......

......

m() 1 m() E(X; ) 1 m() 1 m() E(X; ) 1 m() E 1 (X; ).1 E 1 (X; ) E 2 (X; ).5 m() E 1 (X; ) E 2 (X; ) E 3 (X; ) 1 m() E 1 (X; ) E 2 (X; ) 2 E 2 (X; ) E 3 (X; ).5 m() E 1 (X; ) E 2 (X; ) E 3 (X; ) 2 T CE q (X) T CV q (X) E 1 (X; x q ) E 2 (X; x q ) E 3 (X; x q ) 4 x q 1 T CE q (X) E 1 (X; x q ) E 2 (X; x q ) E 3 (X; x q ) 4 x q 1 T CE q (X) E 1 (X; x q ) E 2 (X; x q ) E 3 (X; x q ) 4 x q 1

T CE q (X) T CV q (X) E 1 (X; x q ) E 2 (X; x q ) E 3 (X; x q ) 4 x q 6 T CE q (X) E 1 (X; x q ) E 2 (X; x q ) E 3 (X; x q ) 4 x q 6

f() F () λ() Λ() (σ) (CRE) (σ) (CRE)

X f() F () F () = P (X > ) f() = F (), F () [, ) F () = 1 F () = S()

F () F ( + x) F () F () F () = x F ( + x) F () x F () F () = x F ( + x) F () x F () = F () F ( + x) F () P (X > ) P (X > + x) = P (X > ) P ( X + x) = P (X > ) =, P (X + x) P (X ) P (X > ) = P (X + x X > ). λ() = f() F () = F () F () = P (X + x X > ) x x (, + x] x X > λ() = F () F () = ( ln F ()),, F () = 1 λ(s)ds = ln F (s) = ln F () F () = ln F (). F () = e λ(s)ds. Λ() = λ(s)ds,,

λ() = Λ (), F () = e Λ() Λ() = ln F (). f() F () λ() Λ() f() F () λ() Λ() ( ) f() f() f(x)dx f(x)dx f(x)dx ln F () F () F () ln F () ( F () λ() e ) λ(x)dx e λ(x)dx λ(x)dx Λ() ( e Λ()) e Λ() Λ () λ() µ()

IF R DF R X F λ() X IF R F IF R λ() X DF R F DF R X [, ] X = X X >

X F X (x) = P (X > x) = P (X > x X > ) = P (X > x, X > ) P (X > + x) = P (X > ) P (X > ) = F ( + x), F (), x, x (, + x] x X dx λ X (x) = F X d (x) F X (x) = = F (x + ) F (x + ) F (x+) F () F (x+) F () = λ(x + ), λ() X X m() = E(X ) = = F X (x)dx F (x + ) dx = F () F (x) F () dx, X = m() = E(X) = µ, E(X) = µ X m () = d d = F () 1 F () F () = λ() m() 1 F (x) F () dx = F () F (x) dx + 1 F () 2 F () F (x) dx 1 d F () d F (x)dx d d F (x)dx

d d F (x)dx = d d m() + 1 = λ() m() F (x)dx = F (). λ() λ() = m() + 1 m(). m() λ() X F m() X IMRL F IMRL m() X DMRL F DMRL X F m() µ m() µ X NBUE F NBUE

m() µ X NW UE F NW UE IF R DF R DMRL IMRL NBUE NW UE IF R (DF R) DMRL (IMRL) IF R (DF R) NBUE (NW UE). (α, β) (α, β) U(α, β) f() = { 1 β α α < < β F () = β β α. E(X) = α + β 2 V ar(x) = (β α)2 12.

λ() = 1 β α β β α = 1 β, α < < β. m() = β α β β β x β α dx = β 2, α < β. λ() α < < β IF R DMRL NBUE exp(λ) λ > f() = λe λ,, f() = f() = λ F () = e λ,. E(X) = 1 λ V ar(x) = 1 λ 2. λ() = λe λ e λ = λ,. m() = 1 e λ e λx dx = 1 λ = E(X),.

λ() m() λ() m() IF R DF R IMRL DMRL NBUE NW UE (α, λ) α > λ > f() = Γ(α) = λα Γ(α) α 1 e λ,, x α 1 e x dx, α > α > α F () = α 1 i= F () = (λ) i i! e λ,. λ α Γ(α) xα 1 e λx dx s = λx ds = λ dx, F () = λα s α 1 1 Γ(α) λ λ α 1 e s λ ds = 1 Γ(α) Γ(α, λ) =,, Γ(α) λ s α 1 e s ds

Γ(α, u) = u x α 1 e x dx, α >, u E(X) = α λ V ar(x) = α λ 2. λ() = λα α 1 e λ Γ(α, λ),. λ() α > 1 α < 1 α = 1 IF R DMRL NBUE α > 1 DF R IMRL NW UE α < 1 P areo(α, β) α > β > f() = F () = α β α ( + β) α+1,. ( ) α β,. + β β α > 1 α 1 E(X) = < α 1 α β 2 α > 2 (α 1) V ar(x) = 2 (α 2) < α 2

λ() = α β α (+β) α+1 = α β α (+β) α + β,. m() = = ( β x+β )α dx ( β +β )α = ( + β) α +β α 1 α > 1 < α 1 [ (x + β) 1 α 1 α ] λ() DF R IMRL NW UE α > 1 α > 2 P ower(α, β) α > β > f() = α (β )α 1 β α, < < β. ( ) α β F () =, β. β E(X) = β α + 1 V ar(x) = α β 2 (α + 1) 2 (α + 2). λ() = α (β ) α 1 β α = α ( β β )α β, < < β.

m() = ( β x β = β α + 1. )α dx ( β β )α = 1 (β ) α [ ] β (β x)α+1 α + 1 λ() < < β IF R DMRL NBUE

X ϱ X ϱ[x] X ϱ X ϱ[x] ϱ ϱ[x] ϱ[x] max[x], X. ϱ[x] E(X), X.

ϱ[x + c] = ϱ[x] + c, X c. ϱ[c] = c, c. ϱ[x + Y ] ϱ[x] + ϱ[y ], X, Y. ϱ[x + Y ] = ϱ[x] + ϱ[y ], X, Y. ϱ[cx] = c ϱ[x], X c. P (X Y ) = 1, ϱ[x] ϱ[y ] X, Y.

X p i = P (X = i) f(x) H(X) H(X) = i p i lnp i, X H(X) = f(x) lnf(x)dx, X

X (, α) α 1 H(X) = α ln 1 dx = lnα, α H(X) < α < 1 f(x) F (x) E(X) E(X) = F (x) ln F (x)dx = F (x) Λ(x)dx, F (x) X Λ(x) E(X) m() X m() E(X)

E(X) < E(X) = E(m(X)). X E(m(X)) = m()f()d. m() E(m(X)) = F (x)dx F () f()d E(m(X)) = = = x x f() F () d λ()d F (x)dx F (x)dx ( ln F (x)) F (x)dx = E(X). X (, α) α > m() = α 2. ( ) α X E(X) = E = 2 = 1 [αx x2 2α 2 α ] α α x 2 1 α dx = α 4.

λ > m() = 1 λ. E(X) = E ( ) 1 = 1 λ λ. α > 1 β > m() = + β α 1. E(X) = E = = ( ) X + β = E(X) + β α 1 α 1 β + β α 1 α 1 α β (α 1), 2 E(X) α > β > m() = β α + 1.

( ) β X E(X) = E α + 1 = β E(X) α + 1 = β β α+1 α + 1 α β = (α + 1), 2 E(X) NBUE NW UE X µ E(X) X NBUE NW UE E(X) ( ) µ. X NBUE NW UE m() ( ) µ >. m() f()d ( ) f()d = 1 µ f()d = µ, m() f()d = E(m(X)). E(X) ( ) µ. NBUE

X NW UE f(x) g(x) f(x) ln f(x) g(x) dx f(x)dx ln f(x)dx g(x)dx. X g g() = E(g(X)) = g (x) F (x)dx. g(x) = x g (x) = 1 X E(X) = F (x)dx. g(x) = x 2 g (x) = 2x X E(X 2 ) = 2 x F (x)dx. X E(X) E(X 2 ) E(X) E(X2 ) 2E(X).

Y λ > F (x) ln F (x) Ḡ(x) dx F (x) ln F (x)dx E(X) + λ Ḡ(x) F (x)dx ln F (x)dx Ḡ(x)dx. F (x) lne λx dx E(X) ln E(X) E(Y ) x F (x)dx E(X) ln(λe(x)) E(X) λ 2 E(X2 ) + E(X) ln(λe(x)), λ >. λ λ = 2E(X) E(X 2 ). λ E(X) E(X) + E(X) ln 2(E(X))2 E(X 2 ) lnx 1 1 x, x >, ( ) E(X) E(X) + E(X) 1 E(X2 ) 2(E(X)) 2 E(X) E(X2 ) 2E(X). NBUE NW UE

X NBUE E(X) µ E(X) E(X2 ) 2E(X). X NW UE µ E(X) E(X2 ) 2E(X). σ µ CV = σ µ = E(X2 ) E(X) 2 E(X), NBUE NW UE X NBUE E(X2 ) E(X) 2 < 1 E(X) E(X 2 ) E(X) 2 < E(X) 2 E(X2 ) 2E(X) < E(X) = µ. NBUE X λ 1 > λ 2 > f() = p λ 1 e λ1 + (1 p) λ 2 e λ2, >, p (, 1). F () = p e λ1 + (1 p) e λ2,.

λ() = p λ 1e λ 1 + (1 p) λ 2 e λ 2 p e λ 1x + (1 p) e λ 2, >. DF R DF R DF R X DF R X NW UE E(X) = (p e λ 1 + (1 p) e λ 2 )d = p 1 λ 1 + (1 p) 1 λ 2 E(X 2 ) = 2 (p e λ1 + (1 p) e λ2 )d ( = 2 p 1 + (1 p) 1 ), λ 2 1 λ 2 2 E(X 2 ) 2E(X) = p 1 + (1 p) 1 λ 2 1 λ 2 2 p 1 λ 1 + (1 p) 1. λ 2 p 1 λ 1 + (1 p) 1 λ 2 E(X) p 1 λ 2 1 + (1 p) 1 λ 2 2 p 1 λ 1 + (1 p) 1 λ 2. X α > λ > E(X) = α λ E(X2 ) = E(X 2 ) 2E(X) = α + 1 2λ. α (α + 1) λ 2, < α < 1 X DF R X NW UE α λ E(X) α + 1 2λ.

α > 1 X IF R X NBUE E(X) α + 1 2λ E(X) α λ. α > 1 α + 1 α 2λ λ, X E(X 2 ) < α > 2 α > 1 X E(X; ) E(X; ) = = F X (x) ln F X (x)dx F (x) F () ln F (x) F () dx. E(X; ) = 1 F () = 1 F () F (x) ln F (x)dx + ln F () F (x) ln F (x)dx + m() ln F (). F (x) F () dx = E(X; ) = E(X).

X E(X; ) m() X m() E(X; ) E(X; ) <, E(X; ) = E(m(X) X ). X E(m(X) X ) = 1 F () E(m(X) X ) = 1 F () x m(x) f(x)dx, F (s)ds F (x) f(x)dx. < x < s 1 ( s ) f(x) E(m(X) X ) = F () F (x) dx F (s)ds 1 ( s ) = λ(x)dx λ(x)dx F (s)ds F () ( ) F (s) = ln F (s) + ln F () F () ds = = E(X; ). F (x) F () ln F (x) F () dx = E(X; ) = E(X) = E(m(X) X ) = E(m(X)).

X U(, α) α > ( ) α X E(X; ) = E X = 1 α α x 1 α 2 α 2 α dx = α. 4 X exp(λ) λ > E(X; ) = E ( ) 1 λ X = 1 λ. X P areo(α, β) α > 1 β > ( ) X + β E(X; ) = E α 1 X = = 1 α 1 1 β α (+β) α (x + β) α β α dx (x + β) α+1 α ( + β) (α 1) 2. X P ower(α, β) α > β > ( ) β X E(X; ) = E α + 1 X = = 1 α + 1 1 ( β β )α β (β x) α (β x)α 1 dx β α α (β ) (α + 1) 2.

X X m() µ m() m() = (c 1) + µ, c >, X c = 1 c > 1 < c < 1 X m() E(X; ) E(X; ) = c m(), c >, X c = 1 c > 1 < c < 1 1 F () F (x) ln F (x)dx + m() ln F () = c m().

c m () = m () ln F () m() λ() + ln F 1 () λ() F (x) ln F () F (x)dx = m () ln F () m() λ() + ln F () + λ() [ c m() m() ln F () ], d d F (x) ln F (x)dx = d F (x) ln d F (x)dx = F () ln F () d d 1 F () = f() F () = λ() 2 F (). m () c (m() λ() 1 ) = (m() λ() 1) ln F () λ() m() + ln F () + c λ() m() λ() m() ln F (). x > m(x) λ(x) = c, m (x) = c 1. x (, ) m() m() = (c 1) + m() = (c 1) + µ c E(X; ) = m(), c = 1

E(X; ) = α ( + β) (α 1) 2 = α α 1 m(), c = α α 1 > 1. E(X; ) = α (β ) (α + 1) 2 = α α + 1 m(), c = α α + 1 < 1. X F E(X; ) X IDCRE F IDCRE E(X; ) X DDCRE F DDCRE E(X; )

X E(X; ) E (X; ) E (X; ) = λ() [E(X; ) m()], λ() X m() E 1 (X; ) = λ() F (x) ln F () F (x)dx + ln F () + m () ln F () m() λ(). m () E (X; ) = ( λ() 1 ) F (x) ln F () (x)dx + ln F () + ln F () [λ() m() 1] m() λ(). 1 F () F (x) ln F (x)dx = E(X; ) m() ln F (), E (X; ) = λ() [ E(X; ) m() ln F () ] + λ() m() ln F () m() λ() = λ() [E(X; ) m()]. X m() E(X; ) E(X; ) E(X; ) ( ) m(). E(X; ) E (X; ) ( ),

λ() > > E(X; ) m() ( ), E(X; ) ( ) m(). IMRL DMRL IDCRE DDCRE X m() E(X; ) m() E(X; ) m() x > m(x) f(x)dx F () m(x) m() m() f(x)dx = m(), F () f(x)dx = F (). X E(X; ) m(), E(X; ) m(),. E (X; ), E(X; ) m()

IMRL DMRL IDCRE DDCRE DF R IMRL IDCRE IF R DMRL DDCRE. DMRL DDCRE IMRL IDCRE IMRL DMRL IDCRE DDCRE IMRL DMRL IDCRE DDCRE X α = 2 κ = 1.3 F () = (1 + 2 ) 1.3, >. m() m() = (1 + x 2 ) 1.3 dx, >. (1 + 2 ) 1.3 m()

1.15 m 1.1 1.5 1..95.2.4.6.8 1. m() 1 m() =.529797 =.529797 m() X IMRL E(X; ) 1 E(X; ) = (1 + 2 ) 1.3 (1 + x 2 ) 1.3 ln (1 + x2 ) 1.3 dx, >. (1 + 2 ) 1.3 m() E(X; )

m 1.5 E_1 X; 1.4 1.3 1.2 1.1 1...2.4.6.8 1. m() E(X; ) 1 > E(X; ) m() X IDCRE X F () = e 5 (2) 1/2,. m() m() = e x5 (2x) 1/2 dx,. e 5 (2) 1/2 m()

m.45.4.35.3.25.2.15.2.4.6.8 1. m() 1 m() =.12786 =.12786 m() X DMRL E(X; ) E(X; ) = 1 e 5 (2) 1/2 e x5 (2x) 1/2 ln e x5 (2x) 1/2 dx,. e 5 (2) 1/2 m() E(X; )

.5 E_1 X; m.4.3.2.2.4.6.8 1. m() E(X; ) 1 > E(X; ) m() X DDCRE

X F (x) Y Ḡ(+x)/Ḡ(x) X = x

F 2 () = = = F () + P (X + Y > X = x ) f(x)dx P (Y > x X = x) f(x)dx + Ḡ() Ḡ(x) f(x)dx, f(x)dx P (X + Y > X = x) = 1 x, f(x) X X n n Fn () F n () = F () n 1 k= [Λ()] k k! = q n ( F ()) n = 1, 2,..., Λ() X q n (x) = x n 1 k= [ lnx] k k! x q n () = q n (1) = 1 F = Ḡ F n X n F X n f n () = [Λ()]n 1 (n 1)! f() n = 1, 2,.... X n µ n = F n (x)dx n 1,

n = 1 µ n+1 µ n = = F (x) n [Λ(x)] k dx k! k= F (x) [Λ(x)]n n! dx. µ 2 µ 1 = F (x) Λ(x)dx = = E(X), F (x) n 1 k= [Λ(x)] k k! F (x) ln F (x)dx dx X X E n (X) n = n = 1 E n (X) = F (x) [Λ(x)]n dx n =, 1, 2,.... n! E (X) = E 1 (X) = F (x)dx = E(X). F (x) Λ(x)dx = E(X). E n (X) F n+1 F n n = E (X) = E(X) F 1 F

E n (X) λ() X X E n (X) λ() ( ) 1 E n (X) = E. λ(x n+1 ) E n (X) = F (x) f(x) [Λ(x)] n n! f n+1 (x) = [Λ(x)]n n! f(x). f(x)dx. 1 E n (X) = λ(x) f n+1(x)dx ( ) 1 = E. λ(x n+1 ) X Y F Ḡ F (x) Ḡ(x) x (, ). X Y X ST Y X Y λ X λ Y λ X () λ Y (),.

X Y X HR Y X Y f() g() g() f() (, ), X Y X LR Y IF R DF R X E n (X) λ() X IF R DF R E n (X) ( ) E n+1 (X) n =, 1, 2... f n+1 () f n () = Λ() n X n LR X n+1. X n ST X n+1. X IF R DF R λ() E n (X) ( ) E n+1 (X) n =, 1, 2.... E n (X) = E(X) = E (X) = E(X) = 1 λ n = 1, 2,....

X Y X HR Y DF R E n (X) E n (Y ) n =, 1, 2,... X Y F Ḡ λ X λ Y X HR Y X ST Y F () Ḡ(),. n = E (X) = F ()d Ḡ()d = E (Y ) q n+1 F n+1 () = q n+1 ( F ()) q n+1 (Ḡ()) = Ḡn+1(), F n+1 () Ḡ n+1 () X n+1 Y n+1 X n+1 ST Y n+1 E(ϕ(X n+1 )) E(ϕ(Y n+1 )) ϕ Y DF R 1 λ Y ( E n (Y ) = E 1 λ Y (Y n+1 ) ) ( ) 1 E. λ Y (X n+1 ) X HR Y 1 1 λ X λ Y

( ) 1 E λ Y (X n+1 ) ( ) 1 E = E n (X) λ X (X n+1 ) E n (X) E n (Y ). X DF R X E n (X; ) E n (X; ) = 1 [ F (x) ln F ] n (x) dx n! F () F () n =, 1, 2.... = n = n = 1 E n (X; ) = E n (X). E (X; ) = E(X ) = m(). E 1 (X; ) = E(X; ). n ( ) n ( 1) n k x k y n k = (x y) n. k k= E n (X; ) E n (X; ) = 1 n! F (x) F () [Λ(x) Λ()]n dx

E n (X; ) = = = 1 n! F () n 1 n! F () 1 n F () k= k= n ( ) n F (x) ( 1) n k [Λ(x)] k [Λ()] n k dx k k= ( n )( 1) n k [Λ()] n k F (x) [Λ(x)] k dx k ( 1) n k k! (n k)! [Λ()]n k F (x) [Λ(x)] n dx F (x) [Λ(x)] n dx = n! F () E n (X; ) k= n 1 ( n )( 1) n k [Λ()] n k k n = 1 [ 1 E(X; ) = Λ() F (x)dx + F () = m() ln F () 1 F () F (x) [Λ(x)] k dx. F (x) ln F (x)dx F (x)[λ(x)] k dx. ] F (x) Λ(x)dx X λ X (x) = λ(x + ) X IF R DF R X IF R DF R E n (X; ) ( ) E n+1 (X; ) > n =, 1,...

( ) 1 E n (X; ) = E, λ( + X,n+1 ) X,n+1 n X X E n (X; ) E n(x; ) E n(x; ) = λ() [E n (X; ) E n 1 (X; )] n = 1, 2,... λ() X E n (X; ) F () = n k= ( 1) n k [Λ()] n k k! (n k)! F (x) [Λ(x)] k dx. E n(x; ) F () E n (X; ) f() = λ() n 1 k= [Λ()] n F () n ( 1) n k [Λ()] n k 1 k! (n k 1)! k= ( 1) n k k! (n k)! F (x) [Λ(x)] k dx d d F (x) [Λ(x)] k dx = F () [Λ()] k.

n k= ( 1) n k k! (n k)! = 1 n! n ( ) n ( 1) n k 1 k = 1 k n! (1 1)n =. k= E n(x; ) F () E n (X; ) f() = λ() n 1 k= ( 1) n k [Λ()] n k 1 k! (n k 1)! F (x) [Λ(x)] k dx λ() n 1 k= ( 1) n k [Λ()] n k 1 k! (n k 1)! F (x) [Λ(x)] k dx = λ() F () E n 1 (X; ). E n(x; ) F () = E n (X; ) f() λ() F () E n 1 (X; ) F () E n(x; ) = λ() [E n (X; ) E n 1 (X; )]. n = 1 E (X; ) = λ() [E(X; ) m()] X E n (X; ) E n (X; ) = c E n 1 (X; ) c > n {1, 2,...} X c = 1 c > 1

< c < 1 n = 1 E(X; ) = c m() n > 1 n 1 E n(x; ) = c E n 1(X; ). E n(x; ) = λ() [E n (X; ) E n 1 (X; )] = λ() [c E n 1 (X; ) E n 1 (X; )] = λ() (c 1) E n 1 (X; ). c E n 1(X; ) = λ() (c 1) E n 1 (X; ). c E n 1(X; ) = c λ() [E n 1 (X; ) E n 2 (X; )]. (c 1) E n 1 (X; ) = c [E n 1 (X; ) E n 2 (X; )], E n 1 (X; ) = c E n 2 (X; ) n 2. E n (X; ) = c E n 1 (X; ) = c 2 E n 2 (X; ) =... = c n m() E n (X; ) = E n 1 (X; ) =... = E(X; ) = m() = µ.

E n (X; ) E n (X; ) = E n 1 (X; ) n {1, 2,...} > c > 1 E n (X; ) E n 1 (X; )... E(X; ) m() m() E n (X; ) = c n m() c c = α α 1 X n =, 1, 2,... E n (X; ) = c n m() = αn ( + β) (α 1) n+1 < c < 1 E n (X; ) E n 1 (X; )... E(X; ) m() m() E n (X; ) c c = α α + 1 X n =, 1, 2,... E n (X; ) = c n m() = αn (β ) (α + 1) n+1

U R n+1 f i : U R ϑf i ϑy j U i, j = 1,..., n (x, z 1,..., z n ) U y 1 = f 1 (x, y 1,..., y n )... y n = f n (x, y 1,..., y n ), y j (x ) = z j j = 1, 2,..., n U X E i (X; ) i =, 1,..., n E n (X; ) E n 1 (X; ). E i (X; ) i =, 1,..., n 1 E n (X; ) X E 1(X; ) = λ() (E 1 (X; ) E (X; )) E 2(X; ) = λ() (E 2 (X; ) E 1 (X; ))... E n(x; ) = λ() (E n (X; ) E n 1 (X; )) λ() λ() = E n(x; ) E n (X; ) E n 1 (X; )

j = 1, 2,..., n 1 E j(x; ) = E j(x; ) E j 1 (X; ) E n (X; ) E n 1 (X; ) E n(x; ) E (X; ) = m() E (X; ) = λ() E (X; ) 1 = E (X; ) E n(x; ) E n (X; ) E n 1 (X; ) 1 y = f (, y,..., y n 1 ) y 1 = f 1 (, y,..., y n 1 )... y n 1 = f n 1 (, y,..., y n 1 ), y j = E j (X; ), j =, 1,..., n 1, f (, y,..., y n 1 ) = y E n(x; ) E n (X; ) y n 1 1, f j (, y,..., y n 1 ) = y j y j 1 E n (X; ) y n 1 E n(x; ), j = 1,..., n 1, E n (X; ) E i (X; ) i =, 1,..., n 1 E n (X; ) E j (X; ) j =, 1,..., n 1 E (X; ) = m()

X µ E(X; ) E(X; ) m(), µ E(X; ) X n = 1 E(X; ) = E (X; ) X E (X; ) = m() = µ. E n (X; ) E n (X; ) = E n 1 (X; ) X IDGCRE n E n (X; ) X IDGCRE n DDGCRE n E n (X; ) X DDGCRE n

n = IDGCRE IMRL DDGCRE DMRL. n = 1 IDGCRE 1 IDCRE DDGCRE 1 DDCRE. DF R IF R IDGCRE n DDGCRE n n {1, 2,...} X IF R DF R E n (X; ) n =, 1, 2,... n = E (X; ) = m() X IF R DF R X DMRL IMRL n 1 E n(x; ) = λ() [E n (X; ) E n 1 (X; )]. E n (X; ) ( ) E n 1 (X; ). E n(x; ) ( ) >, E n (X; ) X DDGCRE n IDGCRE n n = 1 IF R (DF R) DMRL (IMRL) DDCRE (IDCRE).

IDGCRE n 1 DDGCRE n 1 IDGCRE n DDGCRE n X E n (X; ) E n (X; ) < n = 1, 2,... E n (X; ) = E n 1 (X; z) f(z)dz F () X X IDGCRE n 1 DDGCRE n 1 IDGCRE n DDGCRE n X IDGCRE n 1 E n 1 (X; ) E n 1 (X; z) E n 1 (X; ) z >. E n (X; ) = E n 1 (X; z) f(z) dz F () E n 1 (X; )f(z)dz = E n 1 (X; ),. F () E n(x; ),, X IDGCRE n X DDGCRE n 1

IMRL IDCRE IDGCRE 2... IDGCRE n 1 IDGCRE n DMRL DDCRE DDGCRE 2... DDGCRE n 1 DDGCRE n. DMRL DDCRE DDGCRE n n IDGCRE n n m() E(X; ) m() E(X; ) E n (X; ) n 2 E n 1 (X; ) E n (X; ) > E n 1 (X; ), > E n(x; ) >, >, E n (X; ) E n 1 (X; ) E n (X; ) < E n 1 (X; ), >, E n (X; )

n E n (X; ) E n 1 (X; ) E n (X; ) X F () = e 1.5.5,. m() =.455765, E 1 (X; ) =.495448, E 2 (X; ) =.463393, E 3 (X; ) =.427791, E 1 (X; ) > E 2 (X; ) > m() > E 3 (X; ). m() E 1 (X; )

m E_1 X;.52.5.48.46.1.2.3.4.5 m() E 1 (X; ).5 m() E 1 (X; ) m() E 1 (X; ) E 1 (X; ) E 2 (X; ) E_1 X;.49 E_2 X;.48.47.46.45.44.43..1.2.3.4.5 E 1 (X; ) E 2 (X; ).5 E 2 (X; ) < E 1 (X; ) E 2 (X; ) < E 1 (X; ) E 2 (X; )

E 2 (X; ) E 1 (X; ) 3.2.2 = E 2 (X; ) X E n (X; ) n =, 1, 2, 3.5 m E_1 X;.45 E_2 X; E_3 X;.4.2.4.6.8 1. m() E 1 (X; ) E 2 (X; ) E 3 (X; ) 1 E 3 (X; ) X X α = 3 κ = 2.5 F () = (1 + 3 ) 2.5,. m() =.72757, E 1 (X; ) =.323797, E 2 (X; ) =.284452, E 3 (X; ) =.287916, m() > E 1 (X; ) > E 3 (X; ) > E 2 (X; ). m() E 1 (X; ) E 2 (X; )

m E_1 X; E_2 X;.45.4.35.3.5 1. 1.5 2. m() E 1 (X; ) E 2 (X; ) 2 m() E 1 (X; ) E 2 (X; ) E 2 (X; ) E 3 (X; ) E_2 X;.29 E_3 X;.288.286.284.282.28..1.2.3.4.5 E 2 (X; ) E 3 (X; ).5 E 3 (X; ) > E 2 (X; ) E 3 (X; ) X IDGCRE 3 3.2.2 = E 3 (X; ) X

E n (X; ) X n =, 1, 2, 3.5.45.4 m E_1 X; E_2 X;.35 E_3 X;.3.5 1. 1.5 2. m() E 1 (X; ) E 2 (X; ) E 3 (X; ) 2

X f(x) F (x) σ 2 X σ 2 = V ar(x) = E[(X µ) 2 ] = (x µ) 2 f(x)dx, E(X 2 ) < σ E(X 2 )

E n (X) n = 1 E 1 (X) E(X2 ) 2E(X) E(X 2 ) < E 1 (X) < X IF R E n (X) E n+1 (X), n, E(X 2 ) < E n (X) < n 1 < α < 2 NBUE E(X) µ IF R DDGCRE n n n E n (X) µ n 1

X (, α) α > σ 2 = α2 12 σ = α 2 3, E 1 (X) = α 4 = 3 α σ2 E 1 (X) = 3 2 σ, E 1 (X) < σ E n (X) = 1 2 E n 1(X) =... = 1 2 n 1 E 1 (X) = 1 n 1 3 2 2 σ = n = 1, 2,... n = 1, 2,... E n (X) < E n 1 (X) <... < E 1 (X) < σ 3 2 n σ X λ > σ 2 = 1 λ 2 σ = 1 λ,

E 1 (X) = 1 λ = λ σ2 E 1 (X) = σ, E n (X) = σ n 1. X α > 2 β > σ 2 αβ 2 α β = σ = (α 1) 2 (α 2) (α 1) α 2, κ 1 = 1 1 (α 1) 2 E 1 (X) = E 1 (X) = αβ (α 1) = α 2 2 β 1 σ 2 1 (α 1) 2 σ = κ 1 σ, < 1 E 1 (X) < σ = E n (X) = c E n 1 (X) =... = c n 1 E 1 (X) = c n 1 κ 1 σ n = 1, 2,... c = c n 1 κ 1 < 1 n α > 1 E α 1 n(x) < σ n < ln c κ 1 + 1, n < 1 2 ln α α 2 + 1. α = 3 n = 1.54931 n = 2, 3,... E 1 (X) < σ < E 2 (X) <... < E n (X) n = 2 = E 2 (X) = α2 β (α 1) 3,

E 2 (X) > σ α 2 β (α 1) 3 > α β (α 1) α 2, α 3/2 (α 2) (α 1) 2 >, α > 2.6183 α > 2.6183 X α > β > σ 2 = αβ 2 (α + 1) 2 (α + 2) σ = α β (α + 1) α + 2, κ 2 = 1 1 (α+1) 2 E 1 (X) = E 1 (X) = αβ (α + 1) = α + 2 2 β 1 σ 2 1 (α + 1) 2 σ = κ 2 σ, < 1 E 1 (X) < σ E n (X) = c E n 1 (X) =... = c n 1 E 1 (X) = c n 1 κ 2 σ n = 1, 2,... c = n = 1, 2,... α α+1 < 1 E n (X) < E n 1 (X) <... < E 1 (X) < σ

(σ) (CRE) σ CRE DJIA NASDAQ SP 5 NY SE HSI T W II ST I N225 F T SE1 DAX3 CAC4 SMI SP 5

(σ) (CRE) σ CRE n

X F (x) V ar q (X) V ar q (X) = x q = F 1 (1 q) F (x q ) = P (X > x q ) = 1 q, q q q X q

X T CE q (X) T CE q (X) = E(X X > x q ) 1 = x f(x)dx, F (x q ) x q x q q x q q T CE q (X) = E(X X > x q ) = x q + E(X x q X > x q ) = V ar q (X) + m(x q ), m(x q ) X x q T CE q (X) V ar q (X) m()

X µ = E(X) T CV q (X) T CV q (X) = V ar(x X > x q ) = E[(X µ) 2 X > x q ] 1 = (x µ) F 2 f(x)dx, (x q ) x q F, f X V ar(x) = E[(X µ) 2 ] = E[(X µ) 2 I(X > x q )] + E[(X µ) 2 I(X x q )], 1 x > x q I(X > x q ) = x x q X E[(X µ) 2 I(X > x q )] = x q (x µ) 2 f(x)dx = F (x q ) (x µ) 2 f(x) x q F (x q ) dx = (1 q) E[(X µ) 2 X > x q ] = (1 q) T CV q (X), V ar(x) (1 q) T CV q (X)

T CV q (X) 1 1 q E[(X µ) 2 I(X x q )]. V ar(x), 1 q < 1 T CV q (X) > V ar(x). X g(x) = (µ X) 2 m(x) X E 1 (X; x q ) = E(m(X) X > x q ) n

X α = 3 β = 8 µ = 4 F () = ( ) 3 8,. + 8 n = 1, 2, 3 ( ) 3 xq + 8 T CE q (X) = x 8 x q 8 3 3 (x + 8) 4 dx, ( ) 3 xq + 8 T CV q (X) = (x 4) 2 8 3 3 8 x q (x + 8) dx, 4 E 1 (X; x q ) = α (x q + β) (α 1) 2 = 3 (x q + 8) 2 2, E 2 (X; x q ) = α2 (x q + β) (α 1) 3 = 32 (x q + 8) 2 3, E 3 (X; x q ) = α3 (x q + β) (α 1) 4 = 33 (x q + 8) 2 4, x q q x q

TCE_q X TCV_q X 4 1 6 3 1 6 2 1 6 E_1 X;X_q E_2 X;X_q E_3 X;X_q 1 1 6 x_q 5 6 7 8 9 1 T CE q (X) T CV q (X) E 1 (X; x q ) E 2 (X; x q ) E 3 (X; x q ) 4 x q 1 T CV q (X) T CE q (X) E 1 (X; x q ) E 2 (X; x q ) E 3 (X; x q ) 3 TCE_q X 25 E_1 X;X_q 2 15 E_2 X;X_q E_3 X;X_q x_q 5 6 7 8 9 1 T CE q (X) E 1 (X; x q ) E 2 (X; x q ) E 3 (X; x q ) 4 x q 1 E 1 (X; x q ) < T CE q (X) < E 2 (X; x q ) < E 3 (X; x q ) n = 2, 3

X α = 3/2 β = 4 F () = ( ) 3/2 4,. + 4 µ = 8 T CE q (X) E 1 (X; x q ) E 2 (X; x q ) E 3 (X; x q ) TCE_q X 7 E_1 X;X_q 6 5 4 3 2 E_2 X;X_q E_3 X;X_q 1 x_q 5 6 7 8 9 1 T CE q (X) E 1 (X; x q ) E 2 (X; x q ) E 3 (X; x q ) 4 x q 1 T CE q (X) < E 1 (X; x q ) < E 2 (X; x q ) < E 3 (X; x q ) n = 1, 2,... X IDGCRE n n = 1, 2,... E n (X; x q ) > E n 1 (X; x q ) n = 1, 2,...

X α = 3 β = 8 µ = 2 ( ) 3 8 F () =, < < 8. 8 n = 1, 2, 3 ( ) 3 8 8 T CE q (X) = 8 x q x q x 3 (8 x)2 8 3 dx, ( ) 3 8 8 2 3 (8 x)2 T CV q (X) = (x 2) dx, 8 x q x q 8 3 E 1 (X; x q ) = α (β x q) (α + 1) 2 = 3 (8 x q) 4 2, E 2 (X; x q ) = α2 (β x q ) (α + 1) 3 = 32 (8 x q ) 4 3, E 3 (X; x q ) = α3 (β x q ) (α + 1) 4 = 33 (8 x q ) 4 4, x q < 8 q

TCE_q X 2 TCV_q X 15 1 5 E_1 X;X_q E_2 X;X_q E_3 X;X_q 45 5 55 6 T CE q (X) T CV q (X) E 1 (X; x q ) E 2 (X; x q ) E 3 (X; x q ) 4 x q 6 T CV q (X) T CE q (X) E 1 (X; x q ) E 2 (X; x q ) E 3 (X; x q ) 6 5 4 3 2 TCE_q X E_1 X;X_q E_2 X;X_q E_3 X;X_q 1 45 5 55 6 T CE q (X) E 1 (X; x q ) E 2 (X; x q ) E 3 (X; x q ) 4 x q 6 T CE q (X) > E 1 (X; x q ) > E 2 (X; x q ) > E 3 (X; x q ) X DDGCRE n E n (X; x q ) < E n 1 (X; x q ) n = 1, 2,... n

IDGCRE n n

CRE CV ar DCRE DDCRE DDGCRE n DF R DGCRE DMRL GCRE HR IDCRE IDGCRE n IF R IMRL MRL NBUE NW UE

T CE T CV V ar V ar

ailf x_ : 1 x^ 2^1.3 L x_ : Log ailf x mrl _ : Inegrae ailf x, x,, Infiniy ailf Enro1 _ : L mrl Inegrae ailf x L xailf, x,, Infiniy FindMinimum mrl,,, 1 FindMinimum Enro1,,, 1 Plo mrl,,, 1, AxesLabel"", "m " Plomrl, Enro1,,, 1, PloSyleRGBColor.75,,, RGBColor,.25,, LegendPosiion.95,.4, LegendSize.4, LegendTexSpace 2, PloLegend"m ", "E_1 X; ", AxesLabel"", " " ailf x_ : Expx^ 4 3 x ^ 1 3 L x_ : Log ailf x mrl _ : Inegrae ailf x, x,, Infiniy ailf Enro1 _ : L mrl Inegrae ailf x L xailf, x,, Infiniy FindMaximum mrl,,, 1 FindMaximum Enro1,,, 1 Plo mrl,,, 1, AxesLabel"", "m " Plomrl, Enro1,,, 1, PloSyleRGBColor.5,,, RGBColor,.75,, LegendPosiion.95,.4, LegendSize.4, LegendTexSpace 2, PloLegend"m ", "E_1 X; ", AxesLabel"", " "

Needs "PloLegends " ailf x_ : Expx^ 1.5x^.5 L x_ : Log ailf x mrl _ : Inegrae ailf x, x,, Infiniy ailf Enro1 _ : L mrl Inegrae ailf x L xailf, x,, Infiniy Enro2 _ : L ^2 mrl 2L ailf Inegrae ailf x L x, x,, Infiniy 12 ailf Inegrae ailf x L x^2, x,, Infiniy Enro3 _ : L ^3 mrl 6 L ^22 ailf Inegrae ailf x L x, x,, Infiniy L 2 ailf Inegrae ailf x L x^2, x,, Infiniy 16 ailf Inegrae ailf x L x^3, x,, Infiniy mrl Enro1 Enro2 Enro3 Plomrl, Enro1,,,.1, PloSyleRGBColor 1,,, RGBColor, 1,, LegendPosiion.95,.4, LegendSize.4, LegendTexSpace 2, PloLegend"m ", "E_1 X; ", AxesLabel"", " " PloEnro1, Enro2,,,.5, PloSyleRGBColor, 1,, RGBColor,, 1, LegendPosiion.95,.4, LegendSize.4, LegendTexSpace 2, PloLegend"E_1 X; ", "E_2 X; ", AxesLabel"", " " Plomrl, Enro1, Enro2, Enro3,,, 1, PloSyleRGBColor 1,,, RGBColor, 1,, RGBColor,, 1, RGBColor.2,,, LegendPosiion.95,.3, LegendSize.7, LegendTexSpace 2, PloLegend"m ", "E_1 X; ", "E_2 X; ", "E_3 X; ", AxesLabel"", " "

Needs "PloLegends " ailf x_ : 1 x^ 3^2.5 L x_ : Log ailf x mrl _ : Inegrae ailf x, x,, Infiniy ailf Enro1 _ : L mrl Inegrae ailf x L xailf, x,, Infiniy Enro2 _ : L ^2 mrl 2L ailf Inegrae ailf x L x, x,, Infiniy 12 ailf Inegrae ailf x L x^2, x,, Infiniy Enro3 _ : L ^3 mrl 6 L ^22 ailf Inegrae ailf x L x, x,, Infiniy L 2 ailf Inegrae ailf x L x^2, x,, Infiniy 16 ailf Inegrae ailf x L x^3, x,, Infiniy N mrl N Enro1 N Enro2 N Enro3 Plomrl, Enro1, Enro2,,, 2, PloSyleRGBColor 1,,, RGBColor, 1,, RGBColor,, 1, LegendPosiion.95,.4, LegendSize.4, LegendTexSpace 2, PloLegend"m ", "E_1 X; ", "E_2 X; ", AxesLabel"", " " PloEnro2, Enro3,,,.5, PloSyleRGBColor,, 1, RGBColor.2,,, LegendPosiion.95,.4, LegendSize.4, LegendTexSpace 2, PloLegend"E_2 X; ", "E_3 X; ", AxesLabel"", " " Plomrl, Enro1, Enro2, Enro3,,, 2, PloSyleRGBColor 1,,, RGBColor, 1,, RGBColor,, 1, RGBColor.2,,, LegendPosiion.95,.3, LegendSize.7, LegendTexSpace 2, PloLegend"m ", "E_1 X; ", "E_2 X; ", "E_3 X; ", AxesLabel"", " "

Needs "PloLegends " Alphapar : 3 Beapar : 8 ailf x_ : Beaparx Beapar^ Alphapar Piknoia x_ : Alphapar Beapar^ Alphaparx Beapar ^ Alphapar 1 Mu : Inegrae ailf x, x,, Infiniy Tce _ : Inegrae x Piknoia x, x,, Infiniy ailf Tcv _ : Inegrae x Mu ^ 2 Piknoia x, x,, Infiniy ailf Enro1 _ : Alphapar BeaparAlphapar 1 ^ 2 Enro2 _ : Alphapar^ 2 BeaparAlphapar 1 ^ 3 Enro3 _ : Alphapar^ 3 BeaparAlphapar 1 ^ 4 N Mu PloTce, Tcv, Enro1, Enro2, Enro3,, 4, 1, PloSyleRGBColor,.25,, RGBColor.35,,, RGBColor,.85,, RGBColor,,.5, RGBColor.9,,, LegendPosiion.95,.3, LegendSize 1.1, LegendTexSpace 2, PloLegend"TCE_q X ", "TCV_q X ", "E_1 X;X_q ", "E_2 X;X_q ", "E_3 X;X_q ", AxesLabel"x_q", " " PloTce, Enro1, Enro2, Enro3,, 4, 1, PloSyle RGBColor.35,,, RGBColor,.85,, RGBColor,,.5, RGBColor.9,,, LegendPosiion.95,.3, LegendSize.9, LegendTexSpace 2, PloLegend"TCE_q X ", "E_1 X;X_q ", "E_2 X;X_q ", "E_3 X;X_q ", AxesLabel"x_q", " "

Needs "PloLegends " Alphapar : 1.5 Beapar : 4 ailf x_ : Beaparx Beapar^ Alphapar Piknoia x_ : Alphapar Beapar^ Alphaparx Beapar ^ Alphapar 1 Mu : Inegrae ailf x, x,, Infiniy Tce _ : Inegrae x Piknoia x, x,, Infiniy ailf Enro1 _ : Alphapar BeaparAlphapar 1 ^ 2 Enro2 _ : Alphapar^ 2 BeaparAlphapar 1 ^ 3 Enro3 _ : Alphapar^ 3 BeaparAlphapar 1 ^ 4 PloTce, Enro1, Enro2, Enro3,, 4, 1, PloSyle RGBColor.35,,, RGBColor,.85,, RGBColor,,.5, RGBColor.9,,, LegendPosiion.95,.3, LegendSize 1.1, LegendTexSpace 2, PloLegend"TCE_q X ", "E_1 X;X_q ", "E_2 X;X_q ", "E_3 X;X_q ", AxesLabel"x_q", " "

Needs "PloLegends " Alphapow : 3 Beapow : 8 ailf x_ : Beapow xbeapow ^ Alphapow Piknoia x_ : Alphapow Beapow x ^ Alphapow 1Beapow^ Alphapow Mu : Inegrae ailf x, x,, Beapow Tce _ : Inegrae x Piknoia x, x,, Beapow ailf Tcv _ : Inegrae x Mu ^ 2 Piknoia x, x,, Beapow ailf Enro _ : Beapow Alphapow 1 Enro1 _ : Alphapow Beapow Alphapow 1 ^ 2 Enro2 _ : Alphapow^ 2 Beapow Alphapow 1 ^ 3 Enro3 _ : Alphapow^ 3 Beapow Alphapow 1 ^ 4 N Mu PloTce, Tcv, Enro1, Enro2, Enro3,, 4, 6, PloSyleRGBColor.35,,, RGBColor,.25,, RGBColor,.85,, RGBColor,,.5, RGBColor.9,,, LegendPosiion.95,.3, LegendSize 1.1, LegendTexSpace 2, PloLegend"TCE_q X ", "TCV_q X ", "E_1 X;X_q ", "E_2 X;X_q ", "E_3 X;X_q ", AxesLabel"", " " PloTce, Enro1, Enro2, Enro3,, 4, 6, PloSyle RGBColor.35,,, RGBColor,.85,, RGBColor,,.5, RGBColor.9,,, LegendPosiion.95,.3, LegendSize.9, LegendTexSpace 2, PloLegend"TCE_q X ", "E_1 X;X_q ", "E_2 X;X_q ", "E_3 X;X_q ", AxesLabel"", " "

3 h