2.
   2024
/  
  - IV
: 3     :100
:    -    ,        
             
-
1. (a)    



()      :



3
4
5
6
7
8
9

2
3
5
6
3
2
4

5
6
4
5
6
3
8
2.             :

2001
2003
2004
2006

2
13
35
210
-
3. (a)        
()                ?
4. ()         
2.
()      
-
5.         
6. ()       
(b)  X       P(X = 2) = 9, P(X = 4) = 90 P(X = 1) X  
   
-
7. ()      
()       
8.        
2.
   2024
/  
  - IV
: 3     :100
:    -    ,        
             
-
1. (a)    



()      :



3
4
5
6
7
8
9

2
3
5
6
3
2
4

5
6
4
5
6
3
8
:󹶆󹶚󹶈󹶉  ( )
    -   -  
     :
 ():  ()
   :

       :

   -        
2.
 ():  
       :
   :
1.  - 

2. - 

󹺢       ( )
󷄧󷄫 - ()
      :   
󷄧󷄬 - ()

          
󷷑󷷒󷷓󷷔:

 1   2  
 3   
󷄧󷄭- ()

             '  
󷷑󷷒󷷓󷷔:

 1  
 2  3   
󽆐󽆑󽆒󽆓󽆔󽆕 (a): 
2.
 :

  :

󼩏󼩐󼩑 ( )
1     -       
 ,   1    2  3      


            
     ,            
   
󷷑󷷒󷷓󷷔 :
  1    2  3 '  ,
  2  1  3 '       
󷄧󼿒  

   
󼪔󼪕󼪖󼪗󼪘󼪙 ():  
  

3
4
5
6
7
8
9

2
3
5
6
3
2
4

5
6
4
5
6
3
8
:
2.
 1 =
 2 =
 3 =
󷄧󹻘󹻙󹻚󹻛 1:  -    
       (  ):









󷄧󹻘󹻙󹻚󹻛 2:  -   

:




󰇛

󰇜󰇛

󰇜
 :

󰇛󰇜󰇛󰇜
󰇛

󰇜󰇛
󰇜


󰇛
󰇜󰇛
󰇜




󷄧󼿒:


󷷑󷷒󷷓󷷔:
    ,
       
2.
󷄧󹻘󹻙󹻚󹻛 3:  -   

:








:

󰇛
󰇜
󰇛󰇜
󰇛󰇜󰇛󰇜󰇛
󰇜
󰇛󰇜





󷄧󼿒:


󷷑󷷒󷷓󷷔:
         
2.             :

2001
2003
2004
2006

2
13
35
210
:󷋇󷋈󷋉󷋊󷋋󷋌 
        :

2001
2002
2003
2004
2005
2006

2
5
13
35
85
210
2.
  ',     
   -        
   2  210 !    
,    
   :          
   

        
󹵈󹵉󹵊       ?
      '     :

:
  () ,
  () ,
  
  ,    -  
      
   

""  :
           ,     

󼪔󼪕󼪖󼪗󼪘󼪙--  
 1:    
   , 
  (2001, 2002, )   
 
, 
 :
for 2001 for 2002 for 2006
      :
x ( )
1
2
3
4
5
6
y ()
2
5
13
35
85
210
 2:  

     , 
     
 

2.
        :

:
,
,
.
 3:    
   (   10    ):


 
1
2
0.301
2
5
0.699
3
13
.
4
35
.
5
85
.
6
210
2.322
 4:     
,        -   (      
 )      

  :





   :



 .
 :
1. 
2. 
2.
 5:   
 (1) :

 (2)  :



 :



 :
󰇛󰇜

 6:     
 :








      :
󰇛󰇜
󷈷󷈸󷈹󷈺󷈻󷈼
    :
   ()  0.89 (1  ) 
 ,   2.24    
󷷑󷷒󷷓󷷔: 2001  ():
󰇛󰇜
    !
2.
2006  ():
󰇛󰇜

 !
󷘹󷘴󷘵󷘶󷘷󷘸   
 ,            "  " 
   " "   ,       :
󰇛󰇜
                  
          
󽆪󽆫󽆬 
            ,      
  ,            
   
 -      ,  
      
(2001  2 )        (2006  210 ) 
     
-
3. (a)        
()                ?
:-
                 
 , 
           
  ,               
 ,             
               
       , 
       
,          '  ,   
  
2.
 (a):    
  
      :

A

B



,

A

B








:
󰇛󰇜󰇛󰇜󰇛󰇜
 ,   :
󷷑󷷒󷷓󷷔       =     ×    

      
  (-)
  
            
     
 ,      '   
        
   
           

  :
󰇛󰇜
󰇛󰇜
󰇛󰇜
(provided 󰇛󰇜)
       :󰇛󰇜
󰇛󰇜󰇛󰇜󰇛󰇜
 :
2.
󰇛󰇜󰇛󰇜󰇛󰇜
   
 :  
  A  B  ,  A    B  

 ,
󰇛󰇜󰇛󰇜
 ,
󰇛󰇜󰇛󰇜󰇛󰇜
       
 (b):         
  
    52   
   = 4
     
      = 4
-- 

   , 
        

 1:       
52   4  
2.
󰇛1st ace󰇜

 2:       
    :
   = 51
   = 3
󰇛2nd ace 1st ace󰇜

 3:       
:
   = 50
   = 2
󰇛3rd ace first two aces󰇜

 4:       
:
   = 49
   = 1
󰇛4th ace first three aces󰇜

 5:    
󰇛all four aces󰇜




2.
    :





 
󰇛all four cards are aces󰇜


               
       ,        
  , 
       '      
  , 
  --      '  
      ,      
,   
   
        
       

 
4. ()         
()      
:(a)         
 
    ,  
         
      ,        '
     
     ',       :
1.   
2.   
        , 
 -     :
   (PMF)
   (PDF)
2.
   -         '  
   (PMF)
             ,  
     
 
PMF               

  
    
 :  (H)    (T)
   
 
     :
X =   
:
( = 0) = ½
( = 1) = ½
,              PMF   
   
      
      
    1   
  0  1    
 , PMF   :
"       X       ?"
   (PDF)
2.
             ,  
         
 
PDF        ' 
,         

      '  
   ( , 160.2 , 160.25 , )

   
 
   :
     160      ?
  , 
      
     '    , 
     , 
:
(160  < 165 )
       PDF          
PDF   
     
  '   
   '   
     1 
 , PDF :
"    X       ?"
PMF  PDF   

.. .
PDF
  


-  


  ' 
-

2.

 
  

, 
, , 
()  -   
     :        ,    
   ,         
  ?
 ,               

  :
             ,     , 
   -    
             
  
  '      :
1.   (  )
2.   (  )
1.   (  )
           
   
     
      
            

2.
2.   (  )
            
     
        -   
  
(i)   
   

       
(ii)   
  
      
(iii)   
  
     :
o   ' 
o   ' 
o 
(iv)   
  

      '   
 
 ?
    :
    
   
     
-    
   
  ,     
2.
   
          :
  
     
     
            

  ,          
         PMF     ,
  PDF  '      ,     ,  
              
  ,              
              
-
5.         
:        :       
  ,        -         
      :   ,   
,  
-,         
      
  , 
       ,     
 --              
    
1.    ?
2.
           
          , :
1.          (  )
2.          
3.       

  n =   
  p =   
  q = 1 − p =   
n    x       :
󰇛󰇜󰇡
󰇢

     ,          
 
2.     ()
             
  
   
-- :
     ,     
       
 
n   ,        
   

      p   
 ,
Mean 
:
 
  10         0.5 , :
2.
Mean 
 , ,  5      
3.     ()
        -   
  :
 =󰇛󰇜
n   ,      ,
Variance 
 
  :
 p = 0  1,   0      
        p = 0.5  ,     
4.     
       
Standard Deviation

      ,        
5.    (MGF)   
   (MGF)          
       
  , MGF  :
󰇛󰇜󰇛
󰇜
2.
  :
  '  = np  
   = npq      
    ' MGF       
,
    MGF     ,    
6.    
      
 p = q = 0.5 ,    
 p < 0.5 ,     '   (   )
 p > 0.5 ,     '   (   )
  ',     :
Skewness

7.    
    
  ,    :


 n   ,       
            
 
8.    
       
2.
Mode 󰇛󰇜
         
 
 
  -              

9.   (   )
         :
󰇛󰇜
󰇛󰇜
󰇛󰇜
󰇛󰇜
   -     , --   
    
10. 
           
  
          -  
 
          , 
    (np),
 (npq),  , ,       
             - 
  ,            
   '    ,         
          
6. ()       
(b)  X       P(X = 2) = 9, P(X = 4) = 90 P(X = 1) X  
   
: ()       
2.
-     ""         
  -         -   - - 
   -   
   
 
o :.
󰇛
󰇜
o :       ( ,  
    )
 
o :.
󰇛
󰇜
o :  , ,        
 ( ,    -th    )
  
󰇛󰇜
o : .
o :
󰇛󰇜
󰇛󰇜

󰇛󰇜


   ?󰇛󰇜󰇛󰇜
󰇛󰇜󰇛󰇜
󰇛󰇜
󰇛󰇜(-   )
o : (), ()
o :
󰇛󰇜
󰇛󰇜



  :  󰇛󰇜
   

o : .󰇟󰇠

o : .󰇛󰇜

󰇛
󰇜
󰇛
󰇜
o : ' ; U- ,  (), ,     

󰇛
󰇜

2.
o : (   )󰇟󰇠
o : ( )󰇛󰇜

o :   -;   -      
      .
    
 /        
 ""     
              -
   ""     
 :    
  ;        

        ,   -     
 
󰇛
󰇜󰇛
󰇜
 ()       :   
     ,    
󰇛󰇜


   
󰇟󰇠 󰇛󰇜 
   :
󰇛󰇜,
󰇛󰇜󰇛󰇜.
            '     
 
 1:          

      
     :

 :
󰇛󰇜
󰇛󰇜




  :󰇛󰇜󰇛󰇜
2.
󰇛󰇜
󰇛󰇜




 



 
     ,  
󰇟󰇠 󰇛󰇜 
 2:       
    '  
 , 
  1     
 , ,    1        ,    ,  
 󰇛󰇜󰇛󰇜

  -   ,     ( ,
)       ,            
  
  '      ,      -
   ;  
      ,     
  ( ,    ?)󰇛󰇜󰇛󰇜󰇛󰇜󰇛󰇜
󰇛󰇜
 3:         
,       , 
󰇛󰇜󰇛󰇜

 ,
󰇟󰇠 󰇛󰇜 
          (     ),  
    
           
󰇛󰇜
-
7. ()      
()       
:-
2.
 , 
              
 '    ,        
,  
               
                
 
              
 ,                
  
    ,         
           
(a)    
   
              
         ,  
      

  
              
( )     ,     '  
  
               
      
󷷑󷷒󷷓󷷔    =  
󷷑󷷒󷷓󷷔   =   
 
?
  ,         :
         
      
     
   ,      
  
             
 
2.
          '       :
  6      = 70
  = 65
         
   
           ,     
  

    ,         
    ,         
 :
  = 2     
  = 15     
 ,              

     
1.     
2.          
3.     
 
4. , ,       ' 
 
5.        

    '   
    

 (  )
       
    ,          
2.
()      
     ,          ' 
       


         
    
   

    
   
          
           ,     ,
    

          ,         


     200 
   1,000   
     50 
         
     ,    

    
        
 ,      
        
 
 :
2.
     
        
     




      
  

 





 
  

 
  

 
  

   
1,000     

 
 ?
1.  
 
2.  
 
3.   
4.  
5.       
       ,            
   
    
        ,           
    ,     

               
             
-         
   ,    
                 

 ,                
    ,       , 
2.
        ,      
      
8.        
:󷋇󷋈󷋉󷋊󷋋󷋌  
   
              
,         , 
  ? 
-     
,     ,        
   

        ( )        
     
 
:        
   ( ,    , 
    )
:         ( , 50 , 100 )
󷷑󷷒󷷓󷷔   :      , 
      
  ,         
 
 
?
   
 
         
 
     
󹵍󹵉󹵎󹵏󹵐  
            ,    
  
1.  
                 
2.
:      500  ,  
     
' 50   ,      
: ,   
:            
2.  
, 
     k-th     
:     1,000     
 10   
  ,   100   
:          
 
:       ,       
3.   
 , ,       '   (  
 )              
:   ,  ,     
      
60    ,  
   20  

:       
 
:      
4.  
   , 
    ()  
:   
         
    , 
   ( )        
   
:          
 
:     
        
 
5.  
    '   
          
:           
    
:   
:   ,     
2.
6.  
, 
  
             
:    40%   60%     
 
    
    
 
:        
 
:       
      
7.  () 
                
:  
      , 
  -
    ,     
:    
:    '     ,      
󷈷󷈸󷈹󷈺󷈻󷈼   
 ,            ""      
     
    :
:    
:  k-th 
:    ,    
:    
:       
:      
:    '
󽆪󽆫󽆬 
         ,        
         '        , 
  ,             
"              
 
       ,       
 "