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   2022
/  
  - 
 : 3   :100
:     ,            
         
-
1.()             
 ()          :






2.(a)     ?         
  ,     (b)     :    

 :
1979
1980
1981
1982
1983
 : ('000 )
1.6
4.5
13.8
46.2
125.0
-
3.()     ()       
         ? ()   52   
                 
      ?
4.()        
()            
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-
5.             
      ,       

6.    ?      
-
7.              
       
8.()   ?          
()      
Easy2Siksha.com
   2022
/  
  - 
 : 3   :100
:     ,            
         
-
1.()             
 ()          :






:(a)       
1.  
                   
           
    :        :
👉" (   )            
  ?"
  :              (X₁)  
  :
  (X₂)
 (X₃)
               ,    
                 
      ,       :
₁.₂₃
(   :  1   2  3    )
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 R₁.₂₃    (1  ) ,           
            (0  ) ,   
             
2.  
                  
      
  :
👉"                ,   
     ?"
:    :
X₁ = 
X₂ =   
X₃ = 
               :
 
        
 -               

                 
 ,        
     :
r₁₂·₃
(   : 1  2   , 3    )
:
  =        
  =               
         
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              
   (Y)     :
 (X₁)
 (X₂)
 (X₃)
  
       "           ?"
   : Rᵧ.₁₂
   
        : "          
       ?"
   :rᵧ₂·₁
    ,   
()     
   : r₁₂ = 0.5, r₁₃ = 0.6, r₂₃ = 0.7
        :
₁.₂₃
₂.₁₃
₃.₁₂
       
X   X   X₃








Easy2Siksha.com
   :

󰇛
󰇜
󰇛󰇜
󰇛󰇜󰇛󰇜󰇛
󰇜
󰇛󰇜

  





 (approx)
X   X   X₃









  󰇛󰇜󰇛󰇜󰇛󰇜


 






X   X   X₂









  󰇛󰇜󰇛󰇜󰇛󰇜


 






Easy2Siksha.com
 






 

 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
Easy2Siksha.com
:        
  
     (, ,  ) ,    
                
        " "  " "
:        ,     ;  ,  
  
?:           
     ""  ,      
       -      
          (    ), 
   :        
 ,         :
  
  , ,         

          ,        

        
   :
󰇛
󰇜
1.    
o       :
󰇛
󰇜
  ,         (  ,  
   )
2.      
o         :
Easy2Siksha.com

󰇛
󰇜󰇛
󰇜

󰇛
󰇜
and


    
3.    
o  , .
o   
4.     ()
o               
    
 :      
   :
: 1979, 1980, 1981, 1982, 1983
 (  ): 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   
     :
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
󰇛
󰇜󰇛
󰇜

󰇛
󰇜
  
 
 
 





  
 


 3: a  b     -

and


,        :
 󰇛󰇜
 4:    (       )
x=0 (1979): 
x=1 (1980): 
x=2 (1981): 

x=3 (1982):
x=4 (1983):
           ,      (
 -        )
  
(a)   :         
       :      
          ,     
 
 
()        :      , 
  :
o ,
o ,
o  : . 󰇛󰇜
   ,          -     -  
    
-
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3.()     ()       
         ? ()   52   
                 
      ?
:()       
                
           ,       
, "       ?"          ?
                  
              
       ,   ,    1  
  ,           ,    0
   0  1    
  ,        :
    = (   ) / (  
)
              ,  6 
   (1, 2, 3, 4, 5, 6)   4      ,    
   (4 )   = 1/6
,     :
                 
 
()       
-                  
  :
               :
     
  6   
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          ,       
    
  
 A  B   ,  A  B            :
P(A ∩ B) = P(A) × P(B | A)
   :
 ,  A     ,
,  A         B        
       
    
    : "         ?"
    
                    
   
:
    
     
     (  !)
   ,  B      A    ,   :
P(B | A) = P(B)
     :
P(A ∩ B) = P(A) × P(B)
 ,    ,       :
          =  - 
 
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  :        = 1/2,   6   
= 1/6
      = 1/2 × 1/6 = 1/12
      
()       
             
  52             
                
             - 
 :
4 : , , , 
   13    13     :
1 
1 
:     = 4 (, ,      )   
 = 4 (, ,     )
 :         
    :    +    = 4 + 4 = 8 
 = 52
 :
 =   / 
 = 8 / 52
  : 8 ÷ 4 = 25, 2 ÷ 4 = 13
,
 = 2 / 13
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            ,      
  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.()        
()            
:🌟
       ,        
    ,            
,                 
         ,      (PMF) 
   (PDF)            
  
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👉  :             ,
 /          
🌟()  :   
📖
               
               
👉:     ""  0  ""  1   ,    
      
🌟   
            :
1.   
      (    )
:
o          (0, 1, 2, …)
o      
o     (16)
👉           -    
2.   
        
:
o      (160.2 , 160.25     )
o        
o   
👉                
      
🌟        

  

, 

, , 

    
 
Easy2Siksha.com

 
🌟(b)       
               ,   
               
📊   (PMF)

   (PMF)           
            
  :
󰇛󰇜󰇛󰇜
  ?󰇛󰇜

1. 󰇛󰇜    
2.        1   :
󰇛󰇜

   :
󰇛󰇜

👉         -      
📈   ()

   (PDF)           
               
  :
Easy2Siksha.com
󰇛󰇜 󰇛󰇜
   ?󰇛󰇜

1. 󰇛󰇜  
2.      1   :
󰇛󰇜


               ,  PDF
    10  12        ,    10  12
   
👉               
  
🌟      



 



  

  󰇛
󰇜
       ; 
    


 

  

   
 
  
  
📖  
 ,      
            ,    
  1  6           

       100       , 
    12.01  , 12.02       
             
Easy2Siksha.com
👉                - 
   ,     
🌟 
       
               
   
        ,     
      
       1   ,    
  
📊 





 
   
 
 
(0=, 1=)

 
 
 

 
   
, 


  

󰇛󰇜
 

  
󰇛󰇜
   

🌍 
               
              , 
                
        ,   ,    
        
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-
5.             
      ,       

:🌟   ?
                 ,  
   5    ,  7,   3    , 
        
               ,     
   ,          ,  10   
      
   ,    ,           
 
     
   :
               
          ,      
      ,       
  ,  x            :
󰇛󰇜

:
λ ()     
x     (0,1,2,3…)
e        2.71828   
x!    x  
          ;      
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      
                
      4         :
 
    
          :
  
  
    :      ,      

     
             ,     
     
        
                :
,  3 
,   5     
  λ  
          
   ,  -    
     ,     
🌍      
     ,           :
    
       
        
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100      
      
   "  /     " ,       
📌     
(       )
         -        
    :
        (n    )
        (p    )
       
        
  ,

,
, ,
:
󰇛󰇜

,            " "
🧠    
      
   
    ()   :
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Mean
  λ       
             6 , :  = 6
   
         :
Variance
   :
Mean Variance
             
  λ = 4,   = 4,  = 4
  :
   
    λ   
 λ   ( 1  2),            

 λ   ( 5  6)  ,         
 λ    ,          
 - λ  ,  
        
  
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     :
=       ,  λ      

 - λ  ,     
 
      ,   λ₁  λ₂     :   
    ,     :
        ,        , 
     
🎯    ?
          :
   
        
 , , , ,      
          
  :       
               


                
              ,   
,         
      :
1.               
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2.            
3.      :
Mean Variance
      ,        
6.    ?      
:🎲   
🌟
             
                 , 
          
👉  :          "     10 
 ,   6       ?"
      , ,  ,     
        ,         
🌟   ?
📖
           -   
   ,          
  :
󰇛󰇜󰇡
󰇢
󰇛 󰇜


:
=   
=   
=      
=   
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
󰇛󰇜
  
🌟   
1.          
2.    ,          
3.      ,        
👉          /      
🌟   
             :
   =
   =
,          ?
1.     :
o :    ,   
o  = .

2.      :
o        
o    = .
3. :
󰇛󰇜󰇡
󰇢

     
🌟    
           
1.  ( )
      ?
󰇛󰇜
:
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     = .
   ,     = .
👉:    10   (, )   =  
2. 
    :
󰇛󰇜󰇛 󰇜
:
     = .
   ,  = .
👉:    10    = .  
3.  

     
4.   
 ,    
  ,   

  
  ,   

  
👉:   ,         
      90% ,        
5.    (MGF)
   MGF :
󰇛󰇜󰇛 
󰇜
    (, , )      
6.  
  , :
󰇛
󰇜
󰇛
󰇜
󰇛
󰇜
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👉:              
 
7.  
-     ,        

-           ,    
      (   )
👉               
📖  
               
 :      ()    ()   20   
         40%  ,      8  
  
 = . 
 = .  
👉         /      

🌟   
1. :     
2. :      
3. :       
4. :     
5. :      
📊 

FORMULA


󰇡
󰇢

  


 


  
 

  

 
 

󰇛

󰇜
    
 
/

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🌍 
                
     ,         
    , ,    
   
-
7.              
       
:🌍  ?
 ,         ,     ,
            
     :
        =    
      =      
          =   
 
                
            (     
),   (   ,         )
👥   ?
                
 ?               ?
                   
 ?     ,         !
,   ,         :
 
   
   
Easy2Siksha.com
     
      ,           
     :
 10,000      ,   500     
      ,       
       ,        
       ,           ,
     
   :
 = 
 =             
 
      (  )


    
   
  
  
       
  
 
        
:    
: 2,000  
📊    
          ,          
  :
  
  ( )
              
 :
Easy2Siksha.com
     10      ,    
  , , ,         
 :
      
         
    ()
   :        ,   
          
    :          ,  
   ,       
    :     (    ),   
   
    :  ,       

         
    ( )
   , , ,      
  :             
      :      ,     
    
               , 
                 
  ,         !
🎯  ( )
  ,               
               
   ()
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 :           ,   
      
                
         
    : , ,       ,
    
-  :        - 
      :
        
      
     :
  
  
   
           
    ( )
         ,      
     ,         
 
             ,    
      
🎤    
                 
                
,  ,        ,     
 
         
    
        
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
              
  ,          ,  
 ,     ,        
   
  ,          , 
, ,          
      , , ,          
 
8.()   ?          
()      
:📊   
🌟
       ,      
  ,              
( ,   )            
   ,   ()  
👉    ,              , 
 ( )            
 ,             
   
  ,                 
         
🌟(a)    ?
📖
               
  
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        (  , 
)
            (,  )
              
   
👉:
:        
:   

: 50         
🌟  
1. 

o            
o :         
2.  
o    ()         
        
o :     : .
👉    " "  ,      "
" 
🌟    
             
   
1. 
             
  : .󰇛
󰇜
:  ,      
👉              
2. 
              
   
:    ,        
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👉    "   ,     "
3. 
  ,          
:        ,      
     
👉    " ,  "
4. 
               
 
:             

👉    "    "
5.     ()
MSE      :
󰇛
󰇜󰇛
󰇜 󰇟󰇛
󰇜󰇠
      
👉  ()   ()     
🌟  




  =  
     

      
    

     
 

      
    
 
  +  
 
🌟()    
📖
  ()              
       -   
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👉  :    " "         
           
🌟  
1.     ():
󰇛󰇜
:
=   
=  
2.    :
󰇛󰇜
󰇛 󰇜
3.        :
󰇛
󰇜
🌟
SE   ,      
SE   ,      
     SE   (       )
👉:
 50     170      (SE) = 2  ,  
          ±2     

🌟   
1.   :SE         
2.    :SE       
o : 
3.  :SE     ( t-  z-) 
    
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4.   : SE    
📖  
                 
 30            ,  30   
                
       ( )   
      ,  
            
👉              :   
,        
🌟 
       
    , , ,      
  
            
 
                  

📊 



 
     
    

 
    
  170 
 
, , , , 

 
 
  
  SE =

,  , 

 SE =  
🌍 
            
         ,       
    
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              
       ,      