Курс математики для технических высших учебных заведений. Часть 4. Теория вероятностей и математическая статистика: Учебное пособие [Н. А. Берков] (pdf) читать онлайн

Лауреат второго Всероссийского конкурса НМС по математике
Министерства образования и науки РФ «Лучшее учебное издание по математике
в номинации «Математика в технических вузах»

Н. А. БЕРКОВ, А. И. МАРТЫНЕНКО,
Е. А. ПУШКАРЬ, О. Е. ШИШАНИН

КУРС МАТЕМАТИКИ
ДЛЯ ТЕХНИЧЕСКИХ
ВЫСШИХ УЧЕБНЫХ
ЗАВЕДЕНИЙ
Часть 4
Теория вероятностей
и математическая статистика
Под редакцией
В. Б. Миносцева, Е. А. Пушкаря
Издание второе, исправленное

ДОПУЩЕНО
НМС по математике Министерства образования и науки РФ
в качестве учебного пособия для студентов вузов, обучающихся
по инженерно&техническим специальностям

•САНКТПЕТЕРБУРГ•МОСКВА•КРАСНОДАР•
•2013•

ББК 22.1я73
К 93
Берков Н. А., Мартыненко А. И., Пушкарь Е. А.,
Шишанин О. Е.
К 93
Курс математики для технических высших учебных
заведений. Часть 4. Теория вероятностей и математиче"
ская статистика: Учебное пособие / Под ред. В. Б. Минос"
цева, Е. А. Пушкаря. — 2"е изд., испр. — СПб.:
Издательство «Лань», 2013. — 304 с.: ил. — (Учебники
для вузов. Специальная литература).
ISBN 9785811415618
Учебное пособие соответствует Государственному образовательному
стандарту, включает в себя лекции и практические занятия. Четвертая
часть пособия содержит 17 лекций и 17 практических занятий по разделу
«Теория вероятностей и математическая статистика».
Пособие предназначено для студентов технических, физико"матема"
тических и экономических направлений.

ББК 22.1я73

Рецензенты:
À. Â. ÑÅÒÓÕÀ — äîêòîð ôèçèêî-ìàòåìàòè÷åñêèõ íàóê, ïðîôåññîð,
÷ëåí ÍÌÑ ïî ìàòåìàòèêå Ìèíèñòåðñòâà îáðàçîâàíèÿ è íàóêè ÐÔ;
À. À. ÏÓÍÒÓÑ — ïðîôåññîð ôàêóëüòåòà ïðèêëàäíîé ìàòåìàòèêè è
ôèçèêè ÌÀÈ, ÷ëåí ÍÌÑ ïî ìàòåìàòèêå Ìèíèñòåðñòâà îáðàçîâàíèÿ è
íàóêè ÐÔ; À. Â. ÍÀÓÌΠ— äîêòîð ôèçèêî-ìàòåìàòè÷åñêèõ íàóê,
äîöåíò êàôåäðû òåîðèè âåðîÿòíîñòåé ÌÀÈ; À. Á. ÁÓÄÀÊ — äîöåíò,
çàì. ïðåäñåäàòåëÿ îòäåëåíèÿ ó÷åáíèêîâ è ó÷åáíûõ ïîñîáèé ÍÌÑ ïî
ìàòåìàòèêå
Ìèíèñòåðñòâà
îáðàçîâàíèÿ
è
íàóêè
ÐÔ;
Ó. Ã. ÏÈÐÓÌΠ— ïðîôåññîð, çàâ. êàôåäðîé âû÷èñëèòåëüíîé ìàòåìàòèêè è ïðîãðàììèðîâàíèÿ ÌÀÈ (Òåõíè÷åñêèé óíèâåðñèòåò), ÷ëåíêîððåñïîíäåíò ÐÀÍ, çàñëóæåííûé äåÿòåëü íàóêè ÐÔ.

Обложка
Е. А. ВЛАСОВА

© Издательство «Лань», 2013
© Коллектив авторов, 2013
© Издательство «Лань»,
художественное оформление, 2013

 

              
  

        
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lim P {a  ξ < a+Δx} = lim

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F (a+Δx)−F (a) = F (a)−F (a) = 0.

lim P {a  ξ < a + Δx} = P {a  ξ  a} = P {ξ = a} 

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x1

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ϕ(x)dx = P {−∞ < ξ < +∞} = 1.

−∞

"  # $  *           
   [x1; x2)   /   ϕ(x)         )
  0       .!
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 ϕ(x)       ξ    
      (x; x + Δx)   
  



 

     





 

3 1
1 3 3
dt = t = .
P {0 < ξ < 3} =
4
4 0 4
0
1
   C = 4 ; P {0 < ξ < 3} = 34

    





        
   






              
              
     ξ          [a; b]
    n  !     Δx1, Δx2, . . . , Δxn
     !  "  Ci (i = 1, 2, . . . , n)
#         ξ        
Ci     pi = ϕ(Ci)Δxi $ 
    i%  %
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Ci ϕ(Ci)Δxi .

i

(        "    ) 
 !         

b

xϕ(x)dx

a

 *+       
    ξ        ϕ(x) 
+∞
M(ξ) =
xϕ(x)dx.
−∞

$*+,&

  *+,         
 f (x)        ξ    

+∞
M(f (ξ)) =
f (x) · ϕ(x)dx,
−∞

  ϕ(x)        ξ

        



   
      
           


2
D(ξ) = M ξ − M(ξ) .

          
    !"     # $ 
D(ξ) =

+∞


2
x − M(ξ) ϕ(x)dx.

% !&'

−∞

   
        ( 
  #  )(     *   !
   &            
     (   +   ! ,!-   
    ! . (  +      *  
      (   
   (ξ; ζ)  (      
             
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F (x; y) = P {ξ < x; ζ < y}.

+#   $ )   F (x; y) + 
   ) &! /      (     $ )  
 F (x; y)      (ξ; ζ)(  
  $ )     # Fξ (x)  Fζ (y)
% ! ! &!-'! + ( +# ( !
  !0      ξ ζ  
            F (x; y)   
    (ξ; ζ)         
 
F (x; y) = Fξ (x) · Fζ (y).

1  +           n  
    $ )  
F (x1 ; x2 ; . . . ; xn ) = Fξ (x1 ) · Fξ (x2 ) · . . . · Fξ (xn ), 
1

2

n

F (x1 ; x2 ; . . . ; xn ) = P (ξ1 < x1 ; ξ2 < x2 ; . . . ; ξn < xn ).

.      (      +)  
( *        ! ,!-!







   

              
        
               !
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 *
+∞

2
D(ξ) =
x2 ϕ(x)dx − M(ξ) .

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,                  
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σ(ξ) = D(ξ).
#+)&
 +%         
            
  
. /  * 0 "  #+%&    ϕ(x)  
    

x ∈ [0; 4] M(ξ) =

4
0

1
x2
xdx =
4
8

4

= 2

0

0 "  #+&    ϕ(x)    
   x ∈ [0; 4]   M(X) = 2   -   
+1  *
4
D(ξ) =
0

1
x3
x · dx − (2)2 =
4
12

4

2

0 "  #+)&  

−4=
0

σ(x)* σ(ξ) =



4
16
−4= .
3
3

4
2
= √ ≈ 1,155.
3
3

2 * M(ξ) = 2, D(ξ) = 34 ≈ 1,333, σ(ξ) ≈ 1,155
   +         
              !
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   ξ      
               
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P {ξ = 0} = 1 −

5
1
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6
6

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6

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ξ
p

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x4 = 3.

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C10
2
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3
C15
91

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1
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C10
20
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3
C15
91

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2
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· C51
45
= ,
3
C15
91

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3
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C10
24
= .
3
C15
91

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ξ

3
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0 + 1, 0 + 2, 0 + 3, 1 + 2, 1 + 3, 2 + 3.

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ξ 
) # * +
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!   !  ξ " !             
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 1   $ 2        "
 p = 0,7, q = 0,3. P {ξ = 0}      "    
     3 " $$        $ / 1 
   "  '"  -"4$  " P {ξ = 0} = 0,7. .
     " P {ξ = 1}       (  & 0
        3         $
. "
P {ξ = 1} = p · q = 0,7 · 0,3 = 0,21.

5  " "
2

P {ξ = 2} = p · q = 0,063,


P {ξ = 3} = p · q 3 = 0,0189,

P {ξ = 4} = q 4 = 0,0081.

% & " &6    
ξ

)
#
*
p -"4--- -")-- -"-!#- -"-7, -"--7
 ,-$! $ !        

    %         p  
            %   
%           !   %   
       









   

    

      
      
              
           
1 − p = q   P {ξ = 1} = q  !        
          "  
 P {ξ = 2} = p · q. #
P {ξ = 3} = p2 · q, ..., P {ξ = m} = pm−1 · q, ...

$    %    
ξ & ' ( 
p q pq p2 q 
) *  +  *     
p  q       
              +  

 ,-.     ξ        
       
ξ     
p     
        ! "         ! 
  !         #$
      /       
+      +     
M(ξ) =

n


x i · pi

i=1



M(ξ) = 1,2 · 0,2 + 1,6 · 0,4 + 2,3 · 0,1 + 3,2 · 0,2 + 4,5 · 0,1 = 2,2.

0      1 

D(ξ) = M(ξ 2 ) − [M(ξ)]2 .

2 + M(ξ 2 )

M(ξ 2 ) = 1,22 · 0,2 + 1,62 · 0,4 + 2,32 · 0,1 + 3,22 · 0,2 + 4,52 · 0,1 =
= 1,44 · 0,2 + 2,56 · 0,4 + 5,29 · 0,1 + 10,24 · 0,2 + 20,25 · 0,1 = 5,914.

"     

D(ξ) = 5,914 − 2,22 = 1,074.

            

 
   



          

σ(ξ) =

D(ξ) =

1,074 ≈ 1,036.

   M(ξ) = 2,2; D(ξ) = 1,074; σ(ξ) ≈ 1,036
         

 ζ   

ζ = 4ξ + 3η,

M(ξ) = 11,

M(η) = 8.

                  
! 
      !    " ! 

  #               $
  #%
M(ζ) = M(4ξ + 3η) = M(4ξ) + M(3η) =
= 4 · M(ξ) + 3 · M(η) = 4 · 11 + 3 · 8 = 68.

   M(ξ) = 68
    ξ  η     

     ζ = 5ξ − 6η   D(ξ) = 3,

D(η) = 2.

       &  $ %
  #   
 !   ! 
       $!
  #              %   #
$   %   
D(ζ) = D(5ξ − 6η) = D(5ξ) + D(6η) =

  

= 25 · D(ξ) + 36 · D(η) = 25 · 3 + 36 · 2 = 147.
D(ξ) = 147

 '     ξ  
    x1    p1 x2 = 5   
p2 = 0,4  x3 = 4    p3 = 0,5  x1  p1   
    M(ξ) = 6
      (   )*      #  
 #     '% p1 = 1−p2 −p3 = 0,1    ##   
  % 

x1 = 20

   x1 = 20;

M(ξ) = x1 p1 + x2 p2 + x3 p3
  6 = x1 · 0,1 + 5 · 0,4 + 4 · 0,5 

p1 = 0,1

) 









   

    

    
   
    x1 = 2, x2 = 5, x3 = 6     
              M(ξ) = 4,4;
M(ξ 2 ) = 22     p1 , p2 , p3     
  x1, x2, x3
 
        
     
⎧
⎨ p1 + p2 + p3 = 1
2 · p1 + 5 · p2 + 6 · p3 = 4,4
⎩ 2
2 · p1 + 52 · p2 + 62 · p3 = 22.
     !  " #

p1 = 0,3 p2 = 0,4,

p3 = 0,3.

$ p1 = 0,3; p2 = 0,4; p3 = 0,3

 %
   n     !   " 
  #
           ! $
$  "   "      %  &   $ 

 !      M(ξ) D(ξ)   
  ξ 
      &# '!  (     ) 
)"!  p1 = 1/n &# '!  (  
)  '!   )"! (   # 
n−1
1
1
·
= .
p2 =
n
n−1
n
*  (  # '!   + ! 
1
1
n−1 n−2
·
·
=
p3 =
n
n−1 n−2
n
 ,"  )! (  " ( #

ξ 
%
-  .
p /. /. /.  /.
0#  # 

( #   
n(n + 1)
,
1 + 2 + 3 + ... + n =
2
n(n + 1)(2n + 1)
.
12 + 22 + 32 + ... + n2 =
6

            


M(ξ) =

n

i=1

M(ξ 2 ) =

n


x2i pi =

i=1

x i pi =



n+1
1
(1 + 2 + 3 + ... + n) =
.
n
2

1 2
(n + 1)(2n + 1)
(1 + 22 + 32 + ... + n2 ) =
,
n
6

(n + 1)(2n + 1)
n + 1 2 n2 − 1
−(
) =
.
6
2
12
n2 − 1
n+1
; D(ξ) =
M(ξ) =
2
12
D(ξ) =

 

            
               
     
        ξi     
   !"#   $%      ξ   
 ! !"   # &#  ξ = ξ1 + ξ2 + ξ3 
M(ξ) = M(ξ1 + ξ2 + ξ3 ) = M(ξ1 ) + M(ξ2 ) + M(ξ3 ).

'  "( !  ! ξi  ! 
D(ξ) = D(ξ1 + ξ2 + ξ3 ) = D(ξ1 ) + D(ξ2 ) + D(ξ3 ).

)!  ! ξi    "      %
 (  ! ! #  ) * +
"
M(ξ) = 3 · M(ξ1 ),

D(ξ) = 3 · D(ξ1 ).

, & !" &    ! "+ 
"   & & ξ1   
ξ1 
-  . / 0
p 10 10 10 10 10 10
     2


7
1
M(ξ1 ) = (1 + 2 + 3 + 4 + 5 + 6) = ,
6
2
M(ξ) = 3 · (7/2) = 21/2.

3   2 &     !
1
91
M(ξ12 ) = (12 + 22 + 32 + 42 + 52 + 62 ) = ,
6
6

  

  



              
   ξ             x1
    p1 = 0,1  x2  x1 < x2   
  M(ξ)  
 D(ξ)    M(ξ) = 3,9, D(ξ) = 0,09
              
    −2, 1, 4         
  M(ξ) = 1,9,   M(ξ 2 ) = 7,3     
p1 , p2 , p3              
   
  ! "    #       
             
  $ #  $

  

  

 
            
             
    
    

 









   n   !" ! #     $ p  
   %! A   & !  '!   ( )% 
  ξ *   #$  +  $     #  %! A
 n ! "  ,     

P {ξ = m} = P (m) = Cnm pm q n−m ,  q = 1 − p, m = 0,1, . . . , n. ' (

     %           
        &'     & 
 
ξ 


...
k
... n
p q n npq n−1 Cn2 p2 q n−2    Cnk pk q n−k    pk

 p



          



n





   

         
        
(p + q)n =

n

m=0

Cnm pm q n−m .




    (p + q)n = p + (1 − p) n = 1
             
       ξ   n !    "
   ξi (i = 1, 2, . . . , n)   ξi = 1   i"  "
      A  ξi = 0   i"      
  Ā #   
ξ = ξ1 + . . . + ξn .
$%&'
(    )  !      A
n   *       
+   !     ξi !,  "
)
ξi 
p p 1−p

#   
D(ξi ) =

M(ξi2 )

M(ξi ) = 1 · p + 0 · (1 − p) = p,

2
− M(ξi ) = 12 · p + 02 · (1 − p) − p2 = p − p2 =

= p(1 − p) = p · q.

.!              
    !    /  $%&'
M(ξ) = M(ξ1 ) + . . . + M(ξn ) = n · p,
D(ξ) = D(ξ1 ) + . . . + D(ξn ) = n · p · q.

0     ,     ξ "
 
M(ξ) = np; D(ξ) = npq.
$%1'
 %    4       

         
 

  

  



      
      
  n = 4 p = 0,5
P4 (0) = 0,54 ≈ 0,0625;

P4 (1) = 4 · 0,5 · 0,53 = 0,25;

P4 (2) = C42 · 0,52 · 0,52 ≈ 0,375;
P4 (3) = p4 (1) = 0,25;

P4 (4) = p4 (0) ≈ 0,0625.

         
ξ
p


!
"
#
$!% !% "&% !% $!%

'   () *"+ ,  
M(ξ) = n · p = 4 · 0,5 = 2;

-  M(ξ) = 2,

D(ξ) = nqp = 4 · 0,5 · 0,5 = 1.

D(ξ) = 1*

 

  

'. ,  n → ∞, p → 0  /
01    /. * 2)*# . Pn (m) 3
4      5.6   '

np → a*

P {ξ = m} =

am −a
e ,
m!

() *#+

a > 0.

 ) *!          
            
       
7    '   
ξ



p e−a ae−a

!

...

m

...

a2 −a
e
2!

***

am −a
e
m!

***

  '      
84  /   ,    *

a*

  

  



ξ



...
p p p(1 − p) p(1 − p)2 p(1 − p)3 . . .


             
        ! "#
$     % ξ        &  A
    '
     &  A 
p = p(A)   (    # ) (
 !  %

M(ξ) =

1−p
,
p

D(ξ) =

1−p
.
p2

*       +    
     # ,           
      #

 



 

  

 -#   

    
      [a; b]     
                 
 

C  x ∈ [a; b],
ϕ(x) =
/ [a; b].
0  x ∈



. !  /         # - #"   0
   C #

+∞
b
ϕ(x)dx = 1 =⇒
Cdx = 1 =⇒
−∞

=⇒ Cx

b
a

a

= 1 =⇒ C(b − a) = 1 =⇒ C =

1
.
b−a

.    !      0   [a; b]   

   1 %
 1
 x ∈ [a; b],
ϕ(x) =
-#/"
b−a
0
 x ∈
/ [a; b].





   


   

         
x
F (x) =

ϕ(t)dt.
−∞





x
F (x) =

xb

a

b
0dt +

−∞

a

x−a
1
dt =
;
b−a
b−a
1
dt +
b−a

x
0dt =
b

b−a
= 1.
b−a


        
    [a; b]      
⎧
⎪
⎨ 0x − a
F (x) =
⎪
⎩ b−a
1





x < a,

 !"

a  x  b,
b < x.

#            a 
$            %&'()
         
b

ϕ(x)

F(x)

1

a

b

   


x

a

b

x

   

  

  



         
M(ξ) =

+∞
b
xϕ(x)dx =
−∞

D(ξ) =

x
1
x2
dx =
·
b−a
b−a 2

a

a
b

=

a+b
(b − a)2
=
2 · (b − a)
2

+∞
b
x2
(a + b)2
(a + b)2
=
dx −
=
x2 ϕ(x)dx −
4
b−a
4

−∞

a

b

x3
(b − a)3
(a + b)2
(a + b)2
1
=
·
=
−
=
−
b−a 3 a
4
3 · (b − a)
4
(b − a)2
4 · (a2 + ab + b2 ) − 3 · (a + b)2
=
.
=
12
12

         [a; b]    
   
M(ξ) =

a+b
;
2

D(ξ) =

(b − a)2
.
12



 !

   P {x  ξ

a  x < x + Δx  b

 

< x + Δx} 
  
             

x+Δx


P {x  ξ < x+Δx} =

x+Δx


ϕ(x)dt =
x

x

x + Δx − x
Δx
1
dt =
=
.
b−a
b−a
b−a

!    "        x    #$
   # !   [a; b]   !   %  # ! Δx
&  '(         )    * $
     +  ,  ! [a; b]     $
  -     "       
  %    ! M(ξ) = a+b

2
 "            !

  .  /      0 !1 .   $
    !  #     .

[0; 4]

  

  



ϕ(x)

F(x)
λ
1

x

   
 

x

     

+∞
+∞
+∞
−λx
M(ξ) =
xϕ(x)dx =
xλe dt = λ
xe−λx dx =

=

−∞

u=x
dv = e−λx



0

0



du = dx
1
= −xe−λx
v = − e−λx
λ
∞
1
1
= − e−λx = .
λ
λ
0

∞

+
0

+∞
e−λx dx =
0

      

+∞
+∞
1
1
1
2
D(ξ) =
x ϕ(x)dx − 2 =
x2 λe−λx dx − 2 = 2 .
λ
λ
λ
−∞

0

      
 

          

M(ξ) =

1
;
λ

D(ξ) =

1
.
λ2

 !

  "  
     
        ξ , ξ , ξ , . . .    
         λ    
            
    !"         
# #         t $$ $    
            a = λt%
1

2

3

              
     F (x)    
              ! !
⎧
0,
 x  1,
⎪
⎪
⎪
⎪
1/6,  1 < x  2,
⎪
⎪
⎪
⎨ 1/3,  2 < x  3,
1/2,  3 < x  4,
F (x) =
⎪
⎪
⎪
⎪ 2/3,  4 < x  5,
⎪
⎪
⎪
⎩ 5/6,  5 < x  6,
1,
 6 < x.


  

F(x)
1
5/6
4/6
3/6
2/6
1/6
1

2

3

4

5

6

x

   
" #  $     $  %&'!(  )
 $ $         !
 &!     ξ       



⎧
⎨ 0  x  −2,
(x + 2)2  − 2 < x  −1,
F (x) =
⎩
1  x > 1.

          
 ξ        
   (−3/2, −1).
*  +   , -  #  .   ξ     /)
     (−3/2, −1)   0 /    )
$      ,

3
P (−3/2 < ξ < −1) = F (−1) − F (−3/2) = (−1 + 2)2 − (−3/2 + 2)2 = .
4
1, 0,75!







            

  
    ξ   
     ϕ(x) = cos x    (0,π/2)  
  ϕ(x) = 0.       ξ  
    (π/4, π/3)









 b
P (a < ξ < b) =
ϕ(x)dx.
a

   a = π/4, b = π/3, ϕ(x) = cos x  
  
√
√
 π/3
3− 2
π
π
π/3
≈ 0,159.
P( < ξ < ) =
cos xdx = sin x π/4 =
4
3
2
π/4
 ≈ 0,159.



  !"    



    ξ



⎧
⎨ 0  x  1,
(x − 1)2  1 < x  2,
F (x) =
⎩
1  x > 2.

      

ϕ(x)

 
         !
 "#
  
⎧
⎨ 0  x  1,
2(x − 1)  1 < x  2,
ϕ(x) = F  (x) =
⎩
0  x > 2.



  



$ ξ # 
   
ϕ(x)    $%
⎧
0,   x  0,
⎪
⎪
⎨
3
ϕ(x) =
(4x − x2 ),   0 < x  4,
⎪
32
⎪
⎩
0,   x > 4.

% &    
   F (x)

ξ

 

  

(1; 2)

$&

!"

              

       


2

P (1 < ξ < 2) =






3
3
x3
2
2
(4x − x )dx =
2x −
32
32
3

2

ϕ(x)dx =

1

=



1

11
≈ 0,344.
32

   
     

F (x) =

=
1

       

F (x)


2

x

ϕ(t)dt.
−∞

 −∞ < x  0!


F (x) =

x

0dt = 0;

−∞

 0 < x  4!



F (x) =

!  ! x > 4!


F (x) =

0

0


ϕ(t)dt +

−∞

x

ϕ(t)dt =

0



4

0dt +

−∞

3
(4t − t2 )dt +
32

0

6x2 − x3
;
32


x

0dt = 1.

0

" ⎧ P (1 < ξ < 2) = 11/32 ≈ 0,344!
⎨ 0,  x  0,
6x −x
F (x) =
,  0 < x  4,
⎩ 32
1,  x > 4.
 #$%&     ξ
      
        ϕ(x) = a/(1 + x2)    
   a        F (x)
      ' 
2

3



+∞

ϕ(x)dx = 1,

−∞



+∞
−∞

a
dx = a · arctg x
1 + x2

+∞
−∞

=a

 π 
− −
= aπ = 1.
2
2

π







            

   a = 1/π        
ϕ(x) = 1/π(1 + x2 ).     
 x
 x
dt
1 1
= + arctg x.
F (x) =
ϕ(t)dt =
2
2 π
−∞
−∞ π(1 + t )

  a = 1/π ≈ 0,318 F (x) = 0,5 + arctg(x)/π
      ξ     

   ϕ(x) = x/8    (0, 4)      ϕ(x) = 0.
      
      ξ 

   

      !  "! #$


M(ξ) =

+∞



xϕ(x)dx.

−∞

0

 ϕ(x) = x/8   
% 

1
M(ξ) =
8



4

x2 dx =

0



b

D(ξ) =



a

D(ξ) =

1
8



4
0

x3 dx −

4

xϕ(x)dx =

1 x3
·
8 3

4

=
0

8
≈ 2,667.
3

x2 ϕ(x)dx − M 2 (ξ)

 2
8
1 x4
= ·
3
8 4

4

8
64
= ≈ 0,889.
9
9

−
0

  M(ξ) = 8/3 ≈ 2,667, D(ξ) = 8/9 ≈ 0,889.
 &       

  ξ !     "# $ 
    
  

%   

   
'( )* ϕ(x)             
  
⎧
⎨ x + 1  x ∈ (−1, 0),
−x + 1  x ∈ (0, 1),
ϕ(x) =
⎩ 0  x  −1, x  1.
  ϕ(x) (    (−1, 1)    +
 ,-   


M(ξ) =

+∞

−∞



xϕ(x)dx =

0

−1



x(x + 1)dx +
0

1

x(−x + 1)dx = 0.









   

     
 
  

            

   ϕ          


⎧
< 0, 
⎨ 0  x 
x2
x
ϕ(x) = F  (x) =
⎩ 2 · exp − 2
σ
2σ



  

   

      
     

M0 (ξ)

  

ϕ (x) =

 x  0.

 


x2
1
x2
·
1
−
exp
−
σ2
2σ 2
σ2

   x  0 ϕ(x) = 0    x = σ.     ϕ (x)
           x = σ ϕ

 !    "       M0 (ξ) = σ
#   Me(ξ)    x $
     F (x) = 1/2. %  
&

1/2 = 1 − exp(−x2 /2σ 2 ), 1/2 = exp(−x2 /2σ 2 ).
√
√
   x = σ 2 · ln 2  Me (ξ) = σ 2 · ln 2



  

 '(((            

    

ξ
p

   
   !

)  *+   F (x)    , *
 '((- "    ξ       Ox # 

$       F (x) = 1/2 + arctg(x)/π    
%      &
   ξ      
√
'      (0, 3).
 '((. (         &   
 & ξ     (0, 2)    ϕ(x) = Ax3) *%  
   ϕ(x) = 0 +   A     %  ξ
        ,-     (0, 1)  .  
  ,  

 

   





   ξ

   

⎧
⎨ 0  x  0,
1/2 − (1/2) cos 3x
F (x) =
⎩ 1  x > π/3.



   

 

 0 < x  π/3,

       ϕ(x)
           
 ξ     
⎧
⎨ 0  x  0,
ax3  0 < x  4,
F (x) =
⎩
1  x > 1.
      a!       ξ !   
    ξ    (2, 3)
  "         ξ
  

0,  x  0  x > π,
ϕ(x) =
(1/2) sin x,  0 < x  π.
      
#
  $   ξ       
  ϕ(x) = (2/π) · cos2 x  x ∈ (−π/2, π/2)  ϕ(x) = 0  
%           
 ξ 

 




   

         
 

           !  

   

    



"   #$      %!
 %' (# ) *         


!&
   







     






    ξ
  
a σ           

ϕ(x) = √
 



  

(x−a)2
1
e− 2σ2 .
2πσ

    





ξ ∼ N (a; σ)

            a 


! "

+∞
ϕ(x)dx = 1

σ

# 

−∞


+∞
+∞
(x−a)2
1
√
e− 2σ2 dx =
ϕ(x)dx =
2πσ

−∞

+∞
=
−∞

(x−a)
σ

= t =⇒ x = σt + a
dx = σdt

=

−∞

t2
1
1
√
e− 2 σdt = √
2πσ
2π

$  

+∞ 2
t
e− 2 dt.
−∞

  # %  

 

+∞ 2
√
t
e− 2 dt = 2π.

 ! "



−∞

  %     %   &    &   ' 
     
 #

!  "

−∞

 ()  

x
F (x) =
−∞

1
ϕ(t)dt = √
2πσ

+∞
ϕ(x)dx = 1

x
−∞

e−

(t−a)2
2σ 2

dt.

 

   



 
        F (x) 

  
      !   "
   #$
1
F (x) = √
2πσ

x

e−

(t−a)2
2σ 2



dt =

1
=√
2πσ



= z =⇒ z = σz + a
dt = σdz

=

−∞

x−a

σ

(t−a)
σ

2
− z2

e
−∞

1
dz == √
2π

x−a

0

1
e dz + √
2π
−∞


x−a
.
= 0,5 + Φ
σ
2
− z2

0

z2

e− 2 dz =

0

%  &&  & &  & &  & '
   & (    &$
0

)&   
* $

z2

e− 2 dz =

−∞

√
+∞ 2
z
2π
.
e− 2 dz =
2
0

         



x−a
.
σ

+,!
-' &&  *     '  "
' '  
F (x) = 0,5 + Φ


+∞
(x−a)2
1
−
2
2σ
e
M(ξ) =
x√
dx =
2πσ

(x−a)
σ

−∞

1
=√
2π

= t =⇒ x = σσt + a
dx = σt

=

+∞
+∞
+∞ 2
t2
t2
t
σ
a
(σt + a)e− 2 dt = √
te− 2 dt + √
e− 2 dt =
2π
2π

−∞

t2
σ
= − √ e− 2
2π

−∞
+∞

−∞

= 0 + a = a.
−∞

.&&   *& &    σ2 &
+∞
(x−a)2
1
D(ξ) =
x2 √ e− 2σ2 dx − a2 = σ 2 .
2π
−∞







  

        ξ      
   a  σ         
M(ξ) = a; D(ξ) = σ 2 ; σ(ξ) = σ.

!"     " #     $ % 
     &' !"     $   
 $ %  ! 
ϕ(x)

F(x)

1

1

2π σ

0,5

a- σ

a

a+σ

   


x

a

x

   

(       " #     )&
  "            
        %   


x − a
2
P {x1  ξ < x2 } = F (x2 ) − F (x1 ) = 0,5 + Φ
−
σ


x − a
x − a
x − a
1
2
1
=Φ
−Φ
,
− 0,5 + Φ
σ
σ
σ
x − a
x − a
2
1
−Φ
.
P {x1  ξ < x2 } = Φ
σ
σ

*
(          +       "
 Φ(+∞) = 0,5, Φ(−∞) = −0,5  
x − a
2
+ 0,5,
σ
x − a
1
.
P {x  ξ} = 0,5 − Φ
σ

P {ξ < x2 } = Φ



 

  



   

    

 
   
          
                
   σ 

     
     
        
P {|ξ − a| < 2σ} ≈ 0,9544.

    


  

     ξ ∼ N (a; σ)  ζ = kξ + b ∼ N (ka + b; |k| σ)

     !"#     

 

 ζ  k > 0

Fζ (x)



!
x − b"
=
Fζ (x) = P {ζ < x} = P {kξ + b < x} = P ξ <
k

 x−b


−a
x − (ka + σ)
= 0,5 + Φ
.
= 0,5 + Φ k
σ
kσ

$ % &  &   ζ ∼ N (ka + σ; kσ)  k > 0 '
     k < 0
!
x − b"
=
Fζ (x) = P {ζ < x} = P {kξ + b < x} = P ξ >
k


x−b
!
−a
x − b"
= 1 − 0,5 − Φ k
=
=1−P ξ <
k
σ
 x − (ka + b) 
 x − (ka + b) 
= 0,5 + Φ
,
= 0,5 − Φ
kσ
−kσ
k < 0 ζ ∼ N (ka + σ; −kσ)

  
    


+

 (%%) *     & 

     ζ ∼ N (a; σ)  ξ  =

ζ −a
∼ N (0; 1)
σ

 &   ξ = σζ − σa &     ,-. 
1
a
k = & b = − &  &   ξ 
     
σ
σ
 
1
1
a
·a− =0 
· σ = 1.
σ
σ
σ

         



        
       a = 0 σ = 1    
             
           
 
       
    
x
t2
1 − x2
1
2
ϕ(x) = √ e ; F  (x) = √
e− 2 dt = 0,5 + Φ(x).
 
2π
2π
−∞

         
       ! "   

  #       
!   
 0,8 "  #              
    ξ     
!   $  $
$  %    &  '      (  ")
*  "  +       %  " , # )
   ",    ", -  "  !   .
p = 0,8, q = 1 − p = 0,2, n = 3
   /  # '
1
12
, P (1) = C31 · (0,8)1 · (0,2)2 =
,
P (0) = C30 · (0,8)0 · (0,2)3 =
125
125
48
64
, P (3) = C33 · (0,8)3 · (0,2)0 =
.
P (2) = C32 · (0,8)2 · (0,2)1 =
125
125
012341)
53# 63771778 791::3;8
53- ?416@73;3D943176, -, +E, 6, +, -F
59- G+++E, ++ H, +-EI, +J#K
01 0)! 
   %     *  *    
'(( !
      .$ $%%  /& &"  /# ,$
 , * $%  $   ! ) * '  0
$ ' F (t) = 1 − e−λt  ! '% /& &"  /# t 
 &#  ' ) * '       

R(t) = 1 − F (t) = e−λt ,









 λ    
 
  
          
  

        






 

λ = 0,03 

R(200) = e−0,03·200 = e−6 ≈ 0,003.



  

           
          0,9  
              
      k  k = 0, 1, 2, 3, 4).
       !  
   ! "         #
  !        0,3 $ #
     %     0, 1, 2, 3, 4
 !       ! "      #
  !
  &       300  
"      0,01 '    #
  "         !
(  
  ) ! !     4% $
   ! "       !  #
   *+, 
        
 !        0,005 -.    
"  400  /      #
  !   !      0
     ! (  
   1  "    6   
      "  !  ! 
"      2 
 !  "   !!    
        ξ    !  #
   !    "    10 
     3 4 + < ξ 0
    
"#

lim P

n→∞

⎧ n
⎨ 
⎩

i=1
n

n


ξi
−

   

i=1

⎫
⎬

M(ξi )

0.

       #     ζn B− An  $
n
 "           %  & 
'   ( 
 





 )!

  

ξ


 
μk = M







ξ − M(ξ)

k 

k

  
)!

.

* #  +  (    "   "  
# "   (   




μ1 = M ξ − M(ξ) = M(ξ) − M M(ξ) = M(ξ) − M(ξ) = 0,

2 
μ2 = M ξ − M(ξ)
= σ2.

 )!! 

  

ξ

 



k

  

)! 
,  (    "        $
  ν1 = M(ξ)
%$  (   +  (         (
νk = M(ξ k ).



μ2 = ν2 − ν12 ,


2
D(ξ) = M(ξ 2 ) − M(ξ) ,
μ3 = ν3 − 3ν1 ν2 + 2ν13 .

 

        



μ3 = M





       

ξ − M(ξ)

3 



2 
3 
= M ξ 3 − 3ξ 2 M(ξ) + 3ξ M(ξ) − M(ξ)
=

= M(ξ 3 − 3ξ 2 ν1 + 3ξν12 − ν13 ) = M(ξ 3 ) − 3ν1 M(ξ 2 ) + 3ν12 M(ξ) − ν13 =
= ν3 − 3ν1 ν2 + 3ν1 ν1 − ν13 = ν3 − 3ν1 ν2 + 2ν13 .

          
μ4 = ν4 − 4ν3 ν1 + 6ν2 ν12 − 3ν14  
          ! " !   # 
   !#             $
#   
! $ ! ! 
 %&'       
   ξ    
A=

μ3
3/2

μ2

=

M


3 
ξ − M(ξ)
3

.

D(ξ)

!     (   ) !    *
 ! !  !     !     
 !    +  , ! -'.
ϕ (x)

ϕ (x)
A>0

M (ξ)

A 3) = P (3 < ξ < +∞) = Φ
1
1
= Φ(+∞) − Φ(3) ≈ 0,500 − 0,499 = 0,001.
P (−2 < ξ < 3) = Φ




!" #$% %&'()* +,,(-',(./0* -1.('(1.* 2* &"*
!3 /%+4-$(# 5  & − /%+4-$(# −6  &7
!$3 2893
!39 /%+4-$(# "  & − /%+4-$(# −"22  &7
!$9 2:;"
!: /%+4-$(# "22  & − /%+4-$(# 5  &7
!$: 222"50
'2
σ >"
,-$( 5  σ  − ,-$( −6  σ  > 2893
,-$( "  σ  − ,-$( −∞  σ > 2:;"
,-$( ∞  σ  − ,-$( 5  σ) = 0.135 × 10−3
?   P (−2 < ξ < 3) ≈ 0,976; P (ξ < 1) ≈ 0,726;
P (ξ > 3) ≈ 0,001
 855     ξ      

           a = 3,
   P (2 < ξ < 3)  P (|ξ − 3| < 0,1).

σ = 2

@  A      
B CD

860 


2−3
3−3
−Φ
= Φ(0) − Φ(−0,5) =
2
2
= Φ(0) + Φ(0,5) ≈ 0 + 0,192 = 0,192.
 EF FF a = 3   GH   I  I 
|ξ − 3| < 0,1  CDD 863   ε = 0,1 J K  ID
 
0,1
= 2Φ(0,05) ≈ 2 · 0,02 = 0,04.
P (|ξ − 3| < 0,1) = 2Φ
2
P (2 < ξ < 3) = Φ

?   P (2 < ξ < 3) ≈ 0,192; P (|ξ − 3| < 0,1) ≈ 0,04









  

    

  
       
 ξ              
          !  M(ξ) = 3,8,
σ(ξ) = 0,6.

 

        
 ξ         !"
 4 − 3,8 
 6 − 3,8 
−Φ
=
P (4 < ξ < 6) = Φ
0,6
0,6
= Φ(3,67) − Φ(0,33) ≈ 0,500 − 0,129 = 0,371.
#  $$        !"   
1 − 0,371 = 0,629. %      
 ξ $
&' $(  '      )     !"  *
 $  )     (' (    0,62943 ≈ 0,2489.
     p = 1 − 0,249 = 0,751
+ ≈ 0,75



 , "          
        ξ     
 a = 8,46#
            
     
  8,40  8,43   0,25# $   
             8,49  %&' (

      -    $    $*
           )    a 
    
P (8,49 < ξ < 8,52) = P (8,40 < ξ < 8,43) = 0,25

 ! "          ) 
 ξ    )         )*)  
   78  84 # +           
     , 78 !     4%      ,  
        , 84 ! 6%# -   
  a           σ#
      - $  '   78  a  a  84
% $$     $(   0,5 − 0,04 = 0,46 

0,5 − 0,06 = 0,44 -


78 − a
≈ 0,46,
P (78 < ξ < a) = Φ(0) − Φ
σ







   







  

    

  

      ξ 
           

      
          
        ! " a #  
   ! k    !!
 ∞
k
μk = M(ξ − a) =
(x − a)k ϕ(x)dx.
−∞

$      ! % 
      &  
 ∞
(x−a)2
1
μk = √
(x − a)k e− 2σ2 dx.
σ 2π −∞
√
$      ! t = (x − a)/(σ 2) #    
√

(σ 2)k ∞ k −t2
μk = √
t e dt.
π
−∞
'  k = 1, 3, 5, ...      (!# )   
    )      *
+   #     
 
 ! 
    % ,! !!     #   μ3            
+! 
   %
$    k            
 "    !   
  μk = (k − 1)σ 2 μk−2 . . 2
%    μk = σ    μ4 = 3σ 4 , μ6 = 15σ 6   
  
    μ4 /σ 4 = 3σ 4 /σ 4 = 3.
    /!#   
  E = μ4 /μ22 − 3 = 0.



  ϕ(x)

0

           



  

       $ ((# (!#%  & , 
(x−a)2
x−a
ϕ (x) = − √ e− 2σ2 .
3
σ 2π
1((#    " !!  +  
(x − a)2 − σ 2 − (x−a)2 2
√
e 2σ .
ϕ (x) =
σ 5 2π
$   )   %   (x−a)2 = σ 2  ,  x = a±σ 
*
+  
!  ) (! 
    
  !
"
     σ

           



  



      ξ      
           a = 1, σ = 0,5     
 P (−1 < ξ < 1)  P (0 < ξ < 3)  P (|ξ − 1| < 0,1)
               
            !    
              "   110%  #
           2%          #
                " 
  $  101  105%         $  
   !$  $   $ "    $  107
 111%
  %        90 − 95  &
'(           92,7      #
      1,2         #
               
   "     )     *   "
       ξ < 90    +   
   ξ > 95
  ,             #
      -       !-   #
   .!       30   σ = 0,25  (#
-         "       0,95/
       )      
       $ )    )   )   #
                    #
 σ = 10   &        )   
     )     $ 0   -    
5 
      ξ       #
      a = 0,3, σ = 0,5 *  $  $  "
     ξ          
|ξ − 0,3| < ε    0,9642/







  

  
       
              
      ξ      a = 100    
       σ = 0,001        !
              
              0,9973

 

   


 
      
  
            !
 
 "   
      

 





 



  
#  #   $ 
#   
%#  #$ &' 
 $
(#
    
  
)               
  (ξ; ζ) *    $  +  '% ,

     

    - "         
      # #    !#  
$    #     (xi; yj ), i = 1, . . . , n, j = 1, . . . , m
    pij = P {ξ = xi; ζ = yj }
.      &'   +    #$ 
    &'  &   xi , yi      pij 

ζ
y1    yj
ξ\
x1 p11    p1j




 


xi pi1    pij




 


xn pn1    pnj






ym
p1m



   pim




   pnm

 

   



 
      
ζ
y1
yj
ym
P {ξ = xi }
ξ\
x1

p11

p1j

p1m

p1·

xi

pi1

pij

pim

pi·

xn
pn1
P {ζ = yj } p·1

pnj
p·j

pnm
p·m

pn·

  

{ξ = xi , ζ = yi }, i = 1, . . . , n, j = 1, . . . , m

    

    
 
 
    
            
           !"
P {ξ = xi } = P {ξ = xi , ζ = y1 } + P {ξ = xi , ζ = y2 } + . . .
m

pij = pi· .
. . . + P {ξ = xi , ζ = ym } =

#$

j=1

%& '
P {ζ = yi } =

n


#$(

pij = p·j .

i=1

)      *    +     
   ,
 '      ξ   
   ,  , #$ -   
  ,
      ' 
 '     
 ζ  
    , #$
         '   
      "
M(ξ) =

n


xi pi· ,

M(ζ) =

i=1

 #$(     

     

m


#$.

yj p·j .

j=1



M(ξ); M(ζ)



 



 

   

           
             
 
!   " #  P {ζ = yj /ξ = xi} 
P {ξ = xi /ζ = yj } $ " # %&'()  " 
P (B/A) =

P (A · B)
.
P (A)

% )

*+"
P {ζ = yj /ξ = xi } =

- ./

pij
P {ξ = xi , ζ = yj }
=
.
P {ξ = xi }
pi·

% ,)

pij
.
p·j

% 0)

P {ξ = xi /ζ = yj } =
m


 

j=1

n


P {ζ = yj /ξ = xi } = 1



i = 1, . . . , n



    P {ξ = xi/ζ = yj } = 1   j = 1, . . . , m % 
i=1
  1)
2 P {ζ = yj /ξ = xi}   j = 1, . . . , m   " " 
    "   # ζ      ξ 
2      "       ζ 
    ξ /
M(ζ/ξ = xi ) =

m


yj P {ζ = yj /ξ = xi }

j=1



i = 1, . . . , n

% ')

 "       ξ      ζ /
M(ξ/ζ = yj ) =

n

i=1

xi P {ξ = xi /ζ = yj }



j = 1, . . . , m.

% &)

   
    
     ξ  ζ
P {ζ = yj /ξ = xi } = P {ζ = yj }  P {ξ = xi /ζ = yj } = P {ξ = xi }.
3".               
    # ".

 

   


  

ξ

      
 ζ    pij = pi· · p·j  
pij
pi· · p·j
=
= p·j .
pi·
pi·

P {ζ = yj /ξ = xi } =

 !




P {ξ = xi /ζ = yj } =

 "# 
  
( 
ξ\ζ #
# *#
' **

pij
pi· · p·j
=
= pi· .
p·j
p·j

   

  

$%& "'
 "#
)

*' *)
*) *#

             ζ 
  ξ = 2           
  ξ    ζ = 1
+  ,    -  . %      ξ  ζ
        %& %& "'   /
,   %&    0    %&   1 0
% ")1

+, 
ξ\ζ
#
#
*#
'
**
P {ζ = yi } * #

$%& ")
 "#
)  P {ξ = xi}
*' *)
*2
*) *#
*"
* *"
#

3  %    4 
%      
M(ξ) = 1 · 0,6 + 2 · 0,4 = 1,4,
M(ζ) = 1 · 0,1 + 3 · 0,5 + 5 · 0,4 = 3,6.

 







  

          
 P {ζ = yj /ξ = 2}  P {ξ = xi/ζ = 1}
P {ζ = yj , ξ = 2}
P {ζ = yj , ξ = 2}
=
,
P {ξ =2}
0,4
P {ξ = xi , ζ = 1}
P {ξ = xi , ζ = 1}
=
.
P {ξ = xi /ζ = 1} =
P {ζ = 1}
0,1

P {ζ = yj /ξ = 2} =

  !!  !  !"#$  
%  
&"# 
&"# 
ζ
' ( 
ξ
' +
P {ζ = yj /ξ = 2} ) (* '*
P {ξ = xi /ζ = 1} ' )
, !  ! !-. /  
0 1  $  !"#  



M(ξ/ζ = 1) = 1 · 1 + 2 · 0 = 1,
1
3
M(ζ/ξ = 2) = 1 · 0 + 3 · + 5 · = 3,5.
4
4

2.     !!! 34 "  ! !
-. / -3!
5!! M(ξ) = 1,4; M(ζ) = 3,6; M(ξ/ζ = 1) = 1;
M(ζ/ξ = 2) = 3,5
 

 



 

   

6   /   (  .#    
  3"     - - 7 !  .
-! . !  - - ! 3  ! !"
# ' "!!  .!  " -   .#  
   . !  - -
 ( 
     
  (ξ; ζ)   
F (x; y) = P {ξ < x; ζ < y}.
 
    .#   "!  34  
! 
' 0  F (x; y)  18
+ F (−∞; y) = F (x; −∞) = F (−∞; −∞) = 0 F (+∞; +∞) = 18

 


   

F (x; y)

 



      

      





  
 ! 

  "#

Fξ (x) = P {ξ < x} = F (x; +∞),
Fζ (y) = P {ζ < y} = F (+∞; y);
$ %   

 

   !&

      #

P {x1  ξ < x2; y1 ζ < y2 } =


= F (x2 ; y2 ) − F (x2 ; y1 ) − F (x1 ; y2 ) − F (x1 ; y1 ) .
+ & 

  

'()*

), -     & .

 '(   "  ( + &     
  !  & 

  

    

F (x)

( '*(-(
/     ! #

F (x; +∞) = P {ξ < x; ζ < +∞} = P {ξ < x} = Fξ (x)(
y

A

y2

B

C

F

y1

G

D

H

E

x1

  
+  & 

F (x2 ; y2 )



'(



!

  





x2

x

     

$ & , !   .

    



 !

.

ACE
 F DE   .

 , F (x2 ; y1 ) 0
 F (x2 ; y2 ) − F (x2 ; y1 )     
 


ACDF ( )1( 2 ! F (x1 ; y2 ) − F (x1 ; y1 )  
  
  ABGF ( / ,  & 3" .
        BCDG(


 

   



         F (x; y)      
              
F (x; y)           
          !"#   F (x; y)  
    ϕ(x; y)
$       "   G        %
& %'     % Δx Δy (  )* +
     i   ,     %   #
      F (x; y) - % %    .  /'  
 %  0 ' & 


P {x1i  ξ < x2i ; y1i  ζ< y2i } = F (x2i ; y2i ) − F (x2i y1i ) −

− F (x1i ; y2i ) − F (x1i y1i ) = Fxy
(si ; ti )ΔxΔy = ϕ(si ; ti )ΔxΔy,

(!"*
'  . (si; ti )  ,   i' %'   
1.  .        G   &2
  %%      / %'   
P {(ξ; ζ) ∈ G} ≈

n


ϕ(si ; ti )ΔxΔy.

i=1

-,       Δx → 0, Δy → 0 (n → ∞)  . %
 "   ϕ(x; y)
3  # .         
+∞

      
ϕ(x; y)dxdy
 '  4
−∞
   
-          , % 
 .   .   .   2   ϕ(x; y)  % %
(!"*
+∞
ϕξ (x) =
ϕ(x; y)dy;
−∞

+∞
ϕζ (y) =
ϕ(x; y)dx.
−∞

(!"*







  

Fξ (x) = F (x; +∞) =

    
ϕξ (x) =

x +∞


  

d
dFξ (x)
=
dx
dx

ϕ(s; t)dsdt

−∞ −∞

ϕ(s; t)dsdt



−∞ −∞

x y

F (x; y) =



     

 x +∞
 +∞
ϕ(s; t)dsdt =
ϕ(x; t)dt.
−∞ −∞

−∞

   !"#$%    &  ' ' 
           ϕ(x; y)    &   &
     &     (  (x; y)  '  
  Δx, Δy  )    *   &   
+       '  &    
 ' )   ,     &,* ζ   
    ξ     
 !"-    ϕ(y/ξ = x)  
 ζ      ξ = x    
ϕ(y/ξ = x) =

  
  ζ = y    

ϕ(x; y)
ϕξ (x)



ϕ(x/ζ = y)

ϕ(x/ζ = y) =

ϕ(x; y)
ϕζ (y)

ϕξ (x) = 0.

!"#"%

  ξ   



ϕζ (y) = 0.

!"#.%

/   ' !"#"% !"#.%  ,  
!""%    &  ' '    0  
ϕ(x; y)ΔxΔy
ϕ(x; y)Δy
=
= ϕ(y/ξ = x)Δy.
ϕξ (x)Δx
ϕξ (x)

 !"1      
     ξ = x    
+∞
M(ζ/ξ = x) =
yϕ(y/ξ = x)dy.
−∞

ζ

!"#-%

 

   

  
  



      ξ   

ζ = y



+∞
x · ϕ(x/ζ = y)dx.

M(ξ/ζ = y) =



−∞

M(ζ/ξ = x)      x M(ζ/ξ = x) = fζ/ξ (x)
M(ξ/ζ = y)      y 
M(ξ/ζ = y) = ψξ/ζ (y)


 

  

    fζ/ξ (x)        ζ 
         ζ  ξ       
    ζ     ξ = x !   ψξ/ζ (y)
       ξ  ζ 
   "  # ϕ(x; y)     

C  x2 + y 2 < R2 ,
ϕ(x; y) =
0  x2 + y 2 > R2 .
$   #    C     ζ  ξ ξ  ζ 
ξ

!

"

 



 # $% &    

* $ 

 '

+∞


$(

−∞

C dxdy = 1 =⇒ C
x2 +y 2 0, y > 0  !
  ϕ(x, y) = 0  x < 0   y < 0

abe−ax−by  x > 0, y > 0,
* ϕ(x, y) = 0    

 +         
      ϕ(x, y) = (1/2) cos(x + y)   
−π/4  x  π/4, −π/4  y  π/4      ϕ(x, y) = 0
      (ξ, ζ)
      &            %


F (x, y) =

1
2



x



x

dx
−π/4

− cos(x + y)).

y

ϕ(x, y)dxdy.
−π/4

,   -  − π4
F (x, y) =



x

π
4

−π/4

 − π4  y  π4

y

1
cos(x + y)dy = (cos(x − π/4) + cos(y − π/4)−
2
−π/4

π
4

.  x < −   y < −
F (x, y) =

π
4




x

y

dx
−∞

−∞

0dy = 0.

π
π
y
4
4
 π/4
 y
1
1
dx
cos(x+y)dy = (1+cos(y−π/4)−cos(y+π/4)).
F (x, y) =
2 −π/4
2
−π/4
π
π
π
  −  x   y >
4
4
4
1
F (x, y) = (1 + cos(x − π/4) − cos(x + π/4)).
2
π
4

  x >  −





  x >

π
y>
4





  

   



π
4

F (x, y) = 1.

 

  
    
      (ξ, ζ)

ϕ(x, y) = a/(x2 + y 2 + 2)4
   a



        


+∞
−∞



+∞

ϕ(x, y)dxdy = 1.

−∞

       
 " 


+∞
−∞



+∞

a
dxdy = a
(x2 + y 2 + 2)4

−∞

  




2π

∞

dϕ
0

0

   !
1
rdr = 1.
(r2 + 2)4

#  

$     ϕ  r 
πa/24 = 1 ⇒ a = 24/π ≈ 7,639
%  a = 24/π ≈ 7,639

 &      (ξ, ζ)  
     ϕ(x, y) = a/((1 + x2 )(4 + y2 ))     
 a      F (x, y)   
 !  G :
 
      (ξ, ζ) 
x ∈ [0, 1], y ∈ 0, 2]  "      ξ  ζ #



  


+∞

a
−∞

dx
1 + x2
a·

'())*  a    +  ,- 



+∞

dy
= 1,
4 + y2

−∞

π
2

+


a· arctg x

π  π π
·
+
= 1,
2
4
4

"  .

ϕ(ξ, ζ) =



a·

+∞



y
1
arctg
·
2
2
−∞

π2
= 1,
2

2
.
π 2 (1 + x2 )(4 + y 2 )

a=

2
.
π2



+∞

= 1,
−∞





 

 





   



 y  


2



a
0

 

  


8
dx = 1.
2x +
3

 
       
a · 28/3 = 1       a = 3/28      
       
   ϕ(x, y) =
= (3/28)(x2 + xy)
    !    "    ξ
ζ     




2

M(ξ) =
0



xϕ(x, y)dxdy =

0

2

M(ζ) =

2

0



2

yϕ(x, y)dxdy =

0

     # 



3
28



2

xdx
0

3
28





2

dx
0

2

(x2 + xy)dy =

0
2

10
,
7

8
y(x2 + xy)dy = .
7
   (10/7; 8/7).

0

 

 $%&            
      (ξ, ζ)
ϕ(x, y) =

1 −(x2 +2xy+2y2 )
e
.
π

             
           
'  (    )
 
 +   ,$-./0  


ϕξ (x) =

+∞

−∞

12 


ϕ(x, y)dy =

1 −x2 /2
e
π



−∞

1
2
2
e−2(y+x/2) dy = √ e−x /2 .
2π

  y     *2 

+∞

√
2
e−t /2 dt = 2π

−∞



+∞

 ) 

   t = 2(y + x/2).

    
1 −y2
e
π



+∞

1
2
2
e−(x+y) dx = √ e−y .
π
−∞
−∞
2   ζ     ξ = x

2 −(1/2) (x+2y)2
ϕ(x, y)
=
e
.
ϕ(y/ξ = x) =
ϕξ (x)
π

ϕζ (y) =

3 

+∞

  * "   

ϕ(x, y)dy =

           

  





   ξ     ζ = y

ϕ(x/ζ = y) =

ϕ(x, y)
1
2
= √ e−(x+y) .
ϕζ (y)
π

        (ξ, ζ)
            
  O(0, 0), A(0, 6), B(6, 0)        
       
           
 
            
!  !   (x, y)       " 
       ϕ(x, y) = a    AB 
y = 6 − x #  a    $      
   %&




6

6−x

dx
0



ady = 1,

6

a

0

0

(6 − x)dx = 1,

  & '  ( 
     
)& *+,+-     $"    
 

18a = 1, a = 1/18, ϕ(x, y) = 1/18



ϕξ (x) =

1
3
1
dy = − x (0 < x < 6),
16
8 16

6−y

1
3
1
dx = − y (0 < y < 6).
16
8 16

0


ϕζ (y) =

6−x

0

. "   (  /0  
 ,1        
ln2 4 · 4−x−y  x  0, y  0,
ϕ(x, y) =
0  x < 0  y < 0.
            
     .    
M(ξ) =

= ln 4
0

∞

0
∞

∞

0

x · ϕ(x, y)dxdy =

x · 4−x dx =

1
.
ln 4

∞

0

∞

0

x · ln2 4 · 4−x−y dxdy =





 
   





  

   

   x        


M(ζ) =

∞



0

  



∞

D(ξ) =

∞

0



0

∞

0

y · ϕ(x, y)dxdy =

1
.
ln 4

x2 · ϕ(x, y)dxdy − M 2 (ξ).

         

        
D(ξ) = 1/ ln2 4.    D(ζ) = 1/ ln2 4.



 !"##

(ξ, ζ)

 
ξ\ζ




  




 

  

     




  
   
   

    M(ξ), M(ζ)

 !"#$
%

 ! " #        $


F (x, y) =

1 + 5−x−y # x  0, y  0,
0 # x < 0  y < 0.

    & ##    (ξ, ζ) #  '&$
 ' % # % x1 = 2, x2 = 3, y1 = 1, y2 = 2

 !"#% ( !

%)



" #    
2

%  )  $

2

F (x, y) = k(1 − e−x )(1 − e−y ), (x  0, y  0);

* #   
  F (x, y)     +   % 
 #       ,!!"  k -#   &  $
 & ##    . & D   #   $

. 
 &  '   R (x  0, y  0)

           



 



 
ϕ(x, y) =



      (ξ, ζ)   
1+

x2

a
.
+ y 2 + x2 y 2

   a          
 ξ  ζ 
           
         
a(x + y)  0  x  1, 0  y  x,
ϕ(x, y) =
0  ! "  .
#    a         
            
    (ξ, ζ)


ϕ(x, y) =

π
π
1
sin(x + y), x ∈ [0, ], y ∈ [0, ],
2
2
2
0   "  .

#           
$    ξ  ζ 
            
     a  % & 
      $     
  ϕ(ξ, ζ)







  

 

 
   
    
  
     
              
              

  



 





      ! "  #      
       $  % $  $
             
   &    ! $  "   
 
'  $             
&$  % '# $       
      #     $  
      #          $
    $       
    $   $       
 
     



(     
   $  
  %    )%    *   +, $%
  +  
  )     
#  , *    #  %    #
  $ %- $    * # % $ %  
   *         $    
$     .   #       ) $ 
  %   #     
&  ' +         
$ #  ,   /    ) #    
   
 012          



      
 !
" "      " 
 "

 

     



 
            

 
           
     
        
  

    
 









 !        "   # $
   %     $  &
     
    
  &
 &      


   !  #'    #   "     
 #           &
   
 
 
    #   #  
 
 #  
        (     )
 * &
+    
  ,
   #
  #  (  # &
   "   
   +   &$ "  ,
   #
 $ 
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    &

   
        
"  
 %  
  
   
       
  
&
  $     - $  &  


  & 
 .$ 

  

    
 &


 
    
  (   &
         
  
& 10 
12 &  
             


      "    %
      

           
    
     ξ  !  m  

 "  n    " #$
# 
 #      "#  

   

  





% /  &

   n 

 0'

&  " '
 "#  (  )
 
      
* "  " 
  
  +
  ,    '
+
  -
"  .     "#    +   #
&  "    x1 "#  n1  x2 , n2  . . .
k

xk , nk  
ni = n , "/* " 
i=1

 
4 10
⎧
0

x  1,
⎪
⎪
⎪
⎪ 0,1  1 < x  4,
⎪
⎪
⎪
⎨ 0,3  4 < x  6,
∗
0,6  6 < x  7,
F (x) =
⎪
⎪
0,7  7 < x  8,
⎪
⎪
⎪
⎪
⎪
⎩ 0,9  8 < x  10,

x > 10.
1

 

   









F *(x)
1
0.9
0.7
0.6

0.3
0.1
1

4

6

  
    

7

8

x

10

   

     

!

F ∗ (x) $ % &  '
∗
( 0  F (x)  1)
∗
*( F (x)  
  
$%+  )
,(
 x1  
  xk   $
F ∗ (x) = 0  x  x1  F ∗ (x) = 1  x > xk

"#     

 
  
 
/ 

j .

j .



    $
$ # 

(aj−1 ; aj ]

 

  

 



    

 

mj
n





-



 % 

  % 

(m1 + . . . + ms = n)
-

    

 .

s

 .

mj 

 .

  # %  

0  

Pj∗ =





 / 
 
$

/

.

  





   

Pj∗
mj
  
=
Δaj
n · (aj − aj−1 )

              
                 
      
   !      " 
   #$%       &
  

   

mj
n
*)
*,
*+
*)

' ( aj−1 aj mj Pj∗ =
)
+
%


*
+
$
#

+ ,
$ -,
# ),
)- ,

 !      

Pj∗
Δaj
)(+*
,(+*
+(+*
)(+*

    -*

5
30

3
30

1
30

0

3

  

6

9

  

12

x

 

"    
        
            "      
 
.
        
  /    
   0  -)1            
" 
    
     
               

 

     



Pi*
0.3
0.2
0.1

1

4

6

7

8

10

     

x



     
       
            
             
             
       

 !"#     


 
             
                
            n   !  " #
$     %      %  &    #
 
    
   
$   %&          
       '   
    
        (    &   '  
 
            
  
 )                
          
  % *
1
xi .
n i=1
n

x̄ =

+!"#,





  

   

  
1
ni · x i .
n i=1
k

x̄ =



     
    
    !      "# 
$ % & &         '
 ( !       (  



S2 =



1
(xi − x̄)2 = (x − x̄)2
n i=1
n

)

 &  
S2 =

1
ni (xi − x̄)2
n i=1
k

*

 
      
+         ,  (  
&      - & &  ((  '
  "#   ,  (       '

$ .       ( &  & (
/  
√
S = S2
0
              12+
3 &&   4   )   
 (

"
S2 =



1 2
x − (x̄)2 = x2 − x2
n i=1 i
n

 &  
S2 =

 



   

1
ni x2i − (x̄)2
n i=1
k

  

5

 

     

  

     

 

stdev(x) = 15.436





       



Stdev(x) = 15.451

   





     



median(x) = 148.615
  ! "    #  
$  %  & &     #  








  
   


 



  



'   ( )   *  



        

+"  "        *&     
"  ,  -,




  !    


. $ /   01234567581

"    #



 
F

     

  



  

  n + k         ξ1 , . . . , ξn
ζ1 , . . . , ζk  ξi ∼ N (0; 1), i = 1, . . . , n ζi ∼ N (0; 1), j = 1, . . . , k 
          
Fn,k

χ2n
= n2
χk
k

    
     
      n, k   
     
 Fn,k  0
     !   
⎧
⎪
⎪
⎪
⎨

m + k 

 n  n2
Γ
n
n − n+k
2
2
· x 2 −1 1 + x




ϕ(x) =
k
n
k
k
⎪ Γ
·Γ
⎪
⎪
2
2
⎩
0

!

x  0,

! x < 0.
"    !  #  $% !   & '(
%  !    )   ')   !*# !' (
 !   +,- . ' ' ' /0123 4567158 4509:5  !;
 

  

 

  

<         % '  ! 
   ! #     % '    !  (
  !      = ' %       #*(
    # % '           !    (
!  
>    % ' %$  !  !   $?@
$' '  !   % ' '   )  
   %    !   
         !" #

$  a %  (a1; a2)     $ #
& '  !    % ( a1 = a1(x1, . . . , xn), a2 =
= a2 (x1 , . . . , xn )' %!        
  % γ ( P {a ∈ (a1; a2)} = γ  )  γ 



 

  

 

 tγ         γ   
     
P {|t| < tγ } = γ.
 !"
# $        #  ϕst(t) 
  tγ      
P {|t| < tγ } = γ ⇐⇒ P {|t| > tγ } = 1 − γ =⇒ P {t > tγ } =
⇐⇒ 1 − Fst (tγ ) =

1−γ
1+γ
⇐⇒ Fst (tγ ) =
.
2
2

1−γ
⇐⇒
2

%   !"  &   '"  

 ¯



ξ−a
ξ¯ − a
√ < tγ = γ ⇐⇒ P − tγ < ∗ √ < tγ = γ,
P
S ∗/ n
S / n

   a   (      )"
  
     

                 
  1 − Fst (x) !"# $    $ %  1 − γ %   &
 ' & ( )#* +  $#  #%  $ %  tγ
  #  ,&$   $   ,-. /.00 1 $ 
 & + #$  $ #* (    
 ! 2 (   (     $
    % +    a      3(  #% (
(  %   &%   x̄ = 10,5     45
S ∗ = 1,6  & &63 n = 16 2 +    
 γ = 0,99

* +    % , &  -     ( 
 k = n − 1 = 15  α = 1 − γ = 0,01 . tγ = 2,95 %
      )"       
 /   ε

1,6
S∗
ε = tγ √ = 2,95 √ = 2,95 · 0,4 = 1,18.
n
16
 (   (10,5−1,18; 10,5+1,18) = (9,32; 11,68)

0. 
   -12

!"

           

        
  



  
      A 
  5%    B  2%     
  94%  ξ              

   !              
 A ζ   B  "       
     # (ξ, ζ) $  %   
 A B 
 
    ξ       
      A      
! "     ζ  
     "  

 #    B  $   x1 = 1, x2 = 0,
y1 = 1 y2 = 0. %   pij = P {ξ = xi , ζ = yj }. &
' p22 = P {ξ = 0, ζ = 0} = 0,94. ( x2 = 0 y1 = 1
y2 = 0)
"    *    P (x2 ) = p21 + p22  $
 P (x2 ) = 0,95 p22 = 0,94  p21 = P (x2 ) − p22 = 0,01 ! 
"    p12 = P (y2 ) − p22 = 0,98 − 0,94 = 0,04 ( '
P (x1 ) = 0,05     P (x1 ) = p11 + p12   p11 = 0,01
+    *  '     ,-

ξ\ζ y1 . y2 .
x1 .

 /
x2 .
 
 /
-

    01 2   

ξ
p


 3



ζ
p


 

 

 - 




   M(ξ) = 0,05, M(ξ 2 ) = 0,05, D(ξ) = 0,0475,
M(ζ) = 0,02, M(ζ 2 ) = 0,02, D(ζ) = 0,0196.
          (ξ, ζ)  !
"          ξ · ζ #

ξ·ζ
p

 %&  $& %' 

$
 $









       

   
  M(ξ ·ζ) = 0,01.    
     
0,01 − 0,001
0,009
≈ 0,295.
rξζ = √
≈
0,0305
0,0475 · 0,0196

          
   (ξ, ζ) 
 1
cos(x − y)  x ∈ [0, π2 ], y ∈ [0, π2 ],
2
ϕ(x, y) =
0 x∈
/ [0, π2 ]  y ∈
/ [0, π2 ].
            ξ ζ 
  
!  " !
  #% '   
    



  #$ %
  %  ξ

+∞


M(ξ) =

+∞


M(ζ) =

xϕ(x, y)dxdy,
−∞

( ) %#

 %&
ζ &

yϕ(x, y)dxdy.
−∞

 )  x 
M(ξ) =

1
2





π/2

xdx
0

0

 ' 

π/2

cos(x − y)dy =

π
.
4

*"

    %    ! ξ ζ   
cos(x − y) + '  M(ζ) = M(ξ) = π/4 ,  
+∞


D(ξ) =

x2 ϕ(x, y)dxdy − M 2 (ξ).

−∞


D(ξ) =

1
2



π/2
0

x2 dx


0

π/2

cos(x − y)dy −

 π 2
4

=

π2 π
+ − 2.
16 2

- ! )  x %#  % "    ,  
D(ζ) = D(ξ) .   ) '  /$      
&
   "%   #$ %

 ( ) % /0  

+∞


M(ξ · ζ) =

xyϕ(x, y)dxdy,
−∞

           



         

 π/2
π2 π
1 π/2
− + 1.
xdx
y cos(x − y)dy =
M(ξ · ζ) =
2 0
8
2
0
     x   y       

              ! 
 !"
 π 2 − 8π + 16
 π2 π
π2   π2 π
− +1−
/
+ −2 = 2
≈ 0,245.
rξζ =
8
2
16
16 2
π + 8π − 32

 #$         
         
M(ξ) = 2,

M(ζ) = −5,

D(ξ) = 9,

(ξ, ζ)

D(ζ) = 4.

          
   (ξ, ζ)
% &   "   '  '  
   
    (  ) rξζ = 0. *   
  +     +
,-,    '    "
1 − 1 ( (x−2)2 + (y+5)2 )
4
e 2 9
ϕ(x, y) =
.
12π
. '     σξ = 3, σζ = 2   ' !    
 
 x  y
F (x, y) =
ϕ(x, y)dxdy,




−∞

−∞

 y
(x−2)2
(y+5)2
1
e− 18 dx ·
e− 8 dy =
12π −∞
−∞
 x
 y
  0

2
(x−2)
(x−2)2
(y+5)2
(y+5)2
− 18
− 18
e
dx+
e
dx ·
e− 8 dy+
e− 8 dy .

F (x, y) =

x


1  0
12π −∞
0
−∞
0
     
/ +    -',  


 0
 +∞
π
−x2 /2
−x2 /2
.
e
dx =
e
dx =
2
−∞
0
0          !, 1 
 x
z2
1
Φ(x) = √
e− 2 dz.
2π 0

=

 

           



    D(ζ) = 1/25     
         ξ · ζ  
+∞


M(ξ · ζ) =



1

xyϕ(x, y)dxdy = 24

x2 dx

0

−∞



1−x

y 2 dy =

0

2
.
15

  !"#$ %&   &      '
 
 


rξζ =

1 1
·
5 5

2
2 2
− ·
/
15 5 5

2
=− .
3

 !("     ξ ζ       
     M(ξ) = M(ζ) = 0, σ(ξ) = σ(ζ) = 1
       (ξ, ζ)  
G    R = 2     

) *    +  ξ  ζ         
    ϕ(x, y) = ϕ (x) · ϕ (y)  ,
ξ

ζ

1
2
ϕξ (x) = √ e−x /2 ,
2π

1
2
ϕζ (y) = √ e−y /2 .
2π

-   

ϕ(x, y) = ϕξ (x) · ϕζ (y) =

.   


P =

ϕ(x, y)dxdy =
G

1
2π

1 − 1 (x2 +y2 )
e 2
.
2π



1

e− 2 (x

2 +y 2 )

dxdy.

G

/          
P =

1
2π


G

1 2

e− 2 r rdrdϕ =

1
2π





2π

dϕ
0

0

2

1 2

e− 2 r rdr = 1 − e−2 ≈ 0,865.

 !((     ξ ζ       
     M(ξ) = M(ζ) = 0, σ(ξ) = σ(ζ) = 1
    R       
     (ξ, ζ)   0,9





      


P =
0



       

         
R

1 2

1 2

e− 2 r rdr = −e− 2 r

R

1

2

= 1 − e− 2 R .

0

     R     1 − e−R /2 = 0,9.  
   R ≈ 2,145
 !"     (ξ, ζ)      
          M(ξ) = M(ζ) = 0, σξ , σζ , rξζ = 0
           
 G
         a = kσξ , b = kσζ 
      #  $  x2/(kσξ )2 + y2 /(kσζ )2 = 1
 %   
2



P ((ξ, ζ) ∈ G) =



 

ϕ(x, y)dxdy,
G
2

2

− 12 ( x2 + y 2 )
1
ϕ(x, y) =
e σξ σζ .
2πσξ σζ

       
 % &&' (
   %   x = σξ r cos ϕ, y = σζ r sin ϕ %& 
$  &   I = σξ σζ r )    %  
P ((ξ, ζ) ∈ G) =

1
2π



2π

0





0

k

re−r

2 /2

dr = 1 − e−k

  

2 /2

.

 !*             ! 
  
    "      #    
0,4; 0,5; 0,6 
  "     0,12; 0,14   
            ξ    ζ   
ξ\ζ $% $& $'
$() $$& $) $(&
$(% $(& $)& $)
*     +   "    ξ ζ  !
!      +   





  

     
       
      
  
 

   

  

!"  !      ! 
  # !    x  !   y
$  !         
!   y   ! !    x
% &'()
* &'(
x
y

x1 x2 . . . xn
y1 y2 . . . yn

*   # y = f (x)+  ,   
#-# !  y  x
     % ./) ! #    
  !      +  
     !   f (x )  
#- #  ! y     0


f (x ) − y −→ min .
Φ=
%&'()
         f (x) ! 
  1     !  !#
     #
  (x ; y )  
"   , 0 x ; f (x )
    ./   ,  !
0 f (x) = ax + b
2  , ! a  b+ #-    
 Φ+  !         0 
 #  !  a   b   Φ ,  
! +     ! ! a  b   
  Φ
i

i

n

i

i

i=1

2

f (x)

i

i

i

i

     

 
∂Φ
=2
∂a

n

i=1



  
(axi + b − yi )xi = 2a

n

i=1

x2i + 2b

n

i=1

xi − 2

n


xi yi ;

i=1




∂Φ
=2
(axi + b − yi ) = 2a
xi + 2nb − 2
yi .
∂b
i=1
i=1
i=1
n

n

n

         

⎧
⎧



∂Φ
⎪
⎨ 2a x2i + 2b xi − 2 xi yi = 0,
⎨ ∂a = 0,
⇐⇒


⎩
⎪
⎩ ∂Φ = 0,
2a xi + 2nb − 2 yi = 0.
∂b

       

⎧ 
⎧  2


n
n
n


xi yi
⎪
⎪
xi yi ,
⎨ a x2i + b xi =
⎨ a xi + b xi =
,
i=1
i=1
i=1

n
n
n

⇐⇒
n
n


x
y
⎪
⎪
i
i
⎩ a xi + nb =
⎩ a
yi ,
+b=
n
n
i=1
i=1

 !"#

  




xi
= x̄,
n

yi
= ȳ,
n



x2i
= x2 ,
n

    !$#    




xi yi
= xy,
n
%&&'  a  b

ax2 + bx̄ = xy,
ax̄ + b = ȳ.

 !$#

  !( 
  y  
  x     x  y  
y = kx + b !  " # k  b
$
N
%
&
'
(
)
*
xi
% %(
( & &( ' '(
yi % +* + ' % )'
* ' ) ()
+' * &
)      *   !$#   +     


8, 875k + 2, 750b = 13, 496,
2, 750k + b = 3, 625,





 

  
k = 2,687 b = −3,765      

   y = 2,687x − 3,765
     f (xi)     Φ   
  ! "# $% %   %  &  ! '( )
2

ri = f (xi ) − yi
*&  ! '

N
"
+
xi


yi
−1,08 
f (xi) −1,08 
ri



'
,





 
  

.
!
/












    

        ! ! 
"  #$ Φ% Φmin = 0,0081!  &
 '     (     
yi  f (xi)!
)% y = 2,687x − 3,765!
*+,        -#&
#$ a b c     f (x) = ax2 + bx + c%

⎧
⎨ ax4 + bx3 + cx2 = x2 y,
ax3 + bx2 + cx = xy,
⎩ 2
ax + bx + c = y.

. !/

 !     
   

   



 

ORIGIN := 1
n := 12
01  ! " % 10
i := 1 . . . n xi := 0.5 + i · 0.5
01  #$% 10
y := ( −1.79 −0.47 1.74 3.87 4.36 6.56 7.94 8.32 9.34 11.68 13.45 12.87 )T
01%&  $' ()* +,- # $ '.  $/ #0
S·A = Q 1# A 2 $"  3 455"$- S 2 $"
 $  Q 2 $ !" !#3 / !10

     


 

⎡ 
⎤
n
n

(xi)2
xi
⎢
⎥
i=1
S := ⎣ i=1
⎦
n

xi
n
i=1

   



          
S :=

⎛ 
⎞
n
xi · yi
⎜
⎟
Q := ⎝ i=1
⎠
n
yi

S
n

Q :=

Q
n

i=1



S·A=Q  




17.0417 3.75
32.5996
A := lsolve(S, Q) S =
, Q=
,
3.75
1
6.4892


2.7743
A=
−3.9146


     
Y (x) := A1 · x + A2 
  

    
         
    !  "     #$ % ! "
  #$ & 
min :=

n


(yi − Y (xi ))2

min = 5.4591.

i=1

'#  "        
       

16
13

Y(x i )

10
7

yi

4
1
-2
1

2

3

4

5

6

xi
     



7

     



⎡ 
⎤
⎤
⎡ 
n
n
n
n


(xi)2 · yi
(xi )3
(xi )2
(xi )4
⎢ i=1
⎥
⎥
⎢ i=1
i=1
i=1
⎢ 
⎥
⎥
⎢ 
n
n
n

⎢ n (x )3 
⎥
⎢
2
(xi )
xi ⎥
x i · yi ⎥
S := ⎢
Q := ⎢
⎥
i
⎢ i=1
⎥
⎥
⎢ i=1
i=1
i=1
n
n
⎣ 
⎦
⎦
⎣
n


1
2
(xi )
(xi )
n
yi
i=1
i=1
i=1
⎛
⎞
⎛
⎞
877.3 129.5 20.5
171.974
Q
S
Q :=
S = ⎝ 129.5 20.5 3.5 ⎠ Q = ⎝ 25.382 ⎠
S :=
n
n
20.5
3.5
1
3.572
⎛
⎞
0.1627
A := lsolve(S, Q)
A = ⎝ 0.4227 ⎠
−1.2416


         
Y (x) := A1 · x2 + A2 · x + A3
min :=

n


(yi − Y (xi ))2

min = 28.8735

i=1

15
12
9

Y(x i )

6

yi

3
0
-3

-1

0.5

2

3.5

5

xi
   

   

6.5

8





 

15
12
9
6

yi

3

Y(x i )

0
-3

-1

0.5

2

3.5

5

6.5

xi
  
Ȧ

 

 

Ȧ

     

     º¶»


         º¶»

  ! "
Ȧ

º¶»

   

   
 
    

Ȧ

   

     º¶»

#$#$%& '   

( )*+ ##+  !$ ! $ # #$ # $$
#!  #*    ,$ #   ,$$
#
+----$.
Ȧ

        º ¶»

, /

! #$ '!#$&   

8

     
  





  




   n      
      (ξ; ζ)  n   
(xi ; yi )    

     
 !      "   " #  $%&'
(    "   "   xi   
) yi  " !   *  i  
+ " $%&
,"  "
ξ\ζ
x1
x2

y1 y2
n11 n12
n21 n22

ys
n1s
n2s

ni·
n1·
n2·

xk
n·j

nk1 nk2
n·1 n·2

nks
n·s

nk·
n

 j   " )   -  nij      
   (xi; yj ) 
       
"  "s   "     

   ni· =
nij      )    
 " n·j =

k


j=1

nij 

i=1

    

.  !    " 
 

n=

k 
s


nij =

i=1 j=1

k


ni· =

i=1

s


n·j .

j=1

     " "  "   
        ξ   
       
    ζ 
/   !    $0&   -  "  
  !  
k


x=

k


ni· xi

i=1

n

,

x2 =

ni· x2i

i=1

n

,

Sx2 = x2 − x2 ,




s


y=

j=1





s


n·j yj
n


y2 =

,

n·j yj2

j=1

n

Sy2 = y 2 − y 2 .

,

 
∗
rxy
=

xy − x · y
,
S x · Sy
s
k 


xy =

  

 


∗
rxy

 

nij xi yj

i=1 j=1

.
 
n
    
   
    
        
    !"#    
  $   % &
%'(
∗
∗
 rxy
= ryx
)
*
    
 +   $  + 
∗
−1
( −1  rxy
 1)
∗
, |rxy
| = 1      '       - ! % xi  yi
∗
  % ' . - rxy
 !"  +!- 
% ' $$  !  

 *    yx    
     ζ 
   ξ = x
     !" #  yx #
 #    ζ   $ #% ξ = x
S


& $  $
 xy 

yx1 =

j=1

yj n1j

n1 .

 '   #

 
   $    ζ  ξ (
M(ζ/ξ = x) = fζ/ξ (x)  ξ  ζ ( M(ξ/ζ = y) = Ψξ/ζ (y)  $ !  &
!
 *   /  $  +       
0- ! !   +   

     




   

   



  


                
                
!           
   "  ζ  ξ #
yx = ρ∗ζ/ξ · x + b∗ .

$    %    %&&  n    

        & Φ(ρ∗ ; b∗) = (ρ∗ ·xi +b∗ −yi)2
i=1
' (      yi  f (xi)   
∂Φ
   ) ∂ρ∂Φ∗ ∂b
        
∗
ζ/ξ
  *+,   '   %&&   ρ∗ζ/ξ b∗    
 (      & #
ζ/ξ

⎧ ∂Φ
⎪
= 0,
⎪
⎪
⎨ ∂ρ∗ζ/ξ
⎪
⎪
⎪
⎩ ∂Φ = 0,
∂b∗

⇐⇒

ζ/ξ

⎧ ∗ 2
∗
⎨ ρζ/ξ x + b x = xi yi ,
⎩

ρ∗ζ/ξ x + b∗ n = yi ,

⎧
xy − x · y
⎪
∗
⎪
,
⎨ ρζ/ξ = 2
x − x2
⎪
⎪
⎩ b∗ = y − ρ∗ x.
ζ/ξ

⇐⇒

*,

-%&&   ρ∗ )   %&&    " 
 ζ  ξ  . '    )  %&&    
  & #
∗ Sy
ρ∗ = rxy
.
*/,
Sx
0  )         ( #
∗ Sy
yx = rxy
(x − x) + y .
*,
S
ζ/ξ

ζ/ξ

x

 +  

   ζ  ξ

  





 

 

     
 
∗
    rxy
> 0       
∗
 ζ  ξ    rxy
< 0  
   !  "          
 ξ  ζ 
∗ Sx
xy = rxy
(y − y) + x.
#$%&'(
Sy

)      * *+*  *   

Sy
(x − x) ,
Sx
∗ Sx
xy − x = rxy
(y − y) .
Sy
∗
yx − y = rxy

)   ,      (x; y)  *   
 
∗
)   *     "    |rxy
| = 1     !    " !   xi  yi 
    " .       
"*

 $%/  
        
            
  

! "
 ! # $%
$ &  !
$# $  !
n·j # $# ##
ξ\ζ

ni·


#$
$"
$


0  1    )23    n = 60 .   #$45( #$46(
#$%7(  ,     
5620
≈ 93,7;
60
552400
≈ 9206,7;
x2 =
60
Sx2 ≈ 433,2;
x=

3520
≈ 58,6;
60
225600
≈ 3760;
y2 =
60
Sy2 ≈ 318,2;
y=

     
Sx ≈ 20,8;
∗
=
rxy



Sy ≈ 17,8; xy =

xy − x y
≈ −0,55;
S x Sy
∗
·
ρ∗ξ/ζ = rxy

  
 

ρ∗ζ/ξ

317600
≈ 5293,3;
60
sy
∗
= rxy
·
≈ −0,47;
sx

sx
≈ −0,64.
sy

  

      

a)yx = −0,47x + 102,3,
b)xy = −0,64y + 131,3.

  

     !
y

58,7

a
b
0

93,7

x

   

  
       
            ζ  ξ   
        n f (x) = ax2 + bx + c

     
   Φ = (ax2i + bxi + c− yi )2
i=1
             !
 "  #   $%&'(





 

          
      f (x) = ax3 + bx2 + cx + d  
!         " 
#     !      $!  
      z = f (x; y) "%   zi∗ = f (xi; yi)
    % !  xi  yi    
 "  "
      %
 
  zi         &  
' "  %     
z = ax + by + c (  '!   
 " $
"%)  "   *
a=

∗
∗
∗
∗
∗
∗
rxz
ryz
− ryz
· rxy
− rxz
· rxy
Sz
Sz
· , b=
· ,
2
2
∗
∗
1 − rxy
Sx
1 − rxy
Sy
c = z − ax − by.

 +,-  
         
         ! " # $  % &  '
         #    (  ()
* X  Y !
.      -!  Xj  /!  Yi
012341   *
536 7849: * ;:89< =>>:37;>:9? * @< 72 * -< 7A * /<
53B C * 41D9E3F;-G H IG J 3 3 6 72K
5LB M@G NG6GG 6IGO
53@ P * 41D9E3F;IGHIGJQ Q 6 7AK
5L@M-G@GNGO
53, 4 * 41;:32MBG/O MGI6+O M,@GO M6- - GOK
53N 7 * F84F844M3QO Q 6 7A 3 6 72K
53+ 7Q * 41D9E3F;F844M3QO Q 6 7A 3 6 72K
5L+MN I6 6/ 6NO
536G 73 * 41D9E3F;F844M3QO 3 6 72 Q 6 7AK
5L6GMI@ 6I IIO
5366 02 * F847QM3OJCM3O 3 6 72R7K
5L66+/@@@,
536I 0A * F8473MQOJPMQO Q 6 7AR7K
 

5L6/-//III

     

     
 





 

 

     

 
 



 
 



  ! ! ! ! "! ! ! "#"

 $
 %

&''# 

 &
$ %

%# % %#

$ &
 &
   ( % &')'*
   ( % &')'*
 +,- .   ! ! ! ! !! ! !
/",- 0, 123-1! 4 1 / .4*
'2 562.&789:8; t 

  
  

       
 

%
     /  ) ' )  
#$ n  /  '     $ 
 '    '  ' .&&) 
∗
 ) rxy
  '        0  
   '            
   ' .&&)  ) rξζ     /
 ' .&&)  n → ∞
(/         α   
 H0 : rξζ = 0      '   H1 : rξζ = 0
+  H0    .        ' .&&)
 )    
    '    ξ  ζ



 

    


             
   H0             

           
 ξ  ζ  
       
        H0      
 
√
∗
'
T = rxy

n−2

∗ 2
1 − rxy

,

!" #$

∗
% rxy
     !& '$ (   % 
 H0   T       )   
n − 2     *       
%  H1     t  = −t  *    
      α    
t     
 n − 2     +  ,$   -
      )   - 
      .   /0123  α/2  n − 2   
 (  !" #$  
.     
    T
 |T | > t  %  H0  %       
α  |T |  t  4      %   H0

 !" 5     α = 0,1    
             
    

6  7   8 9 n = 60  α = 0,1   +  , 
        . t  = 1,67  
∗
= −0,558
        !" #$  rxy
T = −0,55 ·

√
58

1 − 0,552

≈ −5,015.

(   |T | > 1,67   8 %  H0  % 
      α = 0,1 9%  +   
      
    

 

    

 



       





        n  m  
           σ12  σ22  
         x̄  ȳ      α
     H0       
    !
H0 :

M(ξ) = M(ζ).

H0 :

¯ = M(ζ̄).
M(ξ)

"#   $  %  x̄  ȳ  #  
       !
&  #         
        '     
   !
x̄ − ȳ
Z=
()*+,
.
2
2
σ1 σ2
+
n
m

&          $  
 !
Z=

n


ξ¯ − ζ̄

,

σ12 σ22
+
n
m
 H0#



ξ¯ =

i=1

n

m


ξi
, ζ̄ =

i=1

ζi

m

.

-  
 ξi ∼ N (a; σ1)# ζi ∼ N (a; σ2)#
 Z ∼ N (0; 1)
&  # Z       %   
       .     
  -         !


9



σ12 σ22
+
=
/ n
0n  m
m


9 σ12 σ22
+
=
=
M(ξi )/n −
M(ζi )/m
n
m
i=1
i=1

 na ma 9 σ 2 σ 2
1
=
−
+ 2 = 0,
n
m
n
m
¯ − M(ζ̄)
M(Z) = M(ξ)





  




9 σ12 σ22
¯ + D(ζ̄)
+
=
D(Z) = D(ξ)
n
m
  2


n
m

9 σ1 σ22
=
+
=
D(ξi )/n2 +
D(ζi )/m2
n
m
i=1
i=1
 2
 

nσ1 mσ22 9 σ12 σ22
=
+
= 1.
+
n2
m2
n
m

             
        
•

H0 :

M(ξ) = M(ζ),

H1 :

M(ξ) = M(ζ).

          α/2
               H0!
"    F (Z) = 1 − α/2  F (Z) # $ %   &
          '   &
  Z             &
  Z   |Z | > Z       H0
     α! (  |Z |  Z      &
      H0         H1 !
)       F (Z) = 1 − α/2   
 *+, -   ./0123     %    
4    -56!73 ! !
α
⇐⇒
F  (Z ) = Φ(Z ) + 0,5 =⇒ F  (Z ) = 1 −
2
α
⇐⇒
⇐⇒ Φ(Z ) + 0,5 = 1 −
2
1 α
− ;
2 2
M(ξ) = M(ζ),

-56!73

Φ(Z ) =
•

H0 :

H2 :

M(ξ) > M(ζ).

           α
     8        H0! "   
F  (Z ) = 1 − α  '     Z     $&
 -56!43 Z   Z > Z       
H0      α! (  Z  Z      &
      H0! )    Z  ' 
 *+,     %     4   

 

    



 

F  (Z ) = 1 − α ⇐⇒ Φ(Z ) + 0,5 = 1 − α ⇐⇒
1
− α;
2
M(ξ) = M(ζ),

Φ(Z ) =
•

H0 :


       


!   

   





"



M(ξ) < M(ζ).
 α 





H0 


Z




" 

       


H3 :

      


F  (Z
)=α





    

      



Z      

Z
= −Z  # Z < −Z $ "   H0 
  

 α$  Z  −Z $   
%  " H0 

 &    
    
          
            
      !   "#$%&  ' !  
"#$(&    ) * +
Z=

 
  
(

"












σ2 +
 α 
%



" ,  

!    

.

,  

-

/ 

H0 :


 )   

$ 


2

'

.

   
n





% 

   % % )  % 

H0 :

0 "

2

S1∗
S∗
+ 2
n
m

 )  * 

 %  
  

 

x̄ − ȳ

 "  

M(ξ)

H0



  

  

 





 %
 

a0 -

M(ξ) = a0 .
x̄ 



$

   "  

¯ = a0 .
M(ξ)

   

$ 

 " 



  ! 



  

 










   

  

U=

x̄ − a0 √
x̄ − a0
√ =
· n.
σ/ n
σ



             
!  "     #$"  ! " %
(ξ¯ − a0 )
   U = √
       
n/σ
&
      #$!  '#$ %
    #$(
•
H0 : M(ξ) = a0 ;
H1 : M(ξ) = a0 
)   "
  *  " +   %
$ # ,-. " Z    / 
 0
U " #$0" # !
1 |U | > Z    H0       %
 α  |U |  Z        !   
 H0    !  H1 
•
H0 : M(ξ) = a0 ;
H2 : M(ξ) > a0 
)   "
2 " Z    / 

0 U  1 U > Z    H0   
     α  U  Z        !
   H0 
•
H0 : M(ξ) = a0 ;
H3 : M(ξ) < a0 

)   "
2 " Z    / 
0 U  1 U < −Z    H0   
     α  U  −Z        !
   H0 
1
 "0  2 "       "
    
x̄ − a0
x̄ − a0 √
T = ∗ √ =
· n.
3
S∗
S / n
.       
√      #$"
 ! "    T = (ξ¯ − a0 ) · n/S ∗    4 #%
   n − 1  #   5'#$     
   #$!     #$(
•
H0 : M(ξ) = a0 ;
H1 : M(ξ) = a0 
6 "          !7
 "   t2 ""    α  n − 1 

 

    




         
              
    
      !"#$%& '
  ()&*  T 

|T  | > t2  
H0
α  |T  |  t2 

    
 H1



 

  
  

   
  






  

  


  
n

     
       

p



     

  

 

     

   

   

A

 

n



α

   

 





p

 



 

  

-   

  & 

m/n

 

  

p0





 



m



m/n

 


  



' 

H0
p0 )

 

p = p0 .

)

H0 :



 %       

A

( 

    +  

  &

  

H3 : M(ξ) < a0



  

   

H0 :
* 

H0

    

      

           



H2 : M(ξ) > a0

     

 

  

!"#$

 

 



  

    

  

M

m
n

, 

m/n

p



 

= p0 .

    
.    

 



 

 

    



   

m

n
U= √
1




H0



  

  

  

q0 = 1 − p 0 .

m

  



  

n

m

 2

  

/0 3        

= p0 , D

   

 &     

U  

M

/0 0$

       

    

    + 

   +





       



& 

.

− p0 √
· n,
p 0 q0

n

=

p 0 q0
n

  4 

 

/0 5           

     

  

6 

/0 0$









         

         
 

           
          
!   y = ax2 + bx + c
" 
#     
i

$
%
&
'
xi

(

$
%
yi  $) ( &$& ( $) ( ) ( )




 
n

x2 yi

     
! "
n
n
3
4


x
x
i
i
 y # x# xy # x2 # x2 y =
# x3 =
# x4 =
 $ %
n
n
n
i=1
i=1
i=1
&    ' & & ( ! )# &   ' '
$ '  $   (!
i

*(

i

)

.

xi
"#+++++
+#+++++
#+++++
)#+++++
-#+++++
#+++++

yi
#),++
+#)++
"+#) ,++
"+#, ++
"+# ,++
"+#+.++

! )

  ' ' 
xi yi
x2i
x2i yi
x3i
x4i
"#),++ #+++++ #),++ "#+++++ #+++++
+#+++++ +#+++++ +#+++++ +#+++++ +#+++++
"+#) ,++ #+++++ "+#) ,++ #+++++ #+++++
"#- ,++ #+++++ ")# )++ #+++++ ,#++++
")# .++ #+++++ "#++ )#+++++ #++++
"#)+ -#+++++ ")#-.)+ #+++++  #+++

*  ('$#     
/ ' ' '  
& 
⎧
⎨ 19, 8a + 7b + 3c = −2, 1352,
7a + 3b + c = −1, 1872,
⎩
3a + b + 5c = −0,0540.
     0''# 1  2%3  %"
'
456# 7  a = 0,1739; b = −0,8175; c = 0,0484
*  ('$#  $&  %  & 

y = 0,1739x2 − 0,8175x + 0,0484.

           



   f (xi)         
               
               
  !         "# $
% "# $
!&    "# '
i

'
*
$
.
-

xi

('))))
)))))
'))))
*))))
$))))

yi

'*#+)
).*.)
()*"+)
()+"#)
()"#+)

f (xi )

')$",)).#$"
()-"-*,
()#"'*$
()#$"."

Δ2i

))+)+.
)'.')#
))#"-+
))$,$.
))*'.,
)$-))#

/            (
  Δ2i = (f (xi) − yi)2       01  2 
            Φ : Φmin ≈ 0,35  (
          &    
  yi  f (xi)
34 y = 0,1739x2 − 0,8175x + 0,0484



 "# *  

  

   
 
              
y = ax + b     !"  !#
 "# $  
  $  
          %    
y = ax2 + bx + c

           

 
    




\x





 i
     
      
      
     
    
      
     
     
     
       

    
     
     
     

    
      
      
     
     
     

    
      
       
      

    
      
       
       
       
     

i

yi







  

      
           
    χ2

       
                  

 

   

      

            
             
  F (x) !   " ! "  α  β # 
$   % & $! ' F (x, α, β)( F (x) = F (x, α, β) )$*
   y = F (x, α, β)   !  $"   %
! ! "  α  β  )     
     $   !   u = u(x), v = v(x)
$       y = F (x, α, β)  " 
" (u; v)    V = k · u + b & 
k  b ! *  ! "  α  β ( k = ϕ(α, β)
b = ψ(α, β) +       
y = F (x, α, β)   " (u; v)  %  (
v = k · v + b  v = ϕ(α, β) · u + ψ(α, β).
, %  &        y = F(x) 
  $%. $/'     $!  
      y = F (x, α, β)    ! 
"  &          "
(u; v)  % $!  ,- 0    
(u; v)  '  1* ! (x; y) !1 
           " (u; v)  &
         1   $ 
$   ' "  %       2 
 3     y = F(x)# !   1 4
   $! &   %1 Ou  b  
   %1 Ov     " 
  ! ϕ(α, β)  ψ(α, β) "     ! "

 

    

 α





β



 
k = ϕ(α, β),
b = ψ(α, β).

              
   
     
F (x, m, σ) = Φ
1
 Φ(x) = √
2π

x

z2

e− 2 dz

x − m
σ

+ 0,5,

!   "  

   #

0

$ #   
u = x,

v = Φ−1 (y − 0,5),

%&&'(

 x = Φ (y) !   )  !  "   *   
  (u; v)   !     y = Φ( x−m
) + 0,5
σ
   #$   
−1

 

x − m
Φ
+ 0,5 − 0,5 =
v = Φ (y − 0,5) = Φ
σ
 


u−m
x−m
x−m
= Φ−1 Φ
=
.
=
σ
σ
σ
−1

−1

+  ,   
v=

      
ϕ(m; σ) =

u−m
σ

%&&-(

  %&&.( 
1
,
σ

ψ(m; σ) = −

m
.
σ

/   !  0    !     %&&-(
   k %  0!!  ( b %     # Ov ( 
  m b   
 
⎧
⎨ k = 1,
σ
⎩ b = −m,
σ

⇐⇒

⎧
⎪
⎨ σ

1
,
k
b
⎪
⎩ m = − .
k
=

%&&1(





  

 
         
     !  "     
      #         $  
"  %       &   ! 
v = Φ−1 (y − 0,5)   '     '( !  y 
 )   
       
   (u, v)      !"
 # $ %&      $ % ' 
'''   ()  ! $    &
    * !  $ F#(x)   
      &+   +  #  "
$       $ & , m  σ %"
 (  )-( . /    $ k  b+
    (  !   (u, v)  $
.    Ou  Ov
   *+ k = 1, 0; b = −1, 8 ⇐⇒ m = 1, 8;
σ = 1, 0,
   **+ k = 1, 4; b = −5, 7 ⇐⇒ m = 4, 1;
σ = 0,71,
    ***  !  '   - &   
   &          ***   
     
      +


y = F (x, λ) =

1 − e−λx x  0,
0
x σ22 
' 7    
  
     





 

 

     

  n
  xij (i = 1, . . . , n, j = 1, . . . , k) 
   k          ξ1, . . . , ξk    
  k   F1, . . . , Fk   F 
  !! " #          α 
  $          % H0 :
M(ξ1 ) = . . . = M(ξk )     $ H1    
     
# & !!
'     $  $


( F1 F2
Fk
 
x11 x12   
x1k
)
x21 x22   
x2k

    

n
xn1 xn2   
xnk
$ x  x    
x
 
1

2

k

*       x    $  &
(
  (j = 1, . . . , k)  x +      
 %
j

1
xij ;
n i=1
n

x

j

=

1 
xij .
kn j=1 i=1
k

x=

n

 !! "

, -         *      (
   
  xij         %



x

=

k 
n

(xij − x)2 .
j=1 i=1

 !!)"

.           $  
     x    %

j




=n

k

j=1

(x

j

− x)2 .

 !!/"

     



         xij   
    x     
j




=

n
k 


(xij − x

j )

2

 !

.

j=1 i=1

"#  $%& 




=






+




'!

.

(  ) $ $% 
*  &     $     F  + 
%&     ,    ξ1 , . . . , ξk &    
  x  $%          %
 $    ,   x&   ,  
  F 
-  &    $  $    
  ,  &     $     F & 
     
.     '! #  %   &    
$   
    ξ1 , . . . , ξk  + 
%&     $     &   

          &   
/ $& $      
0$        1!2  !  
 ,     &     , 
 
j





∗2
S

=



nk − 1

;

∗2
S



=





k−1

;

2

∗
S
=





k(n − 1)

.

3!

4    $        #  H0 :
   &   )   
  ,/  5       
$% 6    78    1!  
     &       $ H0
4   #        
      , $   &   
   %8&  $ H0   %&
    2   %
M(ξ1 ) = . . . = M(ξk )







   

             
         

∗
•
 S ∗  < S
       !   H0 
 "               
#        ! 
     
         !   
∗
•
 S ∗  > S
    
∗
F = S ∗  /S
,
$%&&'(
 )  #        * +, 
  $  -(   ! ./0 # t2 (α; k1;
k2 )  k1 = k − 1; k2 = k(n − 1)   F > t2 (α; k1 ; k2 ) 
  H0  
     α 
2

2

2

2

2

F  t2 (α; k1 ; k2 )

2

  



 



H0

  


  
   
 
             
      ! "   
   # $         
 %       α = 0,1   
             




2

S ∗

 

    "

2

∗
S

# $

%"  " &   '




2

∗
S
= 9263,0406,



=

% 
 




+

  









& ( )   " &



2



  

'
2

F



= 27789,1219,

= 27789,1219 + 33348,9583 = 61138,0802.

  

"&   

 !

S ∗ = 438,8021.



%  '





2

S ∗ = 733,8998,

= 33348,9583,

= 61138,0802,

2

S ∗   
n = 20 k = 4 x̄ = −13,655


=

∗
S
2
S ∗

= 21,110.



 !

"     !

     



   

 

        


%
&
$
'
(
)
*



%
&
$
'
(
)
*

x

j


 

#$

 


−81, 6

−2, 16
−8, 31
−7, 16
−6, 58
−7, 16
−6, 42
−8, 16
−8, 42
−10,3
−8, 89
−7, 16
−2, 42
−1, 74
−4, 26
−7, 74
−4, 84
−8, 89
−8, 89
−11, 0
−6, 84

−45, 672

−6, 867

−1, 22
−1, 74
−1, 97
−1, 92
−2, 76
−1, 17
−0,01
−81, 9
−81, 2
−81, 8
−82, 2
−82, 1
−82, 0
−82, 4
−83, 5
−84, 9
−83, 6

&#$

!




" 


−1, 84



−1
−1
−1
−0,58
−1

−1, 58

−1
−1
−2

−1, 11





%#

−1, 69

#&

−1, 69

#)&

#$)
#$)

−1, 58

−1, 16
−2, 42
−1, 16
−1, 58
−3, 26
−1, 26
−2, 53
−3, 69
−3, 69

−0,924

−1, 1565

−0,58



−1
−3
−2, 16
−1, 58



+, #     , - ,
   , k1 = k − 1 = 3#    ,  
 ,    k2 = k(n − 1) = 76#  .,/  
α = 0,1     ,   01/  F 2
  3    -. 456789  ./ .





F   



      

  

   
F  (0,1; 3; 76) = 2,105.

  F > F          
                
  α = 0,1!  "       #"$  %
    !

       
    
 &''!&    n = 20    
      (ξ, ζ)   
  rxy∗ = 0,15.     
 0,05  !  H0" rξζ=0    !
    !# 
 H1 : rξζ = 0$
(
#,-!&$

   ) *   
T



= 0,15 ·

√
20 − 2
1 − 0,152



 + 

≈ 0,429.

∗
   .   H1 : rxy
= 0  /   
 *"   ! 0 * %       
 "      α = 0,05   
* k = 20 − 2 = 18   t (0,05; 18) = 2,10!
  T  < T       "   
       /++ %    % ! 1
   *   /++ %    %    
    !! ξ ζ    !

 &''!2      

  %
 m = 50    %  % 
     x̄ = 25  ȳ = 23$ &
!#    σ12 = 5, σ22 = 4$   
   0,01  !  H0 : M(ξ) = M(ζ) 
!#  H1 : M(ξ) = M(ζ)$
n = 60

            



                 

Z



=

25 − 23

5/60 + 4/50

≈ 54,436.

    ! M(ξ) = M(ζ)"  #
 #$ %# &! # '" % # 
   %#%
Φ(Z ) = (1 − α)/2 = (1 − 0,01)/2 = 0,495.

(  ) ) * # ( +  ,  Z = 2,58
  |Z  | > Z "   % ! %!  - "
 %. . #   #  
 /00'         

      σ2 = 16      n = 80  
       x̄ = 13,12      
  0,05       H0 : a = a0 = 12 

  H1 : a = 12

                 
1
U



=

13,12 − 12 √
· 80 ≈ 2,504.
4
 ! a = a0 "  #

  
%# 2#     %#%

 #$

Φ(Z ) = (1 − α)/2 = (1 − 0,05)/2 = 0,475.

(  ) ) * # ( +  ,  Z = 1,96
  U  > Z "   % ! %!  3!  # %4
" %.   !# ! $. #   #
 
 /005 !         

     "      #  0,03 $ 
  % 500    20 % & 
 '  (

       % ! H0 : p = p0 = 0,03"  # $
#  m/n = 20/500 = 0,04 (  % #% 4
 !. H1 : p > 0,03  %$  # α = 0,05 & 6









     

   q0 = 1 − p0 = 0,97  
   

 

    

 



√
(0,04 − 0,03) · 500
√
≈ 1,311.

0,03 · 0,97
        
    p > p0  
     !    "      
  
=

U

Φ(u ) = (1 − 2α)/2 = (1 − 2 · 0,05)/2 = 0,45.
#  $  $ %  # &  ' ( u = 1,645
  U  < u        !       
  !      )* ++, -   ! .
 & ) !  

 /++0   n = 100       
                
              
  
! " # $ % & ' (
  %()! %()' &*)# & )* & )% &!)! &!)' &")# &#)*
%()' &*)# & )* & )% &!)! &!)' &")# &#)* &#)%
   mj
#
! !& !! * "
+            ξ 

1 *    "    $)       Δaj = 0,6
2  ) &   )        ,'  
 
 $ 
# !    
  

  .
   & )  ! 
   H0        
     !
  3   4   &
  ! $    &     )( 
   !
  
2  5  ! 5   )     ui = (xj−1 +xj )/2, j =
= 1, 2, ..., s  s 6         )   
       
 
s
s


x̄ =
mi ui /n,
S2 =
mi u2i /n − x̄2 .
i=1

i=1

7    S ∗2 = nS 2 /(n − 1) 3  
    x̄ = 71, 876, S ∗ = 0,8982.

! 

).







      

  

         mi − mi, (mi − mi)2, 
       

(mi − mi )2 /mi   
 χ2 = 0,349.

 !        s = 9   !
    " k  # $% k = 9 − 1 − 2 = 6 & '
!     α = 0,05 (      "
   ")      ! χ2  * +  '
    !
χ2 (α, k) = χ2 (0,05; 6) = 12,6.

(    χ2 < χ2          '
       ,    *   '
              '
   -!             
   ξ 



  

 .//   
n = 30    
   rxy∗ = 0,35        
! "      # "  
   #$ "  rξζ = 0.
 .//$   
n = 100    
   !  "  !      
    %
&'      (
ξ\ζ ) * + 
,
- ) ,  

  * ( 
-   ) 
,  , , )
+
.            
  0,1 ! # "      # 
"      #$ " 
rξζ = 0

            
N

   n1 n2

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



 
x1

x2

 
 
 
 
 
 
 
 
 
 
 

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

σ12

 
 
 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

σ22

 
 

 
 
 

 


 
 
 
 
 
 
 
 

 

 
 
 
 
 
 
 
 
 

α

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 









     

    

    
n = 100
   
  x̄ = 210          
m = 90     ȳ = 208     
          σ12 = 80, σ22 = 70
    !      ξ  ζ    
        !     0,05 
 ! H0 : M(ξ) = M(ζ)   !!"# 
M(ξ) = M(ζ)
  $  % &''(         )
 *           
 n1
 n2       x̄1, x̄2     σ12, σ22
+!*       !     α    
   *    !!"#     
 
   ,   
n = 120    
!             
σ = 5      x̄ = 23,54 +   !   
  0,01  !!" ! H0 : a = a0 = 23 
 !!"#  H1 : a = 23
   -         
  %  $      % ! .   
    *  !     0,12 -   
400  !! .      !  60    /*  
           ! .  0
1     !!" ! H0 : p = p0 = 0,12  !
!"#!" 2 H1 : p > 0,12 !     α = 0,05
     n      
!      )   !         
         "   . 
!"#         %! &''3 $  %
&''4          % "     
   !           
  !" " 5!     ! 
   α = 0,05   !    
         !  ξ     6 
     

 

 



 
      
  
     


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!
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!#


#
!

#
!!
#
#
##
#
 
#
#"

 
#

 

 

$   % && '( )*   *% &
+ & & ,- (&  & ( & & ,- (.
& +%(  -/  % 
   





      ξ(t)  
            t  
  t             
   
'     -*  *  -/  %  *.
( 0 )  % * *& 12 -/ 2 ,- (/
       ξ(t) = ζ · sin t  t  0
ζ ∼ N (2; 1)             
  #       
      
    
  t    !  "    
       t    
#   !  "          
     
3  % (( 1*1 -/ / %  4 %   .
%1 5 5 ,(  %61 %7   */ 
%-1 * -  ) 2 5 / 8  %   %.
 -* 1*& * -5& & 91 %  ξ(t)
-*       ( -(: x(t)







 

   
      t = 1 
   ξ(1) = ζ · sin 1    
   ζ
           −3  
   x1(t) = 2 sin t x2(t) = 3 sin t
# $

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       m(t)     
t    
      
mξ (t) = M ξ(t) .

   m(t)          !
 " #  $   %    &  ξ(t) 
 '     f (t) (        m(t) )
   '
   * " $   +

       % "  ,*, M f (t) = f (t)


*., M f (t) · ξ(t) = f (t) ·M ξ(t)
  

*, M ξ1 (t) ± ξ2 (t) = M ξ1 (t) ± M ξ2 (t) 
 /      

    σ (t)    
   


2

 t   

σξ2 (t) = D ξ(t) .

0   $   +     !
     %   % "  
1        "
  !
' 


      %   - σξ (t) = D ξ(t) 
2  '       σξ2 (t)*, σξ2(t) 0
*., Df (t) = 0


*, Df (t) · ξ(t) = f 2 (t)
 · D ξ(t) 
*/, D ξ(t) ± f (t) = D ξ(t) 
%





  



 

  

            
          
   
    
    
      Kξ (t1; t2)
        ξ(t1)  ξ(t2)
Kξ (t1 ; t2 ) = M

  



 

ξ(t1 ) − m(t1 ) · ξ(t2 ) − m(t2 ) .

    
◦

   

ξ (t) = ξ(t) − m(t),

      






◦
◦

Kξ (t1 ; t2 ) = M ξ (t1 ) · ξ (t2 ) .

   

mξ (t) σξ2 (t) Kξ (t1 ; t2 )

     

      ! !  " ""  #  
 $   ! ζ ∼ N (2; 1)$  "


mξ (t) = M ξ(t) = M(ζ · sin t) = M(ζ) · sin t = 2 sin t,


σξ2 (t) = D ξ(t) = D(ζ · sin t) = D(ζ) · sin2 t = sin2 t,


Kξ (t1 ; t2 ) = M (ζ · sin t1 − 2 sin t2 ) · (ζ · sin t2 − 2 sin t2 ) =


= M (ζ − 2)2 sin t1 sin t2 = D(ζ) sin t1 sin t2 = sin t1 sin t2 .

% m (t) = 2 sin t& σ2(t) = sin2 t& Kξ (t1; t2) = sin t1 · sin t2
 "     
 '      (

 Kξ (t1 ; t2) = Kξ (t2; t1) "    $
 Kξ (t; t) = σ2(t) "    $
) |Kξ (t1; t2)|  σ (t1) · σ (t2) "  " *)
+,      
 '   -   . 
 /
0     
 '       "(

"       '   
ξ

ξ

ξ

ξ

ξ

ρξ (t1 ; t2 ) =

Kξ (t1 ; t2 )
.
σξ (t1 ) · σξ (t2 )







 

            
      
         
kξ (−τ ) = kξ (τ ).

       



 K (t1; t2) ! ! "!#
ξ

kξ (t1 ; t2 ) = kξ (t2 ; t1 ) =⇒ kξ (−τ ) = kξ (t1 − t2 ) = Kξ (t2 ; t1 ) =
= Kξ (t1 ; t2 ) = kξ (t2 − t1 ) = kξ (τ ).

$         % &   
 
σξ2 (t) = kξ (0) = σξ2 .

       

$ K (t1; t2 )
σ 2 (t) = K (t; t) = k (t − t) = k (0) = '()*+.
# , -       ./   % 

ξ

ξ

ξ

       
|Kξ (t1 ; t2 )| 

ξ

ξ

|kξ (τ )|  kξ (0).

'





Kξ (t1 ; t1 ) · Kξ (t2 ; t2 ) =⇒ |kξ (τ )| 

# K (t1; t2 )

'

ξ

kξ (0) · kξ (0) =⇒

2

=⇒ |kξ (τ )|  kξ (0) ⇐⇒ |kξ (τ )|  σξ .

0         
  ! - "!#!
ρξ (τ ) =

1   
  

ρξ (τ )

  

k (τ )
kξ (τ )
= ξ 2 .
kξ (0)
σξ

|ρξ (τ )|  1 ρξ (0) = 1!

    

2    .  
ζ(t) = ξ1 · cos wt + ξ2 sin wt,

- ξ1  ξ2 4  .◦ .
w4
◦
D(ξ1 ) = D(ξ2 ) = σ 2 M(ξ1 · ξ2 ) = 0!

 

"!3
M(ξ1 ) = M(ξ2 ) = 0

 

 



            
   M ζ(t) = M(ξ1 ) · cos wt + M(ξ2 ) · sin wt = 0 =⇒ ζ(t) = ζ (t)
◦

◦
◦



Kζ (t1 ; t2 ) = M ζ (t1 ) · ζ (t2 ) = M ζ(t1 ) · ζ(t2 ) =


= M (ξ1 cos wt1 + ξ2 sin wt1 )(ξ1 cos wt2 + ξ2 sin wt2 ) =

= M(ξ12 ) cos wt1 cos wt2 + M(ξ22 ) sin wt1 sin wt2 = σ 2 · cos w(t2 − t1 ).

       !"!#$%   


M ζ(t) = 0;

Kξ (t1 ; t2 ) = kξ (t2 − t1 ).

&    '     
ζ(t) =

n

(ξ1j cos wj t + ξ2j sin wj t),
j=0

  M(ξ1j ) = M(ξ2j ) = 0 wj ( 
D(ξ1j ) = D(ξ2j ) = σj2  M(ξ1j · ξ2k ) = 0   ∀ j, k 
M(ξ1j · ξ1k ) = M(ξ2j · ξ2k ) = 0   j = k #


)   M ζ(t) = 0, ζ(t) = ζ (t)    
              

      !"!#$%   $ # !"!#*%
  
◦

◦

◦

◦

◦

◦

Kζ (t1 ; t2 ) =

n


σj2 cos wj (t2 − t1 ).

j=0

+ ,          -   #
.   ,        
ζ(t) =

∞

(ξ1j cos wj t + ξ2j sin wj t)
j=0

!"!#/%

           


M ζ(t) = 0;

Kζ (t1 ; t2 ) =

∞


σj2 cos wj (t2 − t1 ).

j=0

 !"!#0 
   

                

 

 



σj 2

ω0 ω 1 ω 2 ω 3 ω 4

ωn

   

ω



            
           
ζ(t) =

∞

(ξ1j cos wj t + ξ2j sin wj t),

 !

j=0

"

π
wj = j ,
T

$
kζ (τ ) =

j = 0,2, . . . T # 
t2 − t1 = τ   %
∞


"

σj2 cos wj τ,

j=0



  

π
wj = j , j = 0,2, . . .
T

&!

 &!     '( "  (  
" 2T )% * k (τ )  " +%   % )%  "  ,)
)*  +% 
ζ

σj2

2
=
T

T
kζ (τ ) cos wj τ dτ.

!

0

-   %( *   !  ((     
      (    "% 
 "     Δw = π 
T

 

 




            
  
      
  
   

              

    !"#"$% 
      
        
     
&'

cos wj τ =

eiwj τ + e−iwj τ
;
2

sin wj τ =

eiwj τ − e−iwj τ
.
2i

( )     T → ∞ (Δw → 0)(   
  (    
   kζ (τ )   
  
 Sζ (w)(   *  w ∈ (−∞; +∞)  
 '
Sζ (w) = Sζ∗ (|w|)/2

+∞
Sζ (w) eiwτ dw,

kζ (τ ) =

!"#""+%

−∞

1
Sζ (w) =
2π

+∞
kζ (τ ) e−iwτ dτ.

!"#"",%

−∞

-  !"#""+%( !"#"",% !  .  / 0  % 

*   
*   *   -  1) 
    !"#""2%( !"#""3% /  
*     /
 *   - '

∞
kζ (τ ) = 2

Sζ (w) cos wτ dw,
0

1
Sζ (w) =
π

∞
kζ (τ ) cos wτ dτ.
0

4     
!"% Sζ (w)  0(
!5% Sζ (−w) = Sζ (w)(





!6% D ζ(t) =

+∞


−∞

!"#""2%

Sζ (w) dw = 2

  

∞
0

Sζ (w) dw

 Sζ (w)

!"#""3%

 

 

   
    





      

       ζ(t)  n  
  k   k            n · k
  xi(tj ) i = 1, . . . , n j = 1, . . . , k    

•      ! m
: (tj )  k  "
ζ

n


m
:ζ (tj ) =
•

n

j = 1, . . . , k;

,

#$##%

 S:2(tj )  k  "
ζ

n


:2 (tj ) =
S
ζ
•

xi (tj )

i=1

x2i (tj )

i=1

n


2
− m
:ζ (tj ) ,

j = 1, . . . , k

  & k (t1j ; t2l )  k2  "
ζ

n


k(ζ (t1j ; t2l ) =

xi (t1j ) · xi (t2l )

i=1

n

j = 1, . . . , k,

−m
:ζ (t1j ) · m
:ζ (t2l ),

#$#'$

l = 1, . . . , k.

       m: (t) S:2 (t) K: (t1; t2) (  !
 &     
) *            
         
    
 #$##+ 
   ζ(t) 
ζ

ζ

ζ

           
          
     ! " #  ! " $ %   %
&         
   x(t)  ' %  (    ' ()
  (0; T )"





 

         
             
           k (τ )  τ → ∞ τlim
k (τ ) =
→∞
0
         
           
ζ

m
:ζ =

1
T

T

x(t) dt,

k(ζ (τ ) =

0

1
T −τ

ζ

T −τ


◦

◦

x(t)x(t + τ ) dt,
0

 x(t) = x(t) − m: 
!         " #$   
          [0; T ]    N   
     Δt = T /N  "    
 ti$   %    &       
    
◦

ζ

m
:ζ =
k(ζ (jΔt) =

N
1 
x(ti );
N i=1

N−j
1 
x(ti ) · x(ti+j ) − (m
:ζ )2 ,
N − j i=1

'()(*(+
j = 0, 1, . . . , N − 1.

'()(**+
, '()(*(+    -#$  '()(**+ %  
   -# ,        '()(**+
 j = 0 .  $     j $ "    N 'τ "  
T +$   k( (jΔt)   '   $  
/ N − j +
ζ

        

  

        
 





   º          n1 = 10 n2 = 15
      s21 = 0,98 s22 = 0,56 
                 
            0,05




S1∗ 2
0,98
= 1,75.
=
0,56
S2∗ 2
      k1 = n1 = 10−1 = 9, k2 = n2 −1 = 15−1 =
= 14         F   !
 "# $%  &' (  α = 0,05    !
   F = 2,65
)    F < F  *      ' &+
 &+ 
F



=

  ,           
    !  F "    # $%$$& '  
(         0,05   
(                    
 (      
   
         )        
    #
) 

,
0
2
$

xj

  

 

F1
F2
F3
F4
,
.
,
/
,0
1
,2
3
,$
,
,$
,0
,/
,,
,.
,$
,3
,2
,3
,/
,2*/ ,*2 ,2*3 , */


4 5 n = 5, k = 4 6&          !
   7& " ,%   7& 


x̄ = 457/(4 · 5) = 22,85.







       

          
       
 
 = 246,55 
=
72,15


!
"
#







=
174,4

$


%




#


!     
 &' %  "  ( %) !
 )     #  ! *


174,4
= 10,9.
4 · (5 − 1)
+  (#     F  = 24,05/10,9 ≈ 2, 206. ,
 !   ' k1 = k − 1 = 4 − 1 = 3, k2 = k(n − 1) = 4(5 −
1) = 16.    (#  α = 0,05 -  #   #
∗2
S
=

72,15
= 24,05,
4−1

S ∗2 =

!    ./      ! %  0
F (0,05; 3, 16) = 3,24.

) F  < F       ! (     
# - %     1 (#  # (#  !!' -  (#  



  

           
          
  0,05            
          
  !"!#
N

 




$


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 ϕ(x) = √1

2π


 
 
 




 
 


 
 



 





 
 
 

 


 
 


 
 
 
 

 


 


 
 
 








 
 
 

 
 

 
 

e−x



 
 


 
 
 

 

 
 




2 /2

 
 
 




 











   
 

 
 
 
 



 
 

 
 





































 



 




















 
























































 


 


 

















 





















     



 


 


















 




















 

 
















 






















   
x


































(x)


 
 








 



 

 
 
 
 
 
 
 





 



x









 
 
 
 

 
 


















(x)


 


 
 
 


 


 
 
 
 


 
 
 





 


 



x







 
 
 
 

 
 



 
 
 
 

 
 



 
 
 
 

 

1
(x) = √
2π
(x)


 
 
 


 

 
 
 
 
 

 
 
 
 
 
 
 






 







x
0

x

 






























e−z

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