Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://ms.cs.msu.ru/attachments/037_praktikum_part1.pdf Äàòà èçìåíåíèÿ: Mon Sep 6 15:20:24 2010 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 06:31:43 2016 Êîäèðîâêà:

. ... . . . . , , , , . , , , . . : ; ; ; , . , . " "(- " 1999.) 50 . , 05-01-00535.

3




N (µ, 2 2 2 N2 (µ1 , µ2 , 1 , 2 , (, P (a, b (r, s U (a, b b(n, p ( ¯(k , p b G(p (x

) ) ) ) ) ) ) ) ) ) )

-- -- -- -- -- -- -- -- -- -- --

(x) -- (x, y ) -- (x, y ) -- X F -- X=Y
P d d

; ; -; ; -; ; ; ; ; ; N (0, 1); N (0, 1); N2 (0, 0, 1, 1, ); N2 (0, 0, 1, 1, ); X F ;

-- X Y ;

- -- ; - -- ; an bn -- an /bn 1 n ; I {A} -- A;

4


EX -- X ; DX -- X ; cov (X, Y ) = E(X - EX )(Y - EY ) - X Y ; F (x) = P{X < x} - .. X ; F (x, y ) = P{X < x, Y < y } - .. (X, Y ); F = inf {x : F (x) > 0}; F = sup{x : F (x) < 1}; X
(n) k

-- k-
(n) (n)

( X1 X2 ... Xnn) , X1 , ..., Xn ; (n) = min1in {Xi }; X1 ( Xnn) = max1in {Xi }; n X= i=1 Xi /n; n 2 2 S = i=1 (Xi - X ) /(n - 1); ... -- ( ); .. -- ; [z ] -- z .

5


1. . X ( ) (µ, 2 ), X N (µ, 2 ), X 1 (x - µ)2 f (x; µ, ) = exp - 2 2 2 - < x < , P{X < x} =
x -

=

1 ((x - µ)/ ),

f (u; µ, )du = ((x - µ)/ ), (x) =
x -

1 (x) = exp(-x2 /2), 2

(u)du,

EX = µ, DX = 2 . X N (µ, 2 ), Y = (X - µ)/ N (0, 1). Y N (0, 1), EY 2k-1 = 0, EY 2k = (2k - 1)(2k - 3)...1, k = 1, 2, .... 2 X1 , ...Xk , Xi N (µi , i ), i = 1, 2, ..., k , k=1 ci Xi i k k 2 N ( i=1 ci µi , i=1 c2 i ). i 2. . X = (X1 , X2 ) 2 2 ( ) , X N2 (µ1 , µ2 , 1 , 2 , ), e X f ( x 1 , x 2 ; µ1 , µ 2 , 1 , 2 , ) = exp - 1 2(1 - 2 )
1 2 1 2 1-
2

x1 - µ 1

1

2

+

x2 - µ 2

2

2

-

2(x1 - µ1 )(x2 - µ2 ) 1 2

,

2 µi = EXi , i = DXi , i = 1, 2, =

cov (X1 , X2 ) , || < 1. 1 2
x - y -

(x, y ) = f (x, y ; 0, 0, 1, 1, ), (x, y ) = (u, v )dudv .

3.-. X - (, ), X (, ), X f (x; , ) = () = EX = ,
0 -x -1 ex ()

0 exp(-x)x
-1

, x > 0, , x 0, , > 0, k = 0, 1, ....
k

dx,

DX =

, 2

EX k =

( + k ) , k ()

n- , (n) = (n - 1)!. X1 , ..., X , Xi (, i ), i = 1, ..., k , k=1 Xi (, k=1 i ). i i 6


4. () . X P (a, b), X P (a, b), X f (x; a, b) = b-1 exp (-(x - a)/b) , x > a, 0 , x a,

|a| < , b > 0. , P (0, b) = (1/b, 1). X P (a, b), Y = (X - a)/b P (0, 1). X1 , ..., Xk , Xi P (0, 1), i = 1, ..., k ,

k i=1

Xi (1, k ).

5.-. X - (r, s), X (r, s), X f (x; r, s) = (r + s) x (r)(s)
r -1

(1 - x)s-1 ,

0 < x < 1,

r, s > 0.

Y Z , Y (1, r), Z (1, s). Y (r, s). Y +Z 6. . X U (a, b), X U (a, b), X f (x; a, b) = (b - a)- 0
1

, ,

a x b, x > b x < a,

- < a < b < . X U (a, b), (X - a)/(b - a) U (0, 1). , U (0, 1) = (1, 1). X U (a, b), EX = (a + b)/2, DX = (b - a)2 /12. X1 , ..., Xn U (0, 1), (n) (n) ( X1 X2 ... Xnn) , k (n) Xk - (k , n - k + 1). 7. . X b(n, p), X b(n, p),
k P{X = k } = Cn pk (1 - p)n-k ,

k = 0, 1, ..., n,

0 p 1.

X "" n p "" . X b(n, p). X b(n, p), EX = np, DX = np(1 - p). X1 , ..., Xk , Xi b(ni , p), i = 1, ..., k ,
k k

Xi b
i=1 i=1

ni , p .

8. . X ¯(m, p), X ¯(m, p), b b P{X = k } = C
k m m+k-1 p

(1 - p)k ,

k = 0, 1, ....

7


0 p 1, m 1. X " m - "" p "" . X ¯(m, p). b ¯(m, p), EX = m(1 - p)/p, DX = m(1 - p)/p2 . X b X1 , ..., Xk , Xi ¯(mi , p), i = 1, ..., k , b
k i=1

Xi ¯ b

k

mi , p .
i=1

¯(1, p) G(p). b 9. . C X (), X (), P{X = k } = e- k /k !, k = 0, 1, ..., > 0.

X (), EX = DX = . X1 , ..., Xk , Xi (i ), i = 1, ..., k .
k k

Xi
i=1 i=1



i

.


( X1 X2 ... Xnn) , n ... X1 , ..., Xn .. F (x). (n) k - Xk n . (n) (n) 1) Xk Xn-k k n ( ); 2) k , n , k /n p, 0 < p < 1 ( ); 3) k , n , k /n 0 ( ). F = inf {x : F (x) > 0}, F = sup{x : F (x) < 1}. , . 1. F = > 0, x > 0 (n) (n)

t

lim [1 - F (tx)]/[1 - F (t)] = x- ,

(1)

an > 0,
n ( lim P{Xnn) < an x} =

exp(-x- ) , 0 ,

x > 0, x 0.

an : an = inf {x : 1 - F (x) 1/n}. 8


2. F .. F (x) = F (F -1/x) , x > 0, (1). an > 0 bn ,
n ( lim P{Xnn) < bn + an x} =

1 , exp(-|x| ) ,

x 0, x < 0,

bn = F ,

an = F - inf {x : 1 - F (x) 1/n}.
a

3. a
F

(1 - F (x))dx <

F < t <

F

R(t) : R(t) = (1 - F (t))-
1 t F

(1 - F (y ))dy .

x
tF

lim [1 - F (t + xR(t))]/[1 - F (t)] = e-x ,

an > 0 bn ,
n

lim P{X

(n) n

< bn + an x} = exp(-e-x ),

- < x < .

(2)

an bn : bn = inf {x : 1 - F (x) 1/n}, an = R(bn ).

. z < F , F (x) (z , F ) f (x) = F (x) F (x) < 0 z < x < F . (2) ,
xF

lim

(1 - F (x))F (x) = -1. f 2 (x) 1 - F ( x) > 0, f (x)

a(x) = an an = a(bn ),

bn = inf {x : 1 - F (x) 1/n}.
(n)

X1 , (n) X1 = - max{-X1 , ..., -Xn }. 1. F = - > 0, x > 0 lim F (tx)/F (t) = x- . (3)
t-

9


cn > 0, lim P{X1 n
(n)

< c n x} =

1 - exp(-|x|- ) , 1 ,

x < 0, x 0.

cn : cn = sup{x : F (x) 1/n}. 2. F .. F (x) = F (F -1/x), x < 0, (3). cn > 0 dn ,
n

lim P{X1

(n)

< d n + c n x} =

1 - exp(-x ) , 0 ,

x > 0, x 0.

cn dn : dn = F , 3. cn = sup{x : F (x) 1/n} - F .
a
F

F (y )dy <

a. r(t) = x
tF

1 F (t)

t
F

F (y )dy ,

t > F .

lim

F (t + xr(t)) = ex . F (t)

cn > 0 dn ,
n

lim P{X1

(n)

< dn + cn x} = 1 - exp(-ex ),

- < x < .

(4)

cn dn : dn = sup{x : F (x) 1/n}, cn = r(dn ). z > F ,

. F (x) (F , z ) f (x) = F (x) F (x) > 0 F < x < z . (4) , lim x a(x) = F (x)F (x) = 1. f 2 (x) F (x) > 0, f (x) 10

F


cn cn = a(dn ), dn = sup{x : F (x) 1/n}.

lim n k - p = 0, n

n

.. [5], (n) (Xk - bn )/an an > 0 bn . , n ... X1 , ..., Xn , : (n) P{Xk < x} = P{Zn k }, P{Xn
(n) -k+1

< x} = P{Tn < k },

Zn b(n, P{X1 < x}), Tn b(n, P{X1 x}). Zn Tn . (n) . Uk , n (n) (n) U (0, 1), Uk (k , n - k + 1), Uk 1 /(1 + 2 ), 1 (1, k ), 2 (1, n - k + 1), 1 2 . k n. , , k , n , k = (n + 1)p, 0 < p < 1, 1 = (n + 1)p + Y
n,i d

(n + 1)pY

n,1

, 2 = (n + 1)(1 - p) +

(n + 1)(1 - p)Y

n,2

,

- Yi N (0, 1), i = 1, 2. n + 1(U
(n) k

d - p) - (1 - p) pY1 - p 1 - pY2

p(1 - p)Y , Y N (0, 1).

, X1 , ..., Xn - .. F (x), 0 = F -1 (p0 ), k , n , p = k /(n + 1) p0 , 0 < p0 < 1, F (Xk ) U
(n) (n) k

, Xk

(n)

F

-1

(p +

p0 (1 - p0 )/nYn ), Yn - Y N (0, 1).

d

F -1 p dF -1 (x)/dx = 1/f (F -1 (x)), f (x) = F (x). (n) (n) Xk (Xk - bn )/an , an > 0 bn - , n . 11


X1 , ..., Xn .. F (x) (n) f (x), .. Xk P{Xk
(n) n

< x} =
i=k

i Cn F i (x)[1 - F (x)]

n-i

,

Xk fk (x) =

(n)


k -1

n! F (k - 1)!(n - k )!
(n)

(x)[1 - F (x)]n-k f (x),

(Xk , X fk,r (x, y ) = x y. n! F (k - 1)!(r - k - 1)!(n - r)!

(n) r

), k < r, (x)f (x)(F (y )-F (x))r
-k-1

k-1

f (y )(1-F (y ))n-r ,

U U (0, 1), F (x) - .., X = F -1 (U ) .. F (x). ( ) , , . . , . X , P{X = xi } = pi > 0, i = 1, 2, ..., i1 pi = 1. {X } {U } U (0, 1) , X = xi , p0 +p1 +p2 +...+pi-1 < U p1 +p2 +...+pi , i 1, p0 = 0. . , (a, b). , , , a b. . X f (x), f (x) > 0 x [a, b], f (x) = 0 x [a, b]. f = maxx[a,b] f (x). (U1 , U2 ), U1 U (0, 1), U2 U (0, 1). X = a + (b - a)U2 . U1 < f (X )/f , X , (U1 , U2 ) . , U1 U (0, 1), U2 U (0, 1), U1 U2 , a + (b - a)U2 12


U1 < f (a + (b - a)U2 )/f f (x). . f (x, y ) A = {(x, y ) : a x b, c y d} A. f = max(x,y)A f (x, y ). (U1 , U2 , U3 ), Ui U (0, 1), i = 1, 2, 3. X = a + (b - a)U2 , Y = c + (d - c)U3 . U1 < f (X, Y )/f , (X, Y ) , f (x, y ), (U1 , U2 , U3 ) . . f (x) g (x) (a, b), , f (x) = C g (x)h(x), 0 h(x) 1 x, C - . , g (x) , X g (x) . f (x) . X U, U U (0, 1). U < h(X ), X f (x), (X, U ) . , U U (0, 1), X g (x), U X , f (x) X U < h(X ). C . , -1 e-x e-x f (x) = C , x > 0, 0, C = dx . 1 + x 1 + x 0 (0,1) , U1 U2 , X = - ln U1 . U2 < 1/(1 + X ), X f (x). (U1 , U2 ) . , (1) (U1 , U2 ) (0,1), X1 = (-2 ln U1 )
1/2

cos 2 U2 , X2 = (-2 ln U1 )

1/2

sin 2 U

2

. (2) (U1 , U2 ) (0,1). X = - ln U1 . (X - 1)2 -2 ln U2 , ±X . (U1 , U2 ) . ± . (3) V1 , V2 , ...- , (- 3, 3), Sn = 13


n i=1

Vi / n.
3 Sn + (Sn - 3Sn )/(20n)

n . X1 X2 , Xi N (0, 1), i = 1, 2. (X1 , X1 + 1 - 2 X2 )-

, N2 (0, 0, 1, 1, ) , . X P (a, b), U U (0, 1). X = -b ln U + a. X1 , X2 , ... . (Xi , Xi+1 ) f (x, y ), Xi f (x), Xi Xi-1 = xi-1 f (xi |xi-1 ). f (x1 , x2 , ..., xn ) (X1 , X2 , ..., Xn ) f (x1 , x2 , ..., xn ) = f (x1 )f (x2 |x1 )f (x3 |x2 )...f (xn |xn-1 ). X1 , X2 , ... . X1 f (x), X2 f (x|X1 ), X1 , X3 f (x|X2 ), X2 .. 1. X1 , X2 , ...., f (x) = (x), f (x, y ) = (x, y ), || < 1, f (xi |x
i-1

) = (1 - 2 )-

1/2

((xi - x

i-1

)/ 1 - 2 ).

Y1 , Y2 , ..., EY1 = 0, EY12 = 1, X1 = Y1 , X2 = Y1 + ...... Xk =
k-1



1 - 2 Y2 ,

Y1 +

k-2



1 - 2 Y2 + ... + 1 - 2 Y 14

k -1

+



1 - 2 Yk ,


...... 2. X1 , X2 , ... f (x) = C e-x /(1 + x), f (x, y ) = C e-x-y-xy , x, y > 0, 0, C - , f (xi |xi-1 ) = (1 + xi-1 ) exp(-xi (1 + xi-1 )). Y1 f (x) Y2 , Y3 , ..., P (0, 1). X1 = Y1 , Xk = Yk /(1 + X
k -1

), k 2.

, Y1 , ..., Ym - ... .. G(x). .. 1 Gm (x) = m
m

I {Yi < x},
i=1

I {A} - A. .. (m) (m) ( . Y1 Y2 ... Ymm) , Gm (x) =
0 1
k m

, x Y1 , (m) , Yk < x Y ( , x > Ymm) .

(m)

(m) k+1

,

k = 1, ..., m - 1,

Gm (x) G(x) : Gm (x) - G(x) m . , - , m
-
sup

|Gm (x) - G(x)| 0 1.

g (x) = G (x), g (x). (n) ( . [Y1 , Ymn) ] N (n) (n) h. - [Y1 + k h, Y1 + (k + 1)h) k /(mh), h- , k Y1 , ..., Ym , (n) (n) (n) (n) [Y1 + k h, Y1 + (k + 1)h), k = m I {Yi [Y1 + k h, Y1 + (k + 1)h)}, k = i=1 0, 1, ..., N - 1. Y1 ..., Ym . , .. .. , gm (x) g (x) (n) (n) x [Y1 + k h, Y1 + (k + 1)h) gm (x) = k /(mh). 15


h 0, mh gm (x) g (x). h . . : ) - g 2 (x)dx < ; ) 0 < - (g (x))2 dx < ; m inf E(
h -

(gm (x) - g (x))2 dx) (36

-

(g (x))2 dx)1/3 /(4m

2/3

).

h (6/(
-

(g (x))2 dx))1/3 m

-1/3

.

g (x)- , h 2, 8m
-1/3

.

. , A p, 0 < p < 1. A k p= ^ m p, p - p m . ^ k A m . :
n P

X=
i=1 (n) (n)

Xi /n,

S2 =

n i=1

(Xi - X )2 /(n - 1),

( X1 X2 ... Xnn) , (n) (n) X1 , ..., Xn , Xk - k - , X1 = ( min1in {Xi }, Xnn) = max1in {Xi },

(x) =

x -

(u)du,
x - y

1 (x) = e-x 2
-

2

/2

,

(x, y ) = (x, y ) =

(u, v )dudv , .

1 1 exp - x2 + y 2 - 2xy 2 2(1 - 2 ) 2 1 - 16


1. X1 , ..., Xn - .. F (x), an bn - . : P ( 1) F = . , Xnn) - an - 0 n , : > 0 n(1 - F (an + )) 0, n(1 - F (an - )) . 2) F = -. , X1 - bn - 0 n , : > 0 nF (bn - ) 0, nF (bn + ) . 1.1. X1 , ..., Xn N (0, 1). , an 1 - (an ) = n an 2 ln n - ln ln n + ln 4 . 2 2 ln n
P (n) P

1 , n

( 1, , Xnn) - an - 0 n . X (n) ( n = Xnn) - an n = n , n > 1. an .

1 - (x) x . 1.2. X1 , ..., X ..
n

(x) , x

1 - e- 0
x

F (x) =

, ,

x > 0, x 0,

> 0.

, n
( Xnn) - (ln n)1/ - 0 > 1, P



( Xnn) P (n) - 1 X[n-n] - (ln n) (ln n)1/

1/

- 0

P

> 0.

17


[z ] z . ( Xnn) ( n = , n = Xnn) - (ln n)1/ , (ln n)1/ 1 (n) n = X[n-n] - (ln n)1/ , = , n > 1. 2 2. X1 , ..., Xn .. F (x), x 1 - F (x) ax exp(-bx ), a, b, > 0. an > 0 bn , n Tn = (X .. F (x) = 1 - e-x , x > 0, T .
n
3

(n) n

- bn )/an



3. X1 , ..., Xn .. F (x), x 1 - F (x) ax- , a, > 0.
( an > 0 bn , Tn = (Xnn) - bn )/an n ..

F (x) =

1 1 arctan x + , 2
n

- < x < ,

T .



4. 2, 1, , n P ( Xnn) - an - 0 an . 3, 1, , an , n
( Xnn) - an - 0. P

F (x) = 1 - e-
xx

, x > 0,

( n = Xnn) - an .

5. X1 , ..., Xn .. F (x), 18


x 1 - F (x) ax exp(-bx ), y = yn (t) 1 - F (y ) = 1) ,
n ( lim P{Xnn) < yn (t)} = e-t . 2 2) EX1 = 0, EX1 = 1. , an > 0 bn , ( lim P{(Xnn) - bn )/an < x} = exp(-e-x ), n lim bn /(an n) = 0, yn (t) = an x + bn , n

a, b, > 0,

t , n

t > 0.

,
n

lim P{(X

(n) n

- X )/S < yn (t)} = e-t .

1 (1 - 6x)e 6x , F (x) = 2 1 - 6x 1 - 2 (1 + 6x)e ,

x 0, x > 0,
(n) n

( P{Xnn) < yn (t)} P{(X e-t .

- X )/S < yn (t)}

6. X1 , ..., Xn .. F (x), x 1 - F (x) ax- , y = yn (t) 1 - F (y ) = 1) ,
n ( lim P{Xnn) < yn (t)} = e-t .

a, > 0,

t , n

t > 0.

2) Zn = Zn (X1 , ..., Xn ) Tn = Tn (X1 , ..., Xn ) , nZn n(Tn - 1) n . , an > 0 ,
n ( lim P{Xnn) < an x} =

0 , exp(-x- ) ,

x 0, x > 0,

an x = yn (t) x > 0, ,
n ( lim P{(Xnn) - Zn )/Tn < yn (t)} = e-t .

F (x) = 1 1 arctan x + , - < x < , Zn = X 2 19
(n) [n/2]+1

1 (n) (n) , Tn = (X[3n/4] - X[n/4]+1 ), 2


( P{Xnn) < yn (t)} P{(X e-t . [z ] z .

(n) n

- Zn )/Tn < yn (t)}

7. X1 , ..., Xn U (- 3, 3). an > 0 bn .. H (x),
n ( lim P{(Xnn) - bn )/an < x} = H (x).

, n
( ((Xnn) - X )/S - bn )/an

, ( lim P{ n((Xnn) - X )/S - 3) < x} = (x/ ),
n

> 0.

Vn = (X Tn =
(n) n

- bn )/an 3),

( n((Xnn) - X )/S -

(1/ )(x/ ). 8. X1 , ..., Xn .. F (x), F (x) = 1 - e-x , x > 0, Zn = X1 ,
(n)

Kn = X - X1 .

(n)

an > 0, bn .. H (x),
n ( lim P{(Xnn) - bn )/an < x} = H (x).

,
n ( lim P{((Xnn) - Zn )/Kn - bn )/an < x} = H (x).

Tn = ( ((Xnn) - Zn )/Kn - bn )/an . . , n nZ P (0, 1), n(Kn - 1) n . 9. X1 , ..., Xn .. F (x), F (x) = 1 - (1 - x - µ)3 1 - < µ < , > 0. 20
0

, , ,

µ x -, µ - < x < x 1-µ ,

1-µ

,


1) µ = µ0 = 0 , EX1 = 0,
2 EX1 = 1;

2) an > 0, bn .. H (x),
n ( lim P{(Xnn) - bn )/an < x} = H (x);

3) ,
n ( lim P{((Xnn) - X )/S - bn )/an < x} = H (x),

µ µ0 0 .1). Tn = ( ((Xnn) - X )/S - bn )/an . 10. X1 , ..., Xn .. F (x), F (x) =
0

, 1 - exp(-x/(1 - x)) , 1 ,

x 0, 0 < x < 1, x 1.

an > 0, bn , cn > 0, dn .. H (x) L(x), ( lim P{(Xnn) - bn )/an < x} = H (x),
n n

lim P{(X1 - dn )/cn < x} = L(x).

(n)


( Tn = (Xnn) - bn )/an Dn = (X1 - dn )/cn (n)

. 11. X1 , ..., Xn .. F (x), x F (x) 1 - C (ln x) -1 x an > 0,
n -

,

C , , > 0.

lim P{X

(n) n

< an x} = exp(-x

-

),

x > 0.

2 ln 2 , F (x) = 1 - (x + 1) ln(x + 1) X .
(n) n

x > 1,

/an -

21


12. X1 , ..., Xn .. F (x), F (x) 1 - k (F - x) ,

F x

F

k , > 0.

( an , n an (Xnn) - F ) . f (x) = F (x) = 6x(1 - x), 0 < ( x < 1, a-1 (Xnn) - 1) n .

13. Y1 , ..., Yn .. F (x), k 1. an > 0 bn ,
n

Xk = max(Yk , Y

k+1

),

lim n(1 - F (bn + an x)) = exp(-x).

, F (x) = 1-2 exp(-x)+exp(-x 2), ( x > 0, (Xnn) - bn )/an . {Xn } {Yn }, n = 1, 2, ... 14. X1 , ..., Xn - .. F (x), 0 = F -1 (p0 ). , k , n , p = k /(n + 1) p0 , 0 < p0 < 1, k - (n) Xk n(Xk - ) - =F
(n) d n ( lim P{Xnn) < bn + an x} = exp(- exp(-x)/2).

p0 (1 - p0 )Y /f (0 ),
-1

Y N (0, 1), med X1 , ..., Xn , med
1in

(p),

f (x) = F (x).

{Xi } , ... Xk+1 (n) (Xk + X
(n)

1in

{Xi } =

(n) k+1

, )/2 ,

n = 2k + 1, n = 2k .

n med1in {Xi }, , n med
1in

{Xi } X[

(n) n/2]+1

,

[z ] z . , , . , . 22


15. X1 , ..., Xn .. F (x), , +x , 2 2 F (x) = 1 +x , 2 1 , ,
n

0 1

x -1, -1 < x 0, 1 0 < x 2, x > 1. 2 (2x) , (x) , x 0, x < 0,

lim P{ nX

(n) [n/2]+1

< x} =

(n) Tn = nX[n/2]+1 . 16. X1 , ..., Xn .. F (x),
0 1 1
2 1 2

F (x) = , lim P{n1/4 X[ n

+x 2 +x

2

, , , ,

x -1, -1 < x 0, 1 0 < x 2 , 1 x > 2 .

(n) n/2]+1

< x} =

0 , 2 (2x ) ,

x 0, x > 0.

.. (n) Tn = n1/4 X[n/2]+1 .. 17. X1 , ..., Xn .. F (x),
0 1 +x 2 F (x) = 2 x 1 2+ 2 1

, , , ,

x -1, -1 < x 0, 0 < x 1, x > 1.

,
n

lim P{ nX

(n) [n/2]+1

< x} =

(x) , 1 ,

x 0, x > 0.

.. (n) Tn = nX[n/2]+1 .. n = X
(n) [n/2]+1

.
0

18. X1 , ..., Xn .. F (x), F (x) = , (1 + x)/2 , -1/x (1 + e )/2 , 23 x -1, -1 < x 0, x > 0.


n = , lim P{
(n) (X[n/2]+1

1 2 n + n n - n , n = , a n = , bn = . ln( n/2) n 2 2 , , - bn )/an < x} = 1,
1 2

0

n

x -1, -1 < x 1, x > 1.

.. (n) Tn = (X[n/2]+1 - bn )/an .. 19. X1 , ..., X U (0, 1). ,
n


(n) -1/2

1) nlim P{(X[n] - n 2)X
(n) n

)n3
P

/4

< x} = (x);

- 1,

P

X

(n) [ n]

- 0 n .
-1/2

Tn = (X[n] - n
(n)

)n

3/4

. (n) ( n = Xnn) n = X[n] . [z ] z . 20. X1 , ..., Xn - .. F (x), 0 = F -1 (p0 ), f (x) = F (x), = F -1 (p). Tn = nf ( )(Xi i, n , p=
(n) +k

- Xi ).

(n)

i p0 , 0 < p0 < 1, k . n+1 , Tn - (1, k ). Tn . F (x) = 1 - exp(-2x2 ), x > 0, i = [n/3], k = 2. . In = P{Tn < x} = B= (v - u) < x/(nf ( )), u < v,
(n) B

f (u, v )dudv ,

f (u, v )- Xi In 24

X

(n) i+k

.