x y z.... X Y (cdf) F (x, y) = P (X x, Y y) (X, Y ) 3.1. (X, Y ) 3.2 P (x 1 < X x 2, y 1 < Y y 2 ) = F (x 2, y 2 ) F (x 2, y 1 ) F (x 1, y 2
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- 锈朽 杜
- 6 years ago
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1 xy z.... X Y (cdf) F (x, y) = P (X x, Y y) (X, Y ) 3.. (X, Y ) 3.2 P (x < X x 2, y < Y y 2 ) = F (x 2, y 2 ) F (x 2, y ) F (x, y 2 ) + F (x, y ) 3. F (a, b) 3.2 (x 2, y 2) (x, y 2) (x 2, y ) (x, y ) A(X, Y ). X,, X n F (x, x 2,, x n ) = P (X x, X 2 x 2,, X n x n )
2 X Y x, x 2, y, y 2,. (joint frequency function) p(x, y) p(x i, y j ) = P (X = x i, Y = y j ). X Y. Ω = {hhh, hht, hth, htt, thh, tht, tth, ttt} X Y x y p(, 2) = P (X =, Y = 2) =.. 8 Y. p Y () = P (Y = ) = P (Y =, X = ) + P (Y =, X = ) = 8 + = 8 p Y () = P (Y = ) = P (Y =, X = ) + P (Y =, X = ) = 3 8 Y. p Y Y (marginal frequency function). p X (x) = p(x, y i ) i X.. X,, X m p(x,, x m ) = P (X = x,, X m = x m ) X p X (x ) = p(x, x 2,, x m ) x 2 x m X X 2 p XX 2 (x, x 2 ) = p(x, x 2,, x m ) x 3 x m
3 (). n r p, p 2,, p r. N i n i i =,, r. N = n, N 2 = n 2,, N r = n r p n pn2 2 pnr r..4.2 n! n!n 2! n r! ( ) n p(n,, n r ) = p n n n pn2 2 pnr r r N i n j. N i n p i p i. N i, ( ) n p Ni (n i ) = p ni i ( p i ) n ni n i (histogram). [, ] n,, n. n = p i =. (i =,, ) , X Y F (x, y). (joint density function) f(x, y). f(x, y) y)dydx =. A P ((X, Y ) A) = f(x, y)dydx A f(x,
4 54 3 A = {(X, Y ) X x, Y y} F (x, y) = x y f(x, y) = f(u, v)dvdu 2 F (x, y) x y δ x δ y f (x, y) P (x X x + δ x, y Y y + δ y ) = (X, Y ) (x, y) f(x, y). P (x X x + dx, y Y y + dy) = f(x, y)dxdy x+δx x+δy x y f(u, v)dvdu f(x, y)δ x δ y 3.3. f(x, y) = 2 7 (x2 + xy), x, y 3.4. P (X > Y ) {(x, y) y x } f P (X > Y ) = 2 7 x (x 2 + xy)dydx = f(x, y) = 2 7 (x2 + xy), x, y X (marginal cdf) F X F X (x) = P (X x) = lim y F (x, y) = x X X (marginal density) f X (x) = F X(x) = f(x, y)dy f(u, y)dydu X f X (x) = 2 (x 2 + xy)dy = 2 ( x 2 + x ) Y f Y (y) = 2 ( ) y/2.
5 55... XY Z f(x, y, z). X X Y f X (x) = (Farlie-Morgenstern ) α f XY (x, y) = f(x, y, z)dydz f(x, y, z)dz F (x) G(y) cdf α H(x, y) = F (x)g(y){ + α[ F (x)][ G(y)]}. lim F (x) = lim F (y) = x y H(x, ) = F (x) H(, y) = G(y). [, ] [F (x) = x, x G(y) = y, y ]. α = H(x, y) = xy[ ( x)( y)] = x 2 y + y 2 x x 2 y 2, x, y h(x, y) = 2 H(x, y) = 2x + 2y 4xy, x, y x y 3.5. y ( x ) x. α = H(x, y) = xy[ + ( x)( y)] = 2xy x 2 y y 2 x + x 2 y 2, x, y h(x, y) = 2 2x 2y + 4xy, x, y (copula). H(x, y). cdf C(u, v) P (U u) = C(u, ) = u C(, v) = v. c(u, v) = 2 C(u, v) u v
6 h(x, y) = 2x + 2y 4xy x, y 3.6 h(x, y) = 2 2x 2y + 4xy x, y X Y cdf F X (x) F Y (y). U = F X (x) V = F Y (y) ( 2.3.3). C(u, v) F XY (x, y) = C(F X (x), F Y (y)) C(F X (x), ) = F X (x)f XY cdf F X (x) F Y (y). f XY (x, y) = c(f X (x), F Y (y))f X (x)f Y (y), { λ 2 e λy, x y, λ > f(x, y) =, 3.7. ( 3.8)
7 57 f X (x) = f X (x) = f XY (x, y)dy. x y, f(x, y) = x λ 2 e λy dy = λe λx, x X. f XY (x, y) =, x, x > y Y. f Y (y) = y λ 2 e λy dx = λ 2 ye λy, y. (X, Y ) R A R P ((X, Y ) A) = A R. π f(x, y) = π, x2 + y 2, R. r R r. πr 2 F R (r) = P (R r) = πr2 π R f R (r) = 2r, r. x f X (x) = f(x, y)dy = π x 2 x 2 dy = 2 π = r2 x2, x. ( f(x, y) >.) Y () f(x, y) = 2πσ X σ Y ρ 2 exp f Y (y) = 2 π y2, y ( [ (x µx ) 2 2( ρ 2 ) σx 2 + (y µ Y ) 2 σy 2 2ρ(x µ ]) X)(y µ Y ) σ X σ Y. 5 < µ X < σ X > σ Y > < ρ < < µ Y <
8 58 3 xy. (x µ X ) 2 σ 2 X + (y µ Y ) 2 σ 2 Y 2ρ(x µ X)(y µ Y ) σ X σ Y = f(x, y). (µ X, µ Y ). ρ = x y ρ µ X = µ Y =, σ X = σ Y = X Y N(µ X, σx 2 ) N(µ Y, σy 2 ). X f X (x) = f XY (x, y)dy u = (x µ X )/σ X v = (y µ Y )/σ Y [ ] f X (x) = exp 2πσ X ρ 2 2( ρ 2 ) (u2 + v 2 2ρuv) dv. u 2 + v 2 2ρuv = (v ρu) 2 + u 2 ( ρ 2 )
9 59 [ ] f X (x) = 2 2πσ X ρ 2 e u /2 exp 2( ρ 2 (v ρu)2 dv ) ρu ( ρ 2 ) f X (x) = e (/2)[(x µ X ) 2 /σ 2 X ] σ X 2π. 3.9a d x y µ = σ = c(u, v) = 2 2u 2v + 4uv Farlie-Morgenstern. φ(x) Φ(x) f(x, y) =(2 2Φ(x) 2Φ(y)+4Φ(x)Φ(y))φ(x)φ(y) xy
10 X, X 2,, X n x, x 2,, x n F (x, x 2,, x n ) = F X (x )F X2 (x 2 ) F Xn (x n ).. X Y. F (x, y) = F X (x)f Y (y) x y [ x ] [ y ] F (x, y) = f X (u)f Y (v)dvdu = f X (u)du f Y (v)dv = F X (x)f Y (y) X Y P (X A, Y B) = P (X A)P (Y B) g hz = g(x) W = h(y ). ( ) P (Z z, W w). A(z) g(x) z x B(w) h(y) w y. P (Z z, W w) = P (X A(z), Y B(w)) = P (X A(z))P (Y B(w)) = P (Z z)p (W w)) 3.4. (X, Y ) S = {(x, y) /2 x /2, /2 y /2} S (x, y) f XY (x, y) =.. X Y [ /2, /2]. x /2 x /2 x (). f X (x) =, /2 x /2 f Y (y) =, /2 y /2. X Y. X Y X /2 x /2 Y. f X (.9) > f Y (.9) > f XY (.9,.9) =. X Y. X (X =.9) Y (Farlie-Morgenstern ) α = X Y H F G.
11 X Y ( )ρ = X Y τ. [, T ] T T 2 [, T ] f(t, t 2 ) = T 2 t, t 2 [, T ]. (T, T 2 ) (T τ) 2 /2 T 2 (T τ) 2. f(t, t 2 ) ( τ/t ) t t 2 < τ X Y Y = y j X = x i p Y (y j ) > P (X = x i Y = y j ) = P (X = x i, Y = y j ) P (Y = y j ) = p XY (x i, y j ) P Y (y j ) p Y (y j ) =. p X Y (x y). x. X Y p X Y (x y) = p Y (y) x Y = X y p X Y ( ) = = 2 3, p X Y ( ) = = 3 p XY (x, y) = p X Y (x y)p Y (y)
12 62 3 ( ).. y p X (x) = y p X Y (x y)p Y (y) p. λ N X. N = n X n p. λ n e λ ( ) n P (X = k) = P (N = n)p (X = k N = n) = p k ( ρ) n k n! k n= = (λp)k k! = (λp)k k! e λ n=k n k ( p)n k λ (n k)! e λ e λ( p) = (λp)k e λp k! n=k = (λp)k k! e λ j= λ j ( p) j X λp.. N X X Y X Y : < f X (x) <. f Y X (y x) = f XY (x, y) f X (x). f Y X (y x)dy P (y Y y + dy x X x + dx) P (y Y y + dy x X x + dx) = f XY (x, y)dxdy f X (x)dx j! = f XY (x, y) dy f X (x) x y. f XY (x, y) x y x.. x Y. f XY (x, y) = f Y X (y x)f X (x) f Y (y) = f Y X (y x)f X (x)dx
13 f XY (x, y) = λ 2 e λy, x y f X (x) = λe λx, x f Y (y) = λ 2 ye λy, y. x y. x y x y y x y. ( 3.7.). f Y X (y x) = λ2 e λy λe λx = λe λ(y x), y x X = x Y [x, ) f XY (x, y) = f Y X (y x)f X (x) f XY X Y X(f X ) [x, ) Y (f Y X ). Y. f X Y (x y) = λ2 e λy λ 2 ye λy = y, x y Y = y X [, y]. f XY (x, y) = f X Y (x y)f Y (y) f XY X Y Y [, y] X (). (De- Hoff Rhines 968).. (). f R (r). f X (x). R = r f X R (x r). 3.3 H. H [, r] X = r 2 H 2. R = r X 3.3 H r x
14 64 3 X F X R (x r) = P (X x) = P ( r 2 H 2 x) = P (H r2 r 2 x 2 x ) = 2, x r r f X (x) = f X R (x r) = x r r 2 x 2, f X R (x r)f R (r)dr = x x r x r r 2 x 2 f R(r)dr [ r xf X R (x r) = x.]. f X. f R (). f Y X (y x) = σ Y 2π( ρ2 ) exp 2 X Y [ y µ Y ρ σ Y σ X (x µ X ) σ 2 Y ( ρ2 ) µ Y + ρ(x µ X )σ Y /σ X σ 2 Y ( ρ2 ). X Y Van Atta Chen(968) t t + τ. 3.4 t v t + τ v 2.. v v 2. v v () (rejection method) cdf cdf. f [a, b] (a b ). M(x) [a, b] M(x) f(x) m(x) = b a M(x) M(x)dx. M m. [a, b] m [a, b]. m T. [, ] U T. M(T ) U f(t ) X = T ( T ) ( T ). ] 2
15 v v 2 v v f(x) x. X f P (x X x + dx) =P (x T x + dx ) =. 3.5 P (x T x + dx ) P () P ( x T x + dx)p (x T x + dx) = P () P ( x T x + dx) = P (U f(x)/m(x)) = f(x) M(x)
16 66 3 m(x)dxf(x) M(x) P () = P (U f(t )/M(T )) = = f(x)dx b a b a M(x)dx f(t) M(t) m(t)dt = b M(t)dt m f. f(x)dx (). n X. 2. Θ. [, ] Θ (prior density). Θ [, ] f Θ (θ) =, θ. X Θ. θx n θ ( ) n f X Θ (x θ) = θ x ( θ) n x, x =,,, n x Θ X ( ) n f Θ,X (θ, x) = f X Θ (x θ)f Θ (θ) = θ x ( θ) n x, x x =,,, n, θ θ x. θ X f X (x) = ( ) n θ x ( θ) n x dθ x., ( ) n n! = x x!(n x)! = Γ(n + ) Γ(x + )Γ(n x + ) ( k Γ(k) = (k )!( 2 49). (2.2.4 ) g(u) = Γ(a + b) Γ(a)Γ(b) ua ( u) b, u a
17 67 u a ( u) b du = Γ(a)Γ(b) Γ(a + b) θ ux a n x b Γ(n + ) f X (x) = θ x ( θ) n x dθ Γ(x + )Γ(n x + ) Γ(n + ) Γ(x + )Γ(n x + ) = Γ(x + )Γ(n x + ) Γ(n + 2) = n +, x =,,, n θ X. X = x Θ X = x Θ f Θ X (θ x) = f ( ) Θ,X(θ, x) n = (n + ) θ x ( θ) n x f X (x) x Γ(n + ) = (n + ) Γ(x + )Γ(n x + ) θx ( θ) n x = Γ(n + 2) Γ(x + )Γ(n x + ) θx ( θ) n x xγ(x) = Γ(x + ) ( 2 49). x θ x θ n x Θ. a = x + b = n x Θ U[, ] a = x + = 4b = n x + = θ 2 3 θ. θ <.25. θ % θ a = 4 b =
18 X Y p(x, y) Z = X + Y. Z X = xy = z x Z = z x., Z = z x p Z (z) = p(x, z x) x= X Y p(x, y) = p X (x)p Y (y) p Z (z) = p X (x)p Y (z x) x= p X p Y (convolution).. X Y Z. (X, Y ) 3.7 R z Z z F Z (z) = f(x, y)dxdy = R z z x f(x, y)dydx 3.7 (X, Y ) R z X + Y z y = v x F Z (z) = z f(x, v x)dvdx = z f(x, z x)dx z f Z (z) =. X Y f Z (z) = f(x, z x)dx f X (x)f Y (z x)dx f(x, v x)dxdv f X f Y T T 2 λ S = T + T 2. f S (s) = s λe λt λe λ(s t) dt...
19 69 s f S (s) = λ 2 e λs dt = λ 2 se λs 2 λ ( )... X Y f Z = Y/X. F Z (z) = P (Z z) y/x z (x, y). x > y xz x < y xz. F Z (z) = f(x, y)dydx + xz xz x y = xv F Z (z) = = = z z z () f Z (z) = X Y f Z (z) = xf(x, xv)dvdx + ( x)f(x, xv)dvdx + x f(x, xv)dxdv z z x f(x, xz)dx x f X (x)f Y (xz)dx f(x, y)dydx xf(x, xv)dvdx xf(x, xv)dvdx X Y Z = Y/X. f Z (z) = f Z (z) = π u = x 2 f Z (z) = 2π x 2 2π e x /2 e x2 z 2 /2 dx xe x2 ((z 2 +)/2) dx λ exp( λx)dx = λ = (z 2 + )/2 f Z (z) = π(z 2 + ), e u((z2 +)/2) du < z <
20 7 3 (Cauchy density).. Y/X X X Y f XY (x, y) = 2 2π e (x /2) (y 2 /2) (R, Θ 2π) R = X 2 + Y 2 ( ) Y tan, X > X ( ) Y tan + π, X < Θ = X π sgn(y ), X =, Y 2, X =, Y = ( π 2 < Θ < π.) 2 X = R cos Θ Y = R sin Θ R Θ f RΘ (r, θ)drdθ = P (r R r + dr, θ Θ θ + dθ) 3.8 f XY [x(r, θ), y(r, θ)]. rdrdθ P (r R r + dr, θ Θ θ + dθ) = f XY (r cos θ, r sin θ)rdrdθ f RΘ (r, θ) = rf XY (r cos θ, r sin θ), f RΘ (r, θ) = r 2π e[ (r 2 cos 2 θ)/2 (r 2 sin 2 θ)/2] = 2π re r 2 /2 3.8 rdrdθ
21 7 R Θ Θ [, 2π] R f R (r) = re r2 /2, r (Rayleigh density). T = R 2. f T (t) = 2 e t/2, t 2. R Θ T Θ f T Θ (t, θ) = 2π ( ) e t/2 2 Θ [, 2π] R 2 2. ( tan Θ.) Φcdf Φ. [, ] U U 2. 2 log U 2 2πU 2 [, 2π]. X = 2 log U cos(2πu 2 ) Y = 2 log U sin(2πu 2 ). (polar method). X Y U V u = g (x, y) v = g 2 (x, y) x = h (u, v) y = h 2 (u, v) g g 2 x y, g g ( ) ( ) ( ) ( ) J(x, y) = det x y g 2 g 2 = g g2 g2 g x y x y x y.
22 V. (x, y) u = g (x, y), v = g 2 (x, y) (u, v) U f UV (u, v) = f XY (h (u, v), h 2 (u, v)) J (h (u, v), h 2 (u, v)) u v r θ r x = x x2 + y 2 θ x = y x 2 + y 2 r = x 2 + y 2 ( θ = tan y ) x J(x, y) = x = r cos θ y = r sin θ 3.6. r, θ 2π, r y = y x2 + y 2 θ y = x x 2 + y 2 x2 + y 2 = r f RΘ (r, θ) = rf XY (r cos θ, r sin θ) X,, X n f X X n Y i = g i (X,, X n ), X i = h i (Y,, Y n ), i =,, n i =,, n J(x,, x n ) ij g i / x j Y,, Y n f Y Y n (y,, y n ) = f X X n (x,, x n ) J (x,, x n ) x i y x i = h i (y,, y n ) X X 2 Y = X
23 73 Y 2 = X + X 2 Y Y 2. [ ] J(x, y) = det = x = y, x 2 = y 2 y 3.6.Y Y 2 f YY 2 (y, y 2 ) = [ 2π exp ] 2 [y2 + (y 2 y ) 2 ] = [ 2π exp ] 2 (2y2 + y2 2 2y y 2 ) ( ). y y 2 µ Y = µ Y2 =. ( µ Y y µ Y.) σ Y σ Y2 ρ2 = y σy 2 ( ρ 2 ) = 2 σy 2 2 = 2. y 2 ρ 2 = 2. σ 2 Y 2 ( ρ 2 ) = ρ = / 2. σy 2 =...( 58.) 3.7. X, X 2,, X n F f. U X i V. U V. U u i, X i u. F U (u) = P (U u) = P (X u)p (X 2 u) P (X n u) = [F (u)] n f U (u) = nf(u)[f (u)] n V v i, X i v.
24 74 3 V 3.7. F V (v) = [ F (v)] n F V (v) = [ F (v)] n f V (v) = nf(v)[ F (v)] n n T,, T n λ F (t) = e λt. V T i V nλ f V (v) = nλe λv (e λv ) n = nλe nλv n f U (u) = nλe λu ( e λu ) n.. f U (u) n X i (u, u + du) n X i u u U u + du. [F (u)] n f(u)du n f U (u) = n[f (u)] n f(u) X,, X n f(x). X i X () < X (2) < < X (n) (order statistics). X X (). ( n.) X (n) X (). n n = 2m + X (m+) X i (median) k X (k) f k (x) = n! (k )!(n k)! f(x)f k (x)[ F (x)] n k. ( 66.) x X (k) x + dx k x [x, x + dx] n k x + dx. f(x)f k (x)[ F (x)] n k dx n!/[(k )!!(n k)!] X i [, ] k n! (k )!(n k)! xk ( x) n k, x. x k ( x) n k dx = (k )!(n k)! n!
25 75., x X () x+dxy X (n) y+dy X i [x, x+dx] [y, y+dy] n 2 [x, y]. n(n ) V = X () U = X (n) f(u, v) = n(n )f(v)f(u)[f (u) F (v)] n 2, u v f(u, v) = n(n )(u v) n 2, u v X (),, X (n) R = X (n) X () f R (r) = f(v + r, v)dv [, ] U V. f(v + r, v) = n(n )r n 2 v v + r v r. f R (r) = () r n(n )r n 2 dv = n(n )r n 2 ( r), r F R (r) = nr n (n )r n, r f f(x) (X (), X (n) ).. F (X (n) ) F (X () ) Q F (X i ) Q n U (n) U (). P (Q > α) α% P (Q > α) = nα n + (n )α n n = α = %.96 95% X Y x y
26 76 3 a. X Y. b. Y = X X = Y. 2. p q r n. a.. b.. c d w. r. n (.) 5. () D L D L. 2L/πD. π. 6. x y. 7. cdf 8. X Y x 2 a 2 + y2 b 2 = F (x, y) = ( e αx )( e βy ), x, y, α >, β > f(x, y) = 6 7 (x + y)2, x, y a. (i) P (X > Y )(ii) P (X + Y )(iii) P b. x y. c.. 9. (X, Y ) y x 2 x.. a. X Y. b.. f(x, y) = xe x(y+), x <, y < a. X Y. X Y b. X Y. ( X ). 2. U, U 2 U 3 [, ] U x 2 + U 2x + U a. c. b.. c.. f(x, y) = c(x 2 y 2 )e x, x <, x y < x
27 a. x, y z. b. x y. c. Z = xy. 5. X Y a. c. b.. ( c. P X 2 + Y 2 ). 2 f(x, y) = c x 2 y 2, x 2 + y 2 d. X Y. X Y e.. 6. X [, ] X X 2 [, X ] X X (X, Y ) R = {(x, y) : x + y }. a. R. b. X Y.. c. X Y. 8. X Y f(x, y) = k(x y), y x a.. b. k. c. X Y. d. X Y Y X. 9. T T 2 α β. (a) P (T > T 2) (b) P (T > 2T 2) X. X f(x)... x R(x). Y. Y g(y) = R(y)f(y) R(x)f(x)dx 22. N(t, t 2) (t, t 2). t < t < t 2 N(t, t 2) = n N(t, t ). (). 23. N X N p N m r. X.
28 θ. θ. 25. X f Y = X Y = X Y 2 f Y (y) = f Y ( y). 26. P [, ] P = p X p. X P. 27. x y f X Y (x y) = f X(x) X Y. 28. C(u, v) = uv. 29. Farlie-Morgenstern.. 3. α β C(u, v) = min(u α v, uv β ) (Marshall-Olkin ) (X, Y ). X Y f R(r) [, ]. N. N =. a. N Θ. b Θ. a = b = M(x) f(x). 37. x f(x) = 6x 2 ( x) 2. a.. b x α f(x) = + αx 2 a.. b (D.R.Fredkin). X p p p 2,,, 2, U. U < p X = U p U U p X = U p U p X Y p XY (x, y). X p X Y (x y).
29 79 a. X p X(x). b. p(y X) X. c. X X. a. X, 2,, X = x Y 2 x. Y = 44 X Y = 44 X. E(X Y = 44) 4. X Y 42. a. T λ W T ± X = W T. 2 X (double exponential density). b. c f X(x) = λ 2 e λ x 2π e x2 /2 ce x a. 43. U U 2 [, ]. S = U + U X Y, 2. X + Y A B p A + p B =. λ A p Aλ. 46. T T 2 λ λ 2. T + T X Y. Z = X + Y Z. (.) 48. N N 2 λ λ 2. N = N + N 2 λ + λ X + Y X Y X Y Z = X Y. 5. X Y f(x, y)z = XY. Z ( f Z(z) = f y, z ) y y dy U U 2 U 3 [, ]. U U 2 U XY Z N(, σ 2 ). ΘΦ R (X, Y, Z) x = r sin φ cos θ y = r sin φ sin θ z = r cos φ φ π, θ 2π ΘΦ R. (dxdydz = r 2 sin φdrdθdφ.)
30 R [, ] Θ [, 2π] R. a. X = R cos Θ Y = R sin Θ. b. X Y. c X Y (X, Y ) R Θ. R Θ 57. Y Y 2 µ Y = µ Y2 =, σy 2 =, σy 2 2 = 2 ρ = / 2. x = a y + a 2y 2, x 2 = a 2y + a 22y 2 x x 2. ( ) 58. X X 2 Y = a X + b Y 2 = a 2X 2 + b X X 2 Y Y 2. Y = a X + a 2X 2 + b Y 2 = a 2X + a 22X 2 + b X Y U = a + bx V = c + dy. 62. X Y P (X 2 + Y 2 ). 63. X Y. a. X + Y X Y. b. XY Y/X. c. X Y a b. 64. X + Y X/Y X Y λ. X + Y X/Y. 65. λ i. λi. 66. ( 3.9) λ. cdf n. λ. 68. U U 2 U 3. a. U () U (2) U (3).
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