2.
   2022
/  
  - IV
: 3     :100
:    -    ,        
             
-
1.(a)        -   (b) 
  ,  -    :






2.(a)      ?          
    (b)  :         
:
 :
1979
1980
1981
1982
1983
 : ('000 )
1.6
4.5
13.8
46.2
125.0
-
3.(a)   (b)       
        ?(c)    52     ,
                  
?
4.(a)        
()          
2.
-
5.              
       
6.    ?     
-
7.               
     
8.()     ?          

()       
2.
   2022
/  
  - IV
: 3     :100
:    -    ,        
             
-
1.(a)        -   (b) 
  ,  -    :






:(a)   -  
1. -
 -        
      
            '   
    : -      :
👉" ( )             
  ?"
 :   
          (X₁) 
  :
   (X₂)
 (X₃)
, 
        
  ,     
  
 '          
 -   ,       :
₁.₂₃
(  :  1   2  3   -)
2.
 R₁.₂₃   (1  ),             
       (0  ),          
  

2.  -
 -        
       
       
  :
👉"                 
 ?"
:   :
X₁ = 
X₂ =   
X₃ = 
       '         :
 
        
       '          
 
             
  ,  
 -    
     :
₁₂·₃
(  : 1  2  , 3   )
 :
 - =       
  =           
     
   
       
 
2.
 (Y)    '   :
 (X₁)
 (X₂)
   (X₃)
- 
 
  : "          ?"
 - :Rᵧ.₁₂
 -  
   
   : "            
   ?"
   :rᵧ₂·₁
    '    ,     
()  -   
   : r₁₂ = 0.5r₁₃ = 0.6r₂₃ = 0.7
   -     :
₁.₂₃
₂.₁₃
₃.₁₂
 -   (  )
X  -X   X₃








 :
2.

󰇛
󰇜
󰇛󰇜
󰇛󰇜󰇛󰇜󰇛
󰇜
󰇛󰇜

 





 (approx)
X  -X   X₃









 󰇛󰇜󰇛󰇜󰇛󰇜


 






X  -X   X₂









 󰇛󰇜󰇛󰇜󰇛󰇜


 






2.
 






 

 2  3    1    

 1  3   2     



 1  2   3       
   , X₃           X₁  X₂  
 

  -    :
 -        
 
 -           
 
2.(a)      ?          
    (b)  :         
:
 :
1979
1980
1981
1982
1983
 : ('000 )
1.6
4.5
13.8
46.2
125.0
:          
  
  -  (, ,  ) ,  
               
2.
      '     "  "  "
" :
:   '   ,      ;  ,  
   
"" 
?:
           
  ""  ,      -    
             
( -    ),      :    
   
 , 
 
       :
  
  , ,  .        


-  
  ,        
          
     :
󰇛
󰇜
1.  
o -   :
󰇛
󰇜
    ,   ,    (      )
2.     
o   -    :

󰇛
󰇜󰇛
󰇜

󰇛
󰇜
and


    ?
3.    - 
2.
o  , .
o     
4.     ()
o               
    
  :      
   :
: 1979, 1980, 1981, 1982, 1983
 ('000  ): 1.6, 4.5, 13.8, 46.2, 125.0
  ,    1979     :
1979–1983  
󰇟󰇠.
 1: - 
 ( ):
1979 (x=0):
󰇛󰇜
1980 (x=1):
󰇛󰇜
1981 (x=2):
󰇛󰇜
1982 (x=3):
󰇛󰇜
1983 (x=4):
󰇛󰇜
-   :
x  : 

 : 

Y  : 
    
xY  : 
   

 2: A  B   
  -  :

󰇛
󰇜󰇛
󰇜

󰇛
󰇜
  
 
 
 





  
 


2.
 3: a  b    - 

and


 ,       :
 󰇛󰇜
 4:   (      )
x=0 (1979): 
x=1 (1980): 
x=2 (1981): 

x=3 (1982):
x=4 (1983):
         ,     (
-        )
  
(a)     :     ""  
        : .     ' -
       ,  -   
 
(b)      : -     , 

 :
o ,
o ,
o  : . 󰇛󰇜
                -  -
-   -      
2.
-
3.(a)   (b)       
        ?(c)    52     ,
                  
?
:(a)      
   
        
     

          
 ,   
     
 , "      ?" 
      ?
     
            
              

    ,     ,    1     ,
        ,    0     0  1 
 
  ',       :
    = (   ) / (   
)
         
    ,  6
   (1, 2, 3, 4, 5, 6)  
4      ,   
   (4 )    = 1 / 6
,     :
                
  
()     -     
  
              
  :
            :
  
 
2.
  6
,
      ,   
      
      
  
 A  B   ,  A  B          :
P(A ∩ B) = P(A) × P(B | A)
  :
,  A    ,
,  B        
 A     
       
    
    : "          ?"
   
               


:
    
       

    ( !)
   ,  B     A '  
, :
( | ) = ()
     :
P(A ∩ B) = P(A) × P(B)
 ,   ,       :
2.
        =     

 :          = 1/2  ' 6    
= 1/6
     = 1/2 × 1/6 = 1/12
     
(c)     
           
  : 52        ,       
             
             
 :
4 : , , , 
   13     13  ,  :
1 
1 
  :    = 4 (  , , ,  )    = 4 ( 
, , , )

 :      
     : =    +    = 4 + 4 = 8 
  = 52
  :
 =   /  
 = 8 / 52
  : 8 ÷ 4 = 252 ÷ 4 = 13
 ,
2.
 = 2 / 13
     
     ,    13  2   
   
      
1.            =   / 
   0  1   
2.    
o     A  B :
P(A ∩ B) = P(A) × P(B | A)
3.    
o    :
P(A ∩ B) = P(A) × P(B)
4.  
o  = 4,  = 4 →   = 8
o   = 52
o  = 8 / 52 = 2 / 13
 
     
;       
  
    
   
 ,   ,   
 ,     ,  
          
    ,         
4.(a)        
()          
:🌟-
       ,         
      ,     ,     
  , 
    '   
 '       
    , 
2.

       (PMF)     (PDF)
           
👉  :         
  ,  PMF/PDF            
🌟(a)  :   
📖
              
                

👉:    
"" 0  "" 1  ,    
      
🌟   
    '       :
1.   
    (    )
:
o          (0, 1, 2, …)
o      
o      (16)
👉    '     - 
 -   

2.   
        
:
o       (160.2 , 160.25 ,
   )
o      
o   
👉     '     - 
   ' 
  
    
 
2.
🌟       

  
  

 (   )
,  

 ,   
, , 

PMF    
PDF    

 
 
🌟(b)       
   
         ,   
              PMF  PDF


📊   (PMF)

   (PMF)           
           
  ':
󰇛󰇜󰇛󰇜
...  ?󰇛󰇜

1. 󰇛󰇜   
2.        1   :
󰇛󰇜

    :
󰇛󰇜

👉PMF            
 
📈   (PDF)

2.
   (PDF)          
             
  ':
󰇛󰇜 󰇛󰇜
PDF  ?󰇛󰇜

1. 󰇛󰇜  .
2.      1   :
󰇛󰇜


              ,  PDF  
  10  12        10  12      
👉PDF            , 
  
🌟PMF  PDF   
...
PDF


  󰇛
󰇜
  '  0 ;   
 
 
 
 

   
  
 ' 
📖  
 ,      
           ,      -
 1  6    PMF       
2.
        100      , 
    -  12.01  , 12.02 ,        PDF
            
👉   
   PMF  PDF        - 
  ,    
🌟 
        
PMF  PDF    
 -    '    
    
PMF       ,   PDF    
 '    
   
     1   ,   
   
📊 






    

 
  (0=,
1=)




 
 
...

..
   
, 
PDF
...
   
󰇛󰇜
 
PDF
   
󰇛󰇜
 

🌍 
          
 
           PMF   
,             PDF  
    ,      

 ,   , 
           
2.
-
5.              
       
:🌟   ?
   
   '    
        
      5   ,   7,    3
   ,     '     -  
          
 , 
  '  
  ,          ,  10    
     
   , 
  ,       
    
         
   :
                
        ,      ' 
    ,   -   
  ', x         :
󰇛󰇜

:
λ ()     
x     (0,1,2,3…)
e       2.71828   
x!    x  
          ;    

2.
     

      
           
   
 4       :
       
   
   :
  
    
         
,     
 
   '  
     
      ,     
  

       
     
        :
,   3 
,   5       
  λ   
     
 
 -   ,   
     ,      
🌍      
   ,        '     :
     
        
   
 '    
100      
      
2.
  
"  /    "  ,    
📌    
(      ' )

       -        
       :
        (n   )
        (p   )
       
        
  ',

,
, ,
:
󰇛󰇜

 ,           " " 
🧠   
    
   
 (   )  :
Mean
2.
  λ       
               6 , : 
= 6
   
         :
Variance
  :
Mean Variance
               
   λ = 4,   = 4  = 4
   :
   
    λ '   
 λ   (  1  2),             '
 
 λ   (  5  6),         
 λ  ,              
  - λ  ,            
  
      :
2.
=     '  ,  λ      
=   - λ  ,     
  
    :λ₁  λ₂           
 :
             ,   
 
🎯   
?
        
:
    
        
 , , , ,    
           
 :       
 
      
       

 

       
       
                ,
  ,  ,     ' 
       :
1.              
2.              
3.       :
2.
Mean Variance
     ,         
6.    ?     
:🎲   
🌟-
            
           
   

 ,          -   
👉  :   "    10   ,   6   
  ?"      
  , ,  ,        '
     ,       --

🌟   ?
📖
            
     ,           
  ':
󰇛󰇜󰇡
󰇢
󰇛 󰇜


:
=   
=   
=      
=   
2.

󰇛󰇜
  
🌟   
1.       
2.        ,       
3.      ,        
👉       
/
   
 
🌟   
  
       :
   =
  =
,      ?
1.     :
o :    ,   
o  = .

2.    :
o        
o    = .
3.  :
󰇛󰇜󰇡
󰇢

    
🌟   
        
1.  (  )
     :
󰇛󰇜
:
     =  
2.
  ,     = .
👉:   10  (, )    = . 
2. 
      :
󰇛󰇜󰇛 󰇜
:
     = .
  ,  = . 
👉:   10    = . 
3.  

     
4.   
,   
,     
,     
👉:    ,            
  90% ,       
5.    (MGF)
   MGF  :
󰇛󰇜󰇛
󰇜
   (, , )      
6.  
  , :
󰇛
󰇜
󰇛
󰇜
󰇛
󰇜
2.
👉:               
7.  
 ,       
-        ,     (  )
   
👉           
 
📖  
                
  :     ()       ()   20 
         40%   ,      8
      
 = . 
 = .  
👉 
      - /
     

🌟   
1. :      
2. :      
3. :        
4. :     
5. :      
📊 



...
󰇡
󰇢

  


  


  
 .

  

 
    
...
󰇛

󰇜
   
 
/

🌍 
2.
              -
      
 ,    -    
      - , , ,    - 
   
 
-
7.               
     
:🌍  ?
 ,     
  ,   ,
          
     
 
   :
        =    
      =     
         =    
                
                (   
  ),    (    ,    ' 

    )
👥  ?
   '   
          ?
 
           ?
                  

?     ,      
!
 ,    ,        :
 
   
  ' 
2.
     
   
       ,          
    :
 10,000    , 
 500   
 
      , 
       
      ,
-       
       ,          
,     
   :
 =  
 =              
     (  )


    
   
  
  
    
  
 
     
:    
: 2,000   
📊    
   
     ,  
     
'    :
  
  ( )
                :
   ,   10     ,    -
, ,  , ,       
2.
 :
        
         
   ()
     
       ,    
          
            ,    

,        
        (     ), 
      
           , 
         
    ()
   , , ,      
-           
           ,     
      
         
  ,  
  
            
 , 
   

 !
🎯  ( )
  ,                
  '            
 
   ()
  
         ,      

2.
 
           '     
    
    , ,  -       , 
    
        ,       
      
:
         
       
       :
  
 
   

            
    ()
           
  ,  
   
         ,      

 
       
     ,    
     
🎤    
               
  
      -         ,  , 

   
 ,  
   
 
       
 
  
         
2.

      
        
         ,     ,   
 ,           
  ,           ,
  , ,      '  ' 
         , , ,    
   
8.()     ?          

()       
:📊   
🌟-
     ,       
,   
 '      , 
   (  ,
,  )          
   
    
👉  :             , 
 ( )            
 -   ,
               
   
  ,                 
       
🌟()     ?
📖
                 
   
       ( ,   )
2.
     
  ( ,  ) '   
          '    
  
👉:
:         
 :   

: 50         
🌟    
1.  
o          
o :       
2.  
o      ,   ()      
       
o :    : .
👉   "   "  ,     
" "  
🌟   
       
       
     
1. 
              
  ': .󰇛
󰇜
:         
👉              

2. 
     , -     ,  
    
:    ,         
👉    "   ,     "
2.
3. 
    ,        
  
:      
 ,    
        
👉    " ,  "
4. 
              
 
:              
👉    "    "
5. -    (MSE)
MSE     :
󰇛
󰇜󰇛
󰇜 󰇟󰇛
󰇜󰇠
    MSE   
👉MSE  ()   ()   
🌟  




  =  
   


     
    


      

 

      
    
-
...
  +  
 
🌟()    
📖
2.
  (SE) -         
          -    
👉  :    " "   
  
          
🌟   
1.  (SEM)   :
󰇛󰇜
:
=   
=  
2.    :
󰇛󰇜
󰇛 󰇜
3.       :
󰇛
󰇜
🌟
 SE →     
 SE →     
     SE   (
    

)
👉:
 50     170     SE = 2  , 
            ±2   
   
🌟   
1.   :SE         
2.
2.    :  SE      
o : . 
3.  :SE     (  t-  z-)   
   
4.     : SE    
 
📖  
   
            
  
30             , 
30
                
 
     ( )  
     
            
👉 
           :   
,         
🌟 
        
   , , , ,   MSE    
            

  ,      '       
 
📊 




    

   


   
  170 
 
, , , ,  MSE
 
 
  
  SE =

,  ,  
 SE =  
🌍 
2.
             
       ,         
   
"              
 
       ,       
 "