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   2023
/  
  - 
 : 3   :
:     ,           
           
-
1.(a)            (b)  
     ?        
   








2.    ,        :
X₁
3
5
6
12
14
2
X₂
10
12
7
6
8
3
X₃
9
10
54
42
30
12
-
3.()          ()    
  ?       
4.()            ()   
   ?        
      
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-
5.    ?       -  
     
6.            ?  
                

-
7.()              
 
()          ?
8.()           
   (    )   
     ?
()        
Easy2Siksha.com
   2023
/  
  - 
 : 3   :
:     ,           
           
-
1.(a)            (b)  
     ?        
   








:(a)        
 ,               
          -,      ,
  -,               
        
    ,   
      ,         ,
       :

   ""        ,     'a'
          , ,   
       
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         ?
     : "a, b  c           
    "
     ,          ,
        -         
-- :
  

    
    

    
 
         :


  
         ,         
  
   ,      :
(   )
antilog󰇛󰇜
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 antilog󰇛󰇜
     :

       
,  -     ,       ,  
      ,           
 !
()     
         ,        :
,         
 ,      
       ,       
    X₁  X₂      ,     X₃  ,  
     X₁  X₂           
    X₃    
 ,  X₃            X₁  X₂      
             :
  
  ,

    3       ,  1  2   
   ,               
 

   
Easy2Siksha.com




󰇛

󰇜󰇛

󰇜
 :






-- 




󰇛󰇜  

󰇛

󰇜󰇛

󰇜
󰇛
󰇜󰇛
󰇜
󰇛
󰇜󰇛
󰇜
 
 




 ,        ,  1  2      (0.14) 
        (0.5)     3   
  
                  
      :
            ?
           ?
          
    :

Easy2Siksha.com
    1   2  3    
FORMULA








 








󰇛󰇜󰇛󰇜󰇛󰇜 󰇛󰇜 

 


 






 
  1   (2  3)       0.707 ,    
  
 (  )
              
     -           
,      ,      
               ""

Easy2Siksha.com
                 
 
  :




 ,            ,      
    
2.    ,        :
X₁
3
5
6
12
14
2
X₂
10
12
7
6
8
3
X₃
9
10
54
42
30
12
:🌟  
    -          ,
     ( )       (  )    
     
👉  :              ,  
          
🌟
   :

1
3
10
9
2
5
12
10
3
6
7
54
4
12
6
42
5
14
8
30
6
2
3
12
,   ,    
🌟 
Easy2Siksha.com
 -        :
=  ( )
= 
= 
    ,     :


🌟  
1.  ()

o           
o  ,        
2.  ()

o     ,  0.27     ,  

o       
3.  ()

o     ,  0.15     ,  

o       ,     
🌟   ( 
)
- = 0.317
o            31.7%    
 
o              
 - = -0.138
o        (  6 ),   -
  ,           

- = 0.6974, - = 0.564
o            
o       
👉    :       ,     
,         
🌟   -- 
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1.    :
o  :
o      :
2.   :
o  :
3.     :
o            
o            
4.   :
o
, , .


5.   :
o


.
🌟 
Imagine    ,     ,      
          1        0.27
    
     1        0.15    
      ,   0.94  ()    
                
     
🌟  
   :  6 ,        
 -:   31.7%     
    :-  ,        

👉    ,           
🌟
       :


              ,  
        
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-
3.()          ()    
  ?       
:()  , ,   
                 
     :    ,  ,     ?
            (   
    ,     , )      
     
            ()    
 ()        
,   :
   
    :
"           ,     
  ,               
       "
    :
                  
 
    
    :  A 60%     B 40%  
2%     5%    
           ,    
        :
👉             ?”
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          
    
 
  B₁, B₂, B₃, …, Bₙ             (
                  )
   A      B   
,     :
󰇛
󰇜
󰇛
󰇜󰇛
󰇜
󰇛

󰇜󰇛
󰇜
  ,  A       B      =
Prior probability of Bi Probability of A given Bi
Total probability of A
    (    )
       :
󰇛
󰇜
󰇛
󰇜
󰇛󰇜
,
󰇛
󰇜 󰇛
󰇜󰇛
󰇜
    :
󰇛
󰇜
󰇛
󰇜󰇛
󰇜
󰇛󰇜
  P(A)     A, B₁  B₂  Bₙ      ,
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󰇛󰇜 󰇛
󰇜󰇛
󰇜󰇛
󰇜󰇛
󰇜󰇛
󰇜󰇛
󰇜
          
󰇛
󰇜
󰇛
󰇜󰇛
󰇜
󰇛

󰇜󰇛
󰇜
       !
       
 
 
   
                
      ,  ,  , 
,          
()    
      
  
       ,        
          ,    :
👉,         ?
       
  :
               
  
    X  x₁, x₂, x₃, …, xₙ      p₁, p₂, p₃, …, pₙ ,
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  ,
󰇛󰇜
  = Σ ( × )

   ,    ₹2      ₹1 
  =
󰇛󰇜󰇛󰇜󰇛󰇜󰇛󰇜  
 ,    ₹0.5   
  
        
1.   
 X  Y     a, b   :
󰇛󰇜 󰇛󰇜󰇛󰇜
:              
      
2.    
 C    , :
󰇛󰇜
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   ,      
3.   
󰇛󰇜 󰇛󰇜󰇛󰇜
           
4.   
󰇛󰇜 󰇛󰇜󰇛󰇜
    
5.   
 X  Y  ,
󰇛󰇜 󰇛󰇜󰇛󰇜
  ,       X  Y  
6.      
                  
        ,        
 
    ,      
'              
                 

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             
   " "            
 
4.()            ()   
   ?        
      
:   
   

                
       ,         
,   :
󰇛
󰇜 󰇛󰇜 
󰇛
󰇜
󰇛
󰇜󰇛
󰇜
󰇛

󰇜󰇛
󰇜
 :  =  ×  /           
     ;          


              
   :
󰇛
󰇜
󰇛
󰇜
󰇛󰇜
       :
󰇛
󰇜 󰇛
󰇜󰇛
󰇜
Easy2Siksha.com
         :   
,
󰇛󰇜 󰇛

󰇜 󰇛

󰇜󰇛
󰇜
   :
󰇛
󰇜
󰇛
󰇜󰇛
󰇜
󰇛

󰇜󰇛
󰇜
   
 ,  
          , ,       
( )  , ,            ()    
        ?
󰇛defect
󰇜
󰇛defect
󰇜 󰇛defect
󰇜 
:.󰇛
󰇜 
:.󰇛
󰇜 
:
󰇛󰇜  
 :
󰇛
󰇜



,        ,        
            :   
 
 
  
  (  )          
          
        :󰇛󰇜 󰇛 󰇜
Easy2Siksha.com
󰇟󰇠 󰇛󰇜
    ,     :󰇛󰇜
󰇟󰇠 󰇛󰇜

  ,              
   ( )  
    
 :         
:        
 :  (,     )     
   

  :       ,
󰇟󰇠 󰇟󰇠󰇟󰇠
               
 
   
    :  - ,
󰇟󰇠 
  
         :     
(   1,  0),

󰇟
󰇠 󰇛󰇜
          
  
Easy2Siksha.com
        :   ,
󰇟󰇠 󰇟󰇠󰇟󰇠
  ,      
 
  ()
       :   ,
󰇟󰇛󰇜󰇠 󰇛󰇜󰇛󰇜
(discrete)󰇟󰇛󰇜󰇠 󰇛󰇜󰇛󰇜

(continuous)
           󰇛󰇜󰇟󰇛󰇜󰇠
 
  :   (  )        
   󰇟 󰇠
  (  ):
󰇟󰇟 󰇠󰇠 󰇟󰇠
           
  
         :  ,
󰇛󰇟󰇠󰇜 󰇟󰇛󰇜󰇠
  ,       ()     
  
,       :
󰇛󰇜 󰇟󰇛󰇟󰇠󰇜
󰇠 󰇟
󰇠󰇛󰇟󰇠󰇜
        
  
      0.3    ₹10, 0.6    ₹2   0 
 
Easy2Siksha.com
 :
󰇟󰇠   
    ₹4 ,          ,   
     ,       
      
      ()    
     
             
         (  ,   
 )
 ,         "  "   ,  
      "   "   

             :  
    ,     ()        ,
              
        ,       , 
    ,        
-
5.    ?       -  
     
:🌟   ?
       100           
?  !       ,   ,  
     
        :
  (X-) 
  (-)    
Easy2Siksha.com
     
     ,     ,       
   -   ,  (  )    ,   -
    
 , ,         :
 

 
 
 (     )🔔)
     :
👉             , 
       
               
      ,  :
  
 ( )
   
 
 
    
          
🎨   -  (     !)
       ,       :
Easy2Siksha.com
:
    
     
      
     X-    
         ()   
📏  (    )
          ,   μ   
σ          :
󰇛󰇜

󰇡󰇜

  😄
  :              
     (   )
              
   
  

     
        
   :
  =  
                
       ,         !
          
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 =  = 
  , ,         -  
   :
👉 )
👉 ( )
👉 (     )
         
           
     
      ,          ()
:
       
      
        )
  (σ)        
 σ →     (        )
 σ →     (   )
 σ   ""    😄
     = 1
    :
👉     = 100%
        
Easy2Siksha.com
 689599.7 📊
(      )
           
   :
68%   ±1       (μ − σ  μ + σ  )
95% ±2       (μ − 2σ  μ + 2σ  )
99.7% ±3       (μ − 3σ  μ + 3σ  )
    :         !
:    = 170   σ = 5  , :
68%  165  175      
95%   160  180  
99.7%  155  185  
  ?
     
      X-    
            
   :
👉     
  :           
    σ)
          :
μ →  (  )
σ →   ()
Easy2Siksha.com
       ,         !
 
      ,   :
    
       
     : 170.2 , 170.25 , 170.253 
    
🎯     ?
             

:
       
   
      
    
   
         
    "   "         , 
      
6.            ?  
                

:🌟 : , ,   
🌟
Easy2Siksha.com
             
              
       "   "        
         -   ,     
👉    :             , "
          ?"  "    
 ?"
🌟   
     (lambda)          
      (PMF)      :
󰇛 󰇜


:
            
    ( 2.718)
  
👉:        4    ,    
 2      ?
󰇛 󰇜






🌟   
       :
1.  
  :      👉:   
        
2.       :        
 👉:        
3.         :      
     👉:       

4.       :           
  
Easy2Siksha.com
👉 :           ,    
     
🌟          
             
  :
󰇛 󰇜 󰇡
󰇢
󰇛󰇜


               :
 1: 
       ( )
      ( )
   (   )
 2:     
󰇛 󰇜

󰇛󰇜
󰇛󰇜

 3:   
, ,  .  
󰇜


         
 4: 
󰇛 󰇜

     
👉:       ,      
""   
🌟   
1.   
 () =
 () =
Easy2Siksha.com
👉 :        
2.  
  , :
Poisson󰇛
󰇜
Poisson󰇛
󰇜
Poisson󰇛
󰇜
👉:       3          5,  
   8      ,   (8)     
3.    ()
         ,      
   :             
 
4.   
 =
 =
👉   ,             
5.   
   ,          :
󰇛󰇜
👉:  ,            

6.
  
              , :
    
         
      
🌟     
1.  :         
2.  :       
Easy2Siksha.com
3. :    
4. :        
5. :          
👉              
📖  
                 ,
  -             
  ,         10     
            8, 10  12     
     
🌟
        ,    ,   
                 
               
   ,          
 
-
7.()              
 
()          ?
: ():           
                  
      :
                 
      
         
   ?
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              
  :
 =    
  :
   (        )
        
          
           
   ?
            ,    
 ,    (   )         
 
  :
 =         
  :
             , 
  
            
              
  
        ,       
      
          :
1. 
:       
:       
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2.  
:       10      
    
:             
3. 
:   , , ,     

:         
4. 
:              
 -,           
:              
       ,      
5. 
:             
: 
  ,  ,    
:          /  
:  ,  ,  
6.   
-         (     )
:  ,     
:   ,        
 
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  
         
     (  )  
   
          
            
   
 :
            
              

   ,   ,      
 ():         ?
         
      -  :
  
,     
 ,  ,    
               ,    
     :
      
          
      
        
     (Stratified Random Sampling)  
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    ?
   ,        '' 
             
  :          
      
    
     --  :
 1:    
 ,           :  
  
 2:       
     
       :

 / 
 
(/)
 
      :


    
  :         :
1  
 
 
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         
 3:     
    300    
 4:        
   :
 :      
 :         :    
       ,       
      
 5:      
,                 :
 
  
 
       
 6:      
 ,              
    ?
  :
    
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    
    
     
   
        :
60%  
40%  
 100            :
     :   
    60   40  
         
         ,         80 
  20       

  ,           
          ,       
  ,              
             
               
  
8.()           
   (    )   
     ?
()        
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:🌟
 ,              
         ,     
                
                
   ,             
       -,   -  
              
👉    :          ,     
      
🌟()     
1.  
:         
 :,     
:            
:
o    
o        
o           
👉           ,  
 
2.  
:  ,          
 :   ,       
:          ""  
   
:
o      
o       -   
:
o    
o        
👉          -    
 ,   
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 
 
 

 


      
 
   
 
🌟   
             
                
   :     
1.   ()
:
1.          
2.   ,            

3.                
:    10      3       
 SRSWR     ,        ,   
      ,       
:
o   
o     
👉                  
       
2.    (SRSWOR)
:
1.          
2.      ,         
3.                 
:10         3        
         ,      
:
o      
o      
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👉                 
     
SRSWR  SRSWOR  

  ()
  (SRSWOR)

 
  

    
  
 



   
  
🌟()    

  ()         (  )  
      
👉    :             
 
     
        :

     ,      (   
)
     ,        (  
 )
   
1.   :SE             
 
2.    :        
   
3.  :SE   t-  z-      
  
4.   :SE              
   
   
Easy2Siksha.com
1.       :   SE    
2.       :   SE  

3.   :SE    ,   
4.    :         
🌟     
                30
         160       
                 
      
 SE = 2  ,           ±2   
   
 SE = 10  ,        ,   
  
👉 ,       "  "    
📖  
           
 :            
-       
 :         
          
 :             
    
🌟
         
 ,          
    
            
,    -  
         -  
   
             
       ,       