = S e = i= j= S T = S A + S e 8.6. () a i= j= a i= j= a i= j= (x ij x ) (x i x ) (x ij x i ) (x i x )(x ij i ) = S e a( ) χ - σ e () α = α = + α a = σe S A a χ - (3) σa σ A = a α i E[S A ] = (a )σa a + (a )σ e i=. a () S e = (ε ij ε i ) () S A = i= j= a i= j= (α i + ε i ε ) a a S A = (ε i ε ) = ( ε i ε ) i= i= a (3) S A = (α i + ε i ε ) i= j= 8.7. {x ij } i=,,,a, j=,,, CT S T, S A, S e ( ) CT = a a x ij, S T = x ij a x kl, a a S A = a i= j= ( i= j= l= x il a a k= l= i= j= ) x kl, S e = a i= j= k= l= ( x ij ) x il l=

77 S T = a i= j= x ij CT, S A = a i= j= x ij CT, S e = S T S A CT C T 8.8 (). A, A, A 3, A 4 5 5 3 4 5 A 6 8 3 3 A 37 8 6 6 A 3 8 3 89 93 A a 3 33 39 54 43 () () 95 F (3, 6) F - 8.8. P (F > 6.3) =.5 () i j x ij CT S T S A S e CT = 4 5 x ij S T = S A = 5 4 i= j= i= j= i= 5 x ij CT 4 5 S e = S T S A j= x ij CT CT = 46.3, S A = 59.8, S T = 5.3, S e = 976.5 3 S A 6 S e (3, 6) F -.89 (3, 6) F - 4.77.5 () ( ) / 5(X i µ i ) 6 S e 4 t-

78 9. X, X,, X S = (X j X) E[S ] = V [X ] S = (X j X) E[S ] = V [X ] j= j= 9... X, X,, X N(m, σ ) m V = σ χ - j= (X j µ) 9. (). 3, 78, 98, 8, 9, 69, 9, 85, 76, 83 (mg/dl) 8 () X, X,, X N N(m, σ ) Y = () 95 χ - Y P (Y < 3.5) =.5, P (Y >.48) =.5 N j= σ (X j m) 9.. () χ () 3.5 < Y < 4.8 σ 3.5 < σ (X j 8) < 4.8 j= (X [, X,, X ) = (3, 78, 98, 8, 9, 69, 9, 85, 76, 83) 73 σ 4.8, 73 ] = [6., 39] [6., 39] 3.5

79 9... 4(+4), 85( 5), 95( 5), 9( 3), 83( 7), 9( 8), 88( ), 3(+3), 7( 8), 95( 5), 87( 3), 85( 5), 93( 7), (+), 95( 5), 5(+5), 3(+3), 8( ), 9( 9), 9 X, X,, X N(m, σ ) m X = (X + X + + X ), V = σ (X j X) χ - Z = (X j X) j= 8 χ - P (Z > 3.53) =.5 9 j= (X j X) X, X,, X 9 3.53 4 5 3.53 9. (). j=.37,.7,.5,.9,.33,.8,.3,.4,.9,.37 (g) () () (3) X, X,, X N(m, σ ) Y = σ (X j X), X = X + X + + X j= (4) 99 9 χ - Z P (Z > 3.5) =.5, P (Z <.73) =.5 9.. () (.37 +.7 +.5 +.9 +.33 +.8 +.3 +.4 +.9 +.37) =.3(g) ( () (.37.3) + (.7.3) + (.5.3) + (.9.3) + (.33.3) +(.8.3) + (.3.3) + (.4.3) + (.9.3) + (.37.3) ) =.84 3 (g )

8 (3) 9 χ - (4) () X, X,, X j= (X j X) =.84 P (Y > 3.5) =.5, P (Y <.73) =.5 7.8 4 =.84 3.5 σ.6 =.84.73 [7.8 4,.6 ]

8 Part 6..... 8 8 3 3 8 985 35.99 p(3) =.49865 3 8 3 8 3.. X N(, ) P (X >.96) =.5, P (X >.58) =.5. ( ). () X Y X N(m, σ ) Y N(m, σ ) X + Y N(m + m, σ + σ ) () X N(m, σ ) ax + b N(am + b, a σ ) (3) X m = X E[X] N(, ) σ σ[x] (4) X N(m, σ ) P (X m) =.5 (5) X N(, ) P (X ) =.977....3 ( ). () H (ull hypothesis) () H (alterative hypothesis) H H θ θ Ω Ω.4. 8 3 () H /6 /6 () H /6 /6.5..6 ( ). () H () H

8 (3) (4).7. 8 3 985 35...5,..8 ( ). X X X X (critical regio) () x X H (reject) () x / X H (accept).9. 8 X λ > P ({X > λ}) =.5 X = {X > λ} 3 X H (H ).,, 5. ( ). () H α =.5,. (), (3),.. 8 3. () /6 985 35 35 98499 () /6 /6 3 3.3... ( ). () H H () H H.3 ( ). H : θ = θ H : θ Θ H R R θ Θ p(θ).4. 8 3

83.5. 3 4 p () 3 () H H 5 (3) H H 5 (4) 387 (), (3).5. () 3 = 89 () (a) H : p = / (b) H : p / (c) 89 <.5 H (3) (a) H : p = / (b) H : p > / (c) 89 <.5 H (4) (), (3) H H.6. 45 p log =.3, log.9 =.458, log 7 =.8547 () / 45. () / H H (3) / H H (4) 55 / (), (3).6. () (a) = 45.9 44. = 5.9 45

84 (b) =.9 45, 5..9 45 log 5..9 45 > log 5.458 45 =.699.5.36 >. () (a) H : p = / (b) H : p / (c).374 >.5 =.699 H / (3) (a) H : p = / (b) H : p > / (c).374 >. = H / (4) (), (3) H H (), (3) () =.9 99. =..9 () =.9,..9 45. H.7. N(µ, ) H : µ = H : µ > () X, X,, X N(, ) X = (X + X + + X ) {X > a} () µ = 95 Y P (Y >.3) =., P (Y >.65) =.5.8 (.7 ). () (X + X + + X ) >.3 a =.3 {X > a} () X, X,, X N(, ) ( X + X + + X P >.3 ) =.95 Y = X + X + + X ( X + X + + X P >.3 ) = P (Y >.3 ) =.95.3 =.65 = 6

85..... (). 8.5 8 7.95 5.5 X, X,..., X 8 X = X + X +... + X 8 8 E[X] = 8.5 v.s. E[X] 8.5 X, X,..., X N(8.5,.5) P (X 7.95) E[X] = ( ), V [X] = ( ) Y = X ( ) ( ) Y N(, ) ( ) P (X 7.95) = P (Y ( ) ) = ( ) P (X 7.95).5/ ( ) Yes/No ( ) /.5/ 9 3 = 3 =.53 8.. ( ) E[X] = E[X ] + E[X ] + + E[X 8 ] 8 8.5 ( ) V [X] = V [X + X + + X 8 ] 64 ( ) 8.5 ( ) ( ) =.53 ( ). ( ).49 = 8.5 8 8 = V [X ] + V [X ] + + V [X 8 ] 64 = 8.5 = V [X ] 8 = 9 3

86 ( ) ( )..... 7 6 7, 75, 7, 74, 7, 74 (cm) 5 () 6 () 6 ((7 7) +(75 7) +(7 7) +(74 7) +(7 7) +(74 7) ) (3). (4).. Z 6 χ - P (Z > 4.4) =.5, P (Z >.6) =.5, P (Z <.63) =.5, P (Z <.4) =.5.. () (7 + 75 + 7 + 74 + 7 + 7) = 7.5(cm) 6 () 6 ((7 7) +(75 7) +(7 7) +(74 7) +(7 7) +(74 7) ) = 5 3 (3) 95.4 < 6 (X j 7) < 4.4 6 (X j 7) = σ = j= (4) 95.63 < j= 6 (X j 7) = 5 <.6 6 (X j 7) σ = j= j=.

87... X, X,, X N N(m x, σ x) Y, Y,, Y M N(m y, σ y) σ x, σ y N (X + X + + X N Nm x ) M (Y + Y + + Y M Mm y ) N(, σ ) σ = N σ x + M σ y.. 5 36 365 95 5 5 + 95 5 = 5 4 5 5 X, X,..., X 5 X = X + X +... + X 5 5 5 Y, Y,..., Y 5 Y = Y + Y +... + Y 5 5 E[X] = m, E[Y ] = m m = m v.s. m = m X, X,..., X 5 N(m, ) Y, Y,..., Y 5 N(m, 95 ) P ( X Y a) =.5 a Z = X Y E[Z] = ( ), V [Z] = ( ) W = Z ( ) ( ) W N(, ) P ( Z a) = P ( W ( ) a) =.5 = p(( ) ) a = ( ) ( ) =. 5 365 36 < a ( ) Yes/No ( ) /.. ( ) ( ) ( ) ( ) 4 ( ) /4 ( ), 96 ( ) Yes

88 ( ).. 8 5 7.3 7.45 7.45.38 () () 5.. X, X,..., X X = X + X +... + X () E[X] 7.3 v.s. E[X] 7.3 E[X] = 7.3 X, X,..., X N(7.3,.38) X P ( X 7.3 a) =.5 a 7.45 7.3 < a E[X] = 7.3, V [X] =.38 X 7.3 Y =.38/ Y N(, ).5 =.5 p(.96) P ( X 7.3 a) = P ( Y a /.38) = P (Y a /.38) =.5 a =.96.38/.66.5.66 E[X] = 7.3 () E[X] 7.3 v.s. E[X] 7.3 E[X] = 7.3 X, X,..., X N(7.3,.38) X P (X 7.3 a) =.5 a 7.45 7.3 < a

89 E[X] = 7.3, V [X] =.38 X 7.3 Y =.38/ Y N(, ).5 =.5 p(.645) P (X 7.3 a) = P (Y ( a)/.38) =.5 a =.645.38/.4.5.4 E[X] = 7.3.3. ().79. () 99.79..3. () N(m A,.) N(m B,.) X, X,..., X Y, Y,..., Y X = X + X +... + X, Y = Y + Y +... + Y m A = m B v.s. m A m B m = m A = m B P ( X Y.79).5 X + X +... + X N(m, ) X = X + X +... + X X N(m, /) Y N(m, /) X Y N(, /5) Z = 5(X Y ) Z N(, )

9 P ( X Y.79) = P ( Z.76) =.468 =.96 () N(m A,.) N(m B,.) X, X,..., X Y, Y,..., Y X = X + X +... + X, Y = Y + Y +... + Y m A = m B v.s. m A m B m = m A = m B P ( X Y.79).5 X N(m, /) Y N(m, /) X Y N(, /5) Z = 5(X Y ) Z N(, ) P ( X Y.79) = P ( Z 7.6).999 ()... X, X,, X M N(m x, σ ) Y, Y,, Y N N(m y, σ ) m x, m y X, Y W = M M (X j X) N j= N (Y k Y ) W (M, N ) F - () ().4( ).4. () 5 (a) 5.9(mg/ml).55(mg /ml ) k=

9 (b) 9.9(mg/ml).6(mg /ml ) (a), (b) ( ) 5 ( ) () 3 (c) 8 3.45(mg/ml) 3.5(mg /ml ) (b), (c) ( ) ( ) D, D (4, ), (7, ) F - P (D > 4.8) =.5, P (D > 5.86) =.5 (5, 7) t- D 3, D 4.4. P (D 3 >.947) =.5, P (D 4 > 3.499) =.5 () ( ) X, X,, X 5 N(m x, σ ) Y, Y,, Y N(m y, σ ) X = 5 (X + X + + X 5 ), Y = 5 (Y + Y + + Y ), ( 5 ) ( ) Z = (X i X) (Y i Y ) 4 i= Z (4, ) χ -P (Z > 4.8) =.5 Z ( 5 ) ( ) (X i X) =.55, (Y i Y ) =.6 5 i= i= i= Z = 5 ( ) 4.55.6 < 4.8 5 ( ) m x = m y ( ) ( 5 ) ( U = (X Y ) (X i X) /5 + / 5 + ) (Y i Y ) 5 i= i= t- U ( 5 ) ( ) (X i X) =.55, (Y i Y ) =.6, X =.9, Y = 9.9 5 i= U =. >.947 () ( ) X, X,, X 8 N(m x, σ ) Y, Y,, Y N(m y, σ ) X = 8 (X + X + + X 8 ), Y = (Y + Y + + Y ), i=

9 Z = 7 ( ) 5 (X i X) (Y i Y ) i= Z (7, ) χ -P (Z > 5.86) =.5 Z 8 8 (X i X) = 3.5, i= i= i= Z = 8 ( ) 7 3.5.6 (Y i Y ) =.6 > 5.86 ( ) T = (X Y ) ν = S X 7 ( ) ( ) s x 7 + s y s x 7 3 + s y 3 = 7 + S Y ν t- (3.45 9.9) 3.5 7 +.6 = 5.3 > 3.5.5. 8 69, 65, 7, 58, 6, 69, 66, 6 6 () () 95 7 t- K P (K >.895) =.5, P (K >.365) =.5.5. () H 6 H 6 65 4.3 8 65 6 4.3 () 8 m 65 4.3 =.973 H = ±.365 [6.4, 69.6] () () () 6 t-

93.3...6 (). 5 3 4 5 6 7 8 9 6 7 4 98 8 5 6 9 5 8 7 5 7 () X, X,, X N(5, σ ) X = (X + X + + X ), S = j= S () - (3) - (4) 5 (X j X), T = Xj 5 S t- W P (W >.8) =.95.6. () t- () (3 + 8 + 8 + 7 + + 9 8 3 + + + 4 + ) = ( (3) (3 ) + (8 ) + (8 ) + (7 ) + ( ) + (9 ) + ( 8 ) +( 3 ) + ( ) + ( ) + (4 ) + ( ) ) = 8. (4) 5 8. =.87 >.8 3. 3.. 3 6 7 6 6 34.56 7 3 3.8 () X, X,, X 6 N(m, σ ) Y, Y,, Y 7 N(m, σ ) X = 6 (X + X + + X 6 ), Y = 7 (Y + Y + + Y 7 ) U = 5 6 (X j X), V = 6 j= U/V 7 (Y j Y ) j=

94 () 5 (3) K X, K Y K X = 6 (X j X), K Y = 7 (Y j Y ) 6 7 j= 4 W = 3 (X Y ) 6KX + 7K Y (4) 5 (5, 6) F - D t- D 3.. P (D > 6) =.5, P (D <.4) =.5, P (D >.) =.5 () (5, 6) F - ().4 < 6 ( ) 7 5.56 6 3.8 < 6 (3) 4 σ 3 (X Y ) 6K X + 7K Y χ -W t- 4 6.56 (4) (34 3) + 7 3.8 4 68 > (34 3) = > 3.5 3 3 3 j= 4. 4... 4.. 37 6 3.37 37 37.37.63 6 = 37 () 99 () 95 4.. (i =,,, N = ) i { (i ) X i = (i )

95 X i, i =,,, N P (X i = ) = p X, X,, X N 6,,, 6 H : p =.37 H : p.37 p =.37 X = X + X + + X 6 N(.,.37.63 6) = N(., 3.7) () X [. m,. + m] 99 m 3 m =.58 3.7 > 9 3 [. m,. + m] 6 3 37 () X [. m,. + m] 95 m 3 m =.96 3.7 < 8 3 / [. m,. + m] 6 3 37 4.. 4 5 3 Y P (Y ) =.8 4.. p p = v.s. p = p =.5 4 X P (X ) E[X] =, V [X] = Y = X Y N(, ) P (X ) = P (Y ) =.5.477 =.8 P (X ).5 P (X ).3 P (X ).

96.8.3,.5 p =.8. p = [ ] 5 4.3. 5 84 5 X P (X >.58) =.5 4.3. H : H : B(5,.) 5..88 = 6.88 = 58 P ( > 84) = P (X > (84 6) 58) < P (X >.58) =.5 P (X > ).5 4.4. 4 X E[X], V [X], σ[x] 4 3 Y P (Y.58) =.5 4.4. () E[X] = () V [X] = (3) σ[x] = Y = X Y P (X 3) = P (Y 3) < P (Y.58) =.5

97 4... 4.5. 8 8 /56 F - χ - 5.3

98 Part 7. 5. 5... t- 5.. 3 x, x,, x N(µ y, σ y ) Y, Y,, Y x = x j, Y = Y j, R = (x j x)(y j Y ) (x j x) (Y j Y ) j= j= j= R t- R j= j=. R R R R = (x j x)(y j Y ) (x j x) (Y j Y ) j= j= j= = (x j x)(y j Y ) j= (x j x) (Y j Y ) (x j x)y j j= j= = x j x k= (x k x) (Y j Y ) j= j= j= (Y j Y ) x j x k= (x k x) Y j j= (x j x)(y j Y ) j= (5.) x x x x x x k= (x k x) k= (x k x) k= (x k x)......

99 Z x x x x x x Y Z. = k= (x k x) k= (x k x) k= (x k x) Y. Z...... Y Z (Z, Z,, Z ) k=3 Z k 5. 5. (). X Y 33 9 3 4 4 4 5 9 6 3 7 37 8 8 9 35 39 3 X Y 6 3 5 4 4 5 4 6 9 7 37 8 4 9 45 49 55 5 9 χ - 9.5 5.. R =.3 3R R =.9.9 < 9 R =.99 3R R > 9

5... 5.3 5.3. (X, Y ), (X, Y ),, (X, Y ) P (X [a, b ], X [a, b ],, X [a, b ], Y [c, d ], Y [c, d ],, Y [c, d ]) = P (X [a, b ], Y [c, d ]) P (X [a, b ], Y [c, d ]) P (X [a, b ], Y [c, d ]) C (X, Y ) C exp ( (x µ x) (x µ x)(y µ y ) (y µ y) ) σ x ρ σ y S xx = ρ = E[(X µ x )(Y µ y )] X = (X + X + + X ), Y = (Y + Y + + Y ), (X i X), S yy = i= R = (Y i Y ), S xy = i= (X i X)(Y i Y ), i= S xy, Z = Sxx S yy log + R R, ζ = log + ρ ρ 3 Z N(ζ, ( 3) ) 5.4. X Y 5 3 4 4 9 5 6 6 3 7 37 8 38 9 47 53 63.8 log 3 =. 5.4. ζ = log 3 =. R =.994, Z = 5.7 (Z.) >.58.8 5.3.. 5.5 5.5. (X, Y ), (X, Y ),, (X, Y ) C (X, Y ) C exp ( (x µ x) (x µ x)(y µ y ) (y µ y) ) σ x ρ σ y

(V, W ), (V, W ),, (V m, W m ) C (V, W ) C exp ( (v µ v) (v µ v)(w µ w ) (w µ w) ) σ v ρ σ w V = (V + V + + V ), W = (W + W + + W ), X = m (X + X + + X m ), Y = m (Y + Y + + Y m ), S xx = (X i X), S yy = (Y i Y ), S xy = (X i X)(Y i Y ), i= i= i= m m m S vv = (V i V ), S ww = (W i W ), S vw = (V i X)(W i Y ), R = i= S xy, Z = Sxx S yy log + R R, S vw R =, Z = + R log Svv S ww R R R 3 + N(, ) m 3 5.6. i= i= X Y 5 3 4 4 9 5 6 6 3 7 37 8 38 9 47 53 63 X Y 9 9 3 7 4 39 5 5 6 6 7 67 8 79 9 87 99 8 5.6..96.99 (R R ) R, R 6. 6... (x, y ), (x, y ),, (x, y ) x, x,, x y, y,, y x = x = = x

y = ax + b (y j ax j b) j= a, b x j a x j ab + b x j y j a y j b + y j j= j= j= x j a j= x j j= j= j= ab + b x j y j a b + P ba + Qa Rb Sa = b b( P a + R) + Qa Sa Q P > j= y j j= b + j= = (b + P a R) + Qa Sa P a + P Ra R = (b + P a R) + (Q P )a (S P R)a R a = S P R Q P, b = P S P R P S + QR Q P + R = Q P P = x j, Q = x j, R = j= j= y j, S = j= x j y j x, x,, x Q P > s xx = x j x j, s xy = x j y j x j j= j= x = j= x j, y = j= j= b = s xy s xx, a = y bx y i 6. ( ). x, x,, x Y, Y,, Y N(β + β x i, σe) S xy = x i Y i x i Y j, S yy = Y i Y j i= x, s xx i= j= x = x + x + + x, s xx = i= y j j= x i i= j= j= j= x j j= y j y j

3 B = S xy, B = Y + Y + + Y B x, S e = s xx () S = + x σe s B N(β, S) xx () B N(β, σe/s xx ) (3) B S e B S e (4) σ e S e χ - (5) B + B x N (6) S e (B β ) + x (Y i B B x i ) i= ( β + β x, σ e + (x x) σ e s xx s xx ) t- (7) (8) sxx (B β ) t- S e (B + B x β β x ) S e + (x x) t- s xx. x i = z i + x S xy = (z i + x)y i () B E[B ] = i= B = Y + Y + + Y j= ( V [B ] = B = x (z i + x) Y j, = i= i= s xx i= z i s xx Y i z i Y i = j= j= z i Y i i= ( x ) z j Y j s xx x ) z j (β + β x + β z j ) = β + β x β x s xx s xx j= ( () () () x ) ( ) z j σe = s xx + x σe s xx j= z j

4 S e = (3) S e Z i = Y j β β x j Y j = + = = Y i Y + Y + + Y i= i= j= i= + x x i s xx ( Y i Y ) + Y + + Y + x x i s xx z j Y j j= j= ( Y i Y ) + Y + + Y s xx Y j j= ( Y + Y + + Y z j Y j j= ( Y j Y ) ( + Y + + Y x x j ( z j Y j j= k= ) z j Y j sxx j= z k Y k ) + s xx x x i s xx j= ) z k Y k k= z j Y j j= (,,, ) ( z,, sxx z sxx,, ) z sxx Y S xy S e B S e B S e (4) (3) Se (5) B + B x B + B x = E[B + B x ] = ( j= x x z j s xx ( j= x x z j s xx ) Y j ) (β + β x + β z j ) = β + β x β x + β x = β + β x V [B + B x ] = ( j= x x z j s xx ) ( σe = + (x x) ) σe s xx (5) B + B x 6..

5 X Y 5 58 3 67 4 8 5 95 6 3 7 8 3 9 3 4 6 () y = αx + β () α 95 (3) β 95 6.. () E[X] = 6, E[Y ] =.64, E[X] = 36, E[Y ] = 4657., E[XY ] = 36.36, E[X ] E[X] =, E[XY ] E[X]E[Y ] = 6.54 β = E[X ] E[X] E[XY ] E[X]E[Y ] = 6.54 =.654, α = E[Y ] βe[x] =.64 6.654 = 37.7 y =.654x + 37.7 () S e = 37.37 s xx = B =.654.6 < sxx α 99 (., 9.) (3) S e = 37.37 s xx = B = 37.7.6 < S e (B β) (8, 94.4) S e (B α) <.6 + x s xx <.6 6... (x, y, z) 3 (x j, y j, z j ), j =,,, (z j ax j by j c) j= a, b, c 7. 7... 7.

6 k A, A,, A k p, p,, p k p, p,, p k > k p j = A, A,, A k X, X,, X j= P (X =, X =,, X k = k ) = P (X =, X =,, X k = k ) = = (π) k + = + + + k p + + k + p p k k k (π) k k p p p k (π) k k p p p k + k x j pj + p j = j!!! k! p p p k k p + + + k + p + pk k + k ( ) + ( ) p + p ( ) k + pk k ( ) j+ ( pj pj = ) xj pj+p j+ j x j + p j ( ( exp (x j pj + p j ) log + x )) j pj exp ( x j pj ) x j C = ( C exp x j pj = j= P (X = x p + p, X = x p + p,, X k = x k pk + p k ) ( = exp ) (π) k k p p p k (x + x + + x k ) (π) k k p p p k, Y j = X j p j pj (j =,,, k ) (x + x + + x k ) ( p p x + x + + p k p k ( (Y + Y + + Y p p k ) + Y + Y + + p k p k pk x k p k ) pk Y k p k ) )

7 (k, ) Z Y Z. = A Y. Z k Y k Z + Z + + Z k = (Y + Y + + Y k ) + ( p p Y + Y + + p k p k (Z, Z,, Z k ) ( C exp ) (z + z + + z k ) ) pk Y k p k k χ - 7.. k A, A,, A k p, p,, p k k p, p,, p k > p j = A, A,, A k j= X, X,, X k k χ - W = k (X j p j ) j= 7.. A 5 B 3 O 7 AB 8 A 4 B O 3 AB 5 3 χ - 9.35.5 p j (A 4) (B ) (O 3) (AB ) 7.. W = + + + 4 3 W = 6.75 9.35 7... b\ a a a a m b x x x m x b x x x m x........ b x x x m x x x x m x

8 b\ a a a a m b y y y m x b y y y m x........ b y y y m x x x x m x Z = y ij = X jx i X i= j= m (X ij y ij ) χ - m X ij = X j, X ij = X i + m i= j= y ij X j = X j= + m m m + 7.3 (). Facebook 53 57 4 3 6 35 6 55 5 4 69 3 7 7 6 5 Facebook 6 χ -.59.5 7.3. 4.8 4.8 38.4 34 34 3 5 5 48 5 44. 44. 4.6 3 7 7 6 5 6 W = (X 4.8) 4.8 + + (X 43 4.6) 4.6

9 6 χ - X ij i j W X 43 = 69 5 Facebook 7.3.. X Y a b c d d X Y Z X Y ad = bc d 7.4. d X Y a b c d j =,,, d 7.5. X Y a-j b+j c+j d-j X Y 6 37 9 X Y 63 36 X Y 64 35 X Y a b c d

p XA X A p XB a, b, c, d! a!b!c!d! p XA a p Y A b p XB c p Y B d X, Y A, B p XA = p X p A a, b, c, d! a!b!c!d! p X a+c p Y b+d p A a+b p B c+d a + c = e, b + d = f, a + b = g, c + d = h e + f = g + h a, b, c, d a + c = e, b + d = f, a + b = g, c + d = h! e!f! p X e p f! Y g!h! p A g h p B a, b, c, d a + c = e, b + d = f, a + b = g, c + d = h a, b, c, d e!f!g!h! a!b!c!d!! 7.6 ( ). d X Y a b c d a + c = e, b + d = f, a + b = g, c + d = h () α d e!f!g!h! (a j)!(b + j)!(c + j)!(d j)!! α j= (a, b, c, d) () α d e!f!g!h! (a j)!(b + j)!(c + j)!(d j)!! α j= (a, b, c, d) 7.7. 9 3 4 7.7. 5