2.
   2023
/  
  - IV
: 3     :
:    -    ,        
             
-
1.(a)             (b)  -
     ?      -   - 
 :








2.    , on         :

3
5
6
12
14
2
X₂
10
12
7
6
8
3

9
10
54
42
30
12
-
3.(a)         (b)      
?        
4.()            ()  
    ?           
      
2.
-
5.    ?            
   
6.           ?  
                
-
7.()             
 
() 
         ?
8.()         
  
 (      )        
?
()         
2.
   2023
/  
  - IV
: 3     :
:    -    ,        
             
-
1.(a)             (b)  -
     ?      -   - 
 :








:(a)           
 , 
    -       
    
        ,   
  
,    ,   ,       
          
     ,    
   
  ,         
,    '     :

  ""    
      ,     'a' 
           , ,    
    
 
2.

          ?
    : "a, b,  c             
 
 "

    
,   -        
,        -         
-- :
   

    
     

  '  
 
   -    :


  
 
-      ,       
   ,      :
(  )
antilog󰇛󰇜
 antilog󰇛󰇜
2.
      :

         
 ,   -      , 
  
 ,         ,        
 !
()  -   
        ,        
 :
 ,      '   
 ,    '   
          '    
  
X₁  X₂     ,     X₃ ,  
   X₁  X₂        
    
  X₃     
 , 
X₃   
        X₁  X₂   
              :
 - 
  ',

 3      ,  1  2     
  ,               


 -  
2.




󰇛

󰇜󰇛

󰇜
 :






-- 




󰇛󰇜  

󰇛

󰇜󰇛

󰇜
󰇛
󰇜󰇛
󰇜
󰇛
󰇜󰇛
󰇜
 
 




 ,       ,  1  2   
    (0.14)       (0.5)   3 
 
 - 
    
           
   
   :
         
 ?
        
 ?
  -       
   :
2.

    1  2  3   -









 








󰇛󰇜󰇛󰇜󰇛󰇜 󰇛󰇜 

 


 
 





 
   1     (2  3)  - 0.707 ,    
   
 
 ( )
              
  -      
     , - 

 ,     
2.
            ""  

-             
 
  :




 ,           ,     
       
2.    , on         :

3
5
6
12
14
2
X₂
10
12
7
6
8
3

9
10
54
42
30
12
:🌟 
 on             
 ,      ()       
 ( )        
👉  : 
       
     ,
            
🌟
   :

1
3
10
9
2
5
12
10
3
6
7
54
4
12
6
42
5
14
8
30
6
2
3
12
2.
,   ,      
🌟 
         :
=  ( )
=  
=  
    ,     :


🌟  
1.  ()

o             
o  ,       
 
2. ()  

o     ,   ,  0.27   
o       
 
3. ()  

o     ,   ,  0.15   
o      
 ,     
🌟  ( )
- = 0.317
o       31.7%    
 
o            
  R- = -0.138
o 
       ( 6 ),   R-
 ,  
       
 
F- = 0.6974, p- = 0.564
o     '   '  

o         
👉  :     '   ,      , 
     
 
🌟  -- 
2.
1.    :
o  :
o  :
2.   :
o  :
3.     :
o      -    
o          
4.   :
o
, , .


5.   :
o


.
🌟 
  
 ,   
 ,    

 
        1     0.27
  
    1      0.15   
     ,   0.94  ()  
 
 
       ' -     
  
🌟  
  : 6 ,        
 R-:   31.7%     
  '  
:-  ,       

👉-  ,        
🌟
     :


  
       '   ,   
       
2.
-
3.(a)         (b)      
?        
:(a)    - , ,   
   
             

      :    ,  ,   
  ?
  

 
       (   
'     ,    , )  
   
       
             
 () 
      ( )       
,   :
    
     :
" 
  -      ,     
   ,       
   '  
        "
    :
 
              

    
     :  A 60%  
   B 40%  
 
 A 2%       B 5%     
 
          ,     
     :
👉"           ?"
2.
  
      
      
 
  B₁, B₂, B₃, …, Bₙ            
  (
               
 )
   A       B '   
,     :
󰇛
󰇜
󰇛
󰇜󰇛
󰇜
󰇛

󰇜󰇛
󰇜
  ,  A      Bi     =
Prior probability of Bi Probability of A given Bi
Total probability of A
     (  )

      :
󰇛
󰇜
󰇛
󰇜
󰇛󰇜
,
󰇛
󰇜 󰇛
󰇜󰇛
󰇜
   :
󰇛
󰇜
󰇛
󰇜󰇛
󰇜
󰇛󰇜
  P(A)   
 A B₁  B₂   Bₙ     ,
2.
󰇛󰇜 󰇛
󰇜󰇛
󰇜󰇛
󰇜󰇛
󰇜󰇛
󰇜󰇛
󰇜
        
󰇛
󰇜
󰇛
󰇜󰇛
󰇜
󰇛

󰇜󰇛
󰇜
       !
     
    
 
  
    
      
         
         ,   , 
,  ,     -     

()   -  
     
   
  
     ,       
    
       
     ,  
 :
👉",            ?"
      
  :
          
     
  
2.
    X  x₁, x₂, x₃, …, xₙ   p₁, p₂, p₃, …, pₙ 
 ,
  ,
󰇛󰇜
   = Σ ( × )

       ' ₹2 ,   ' ₹1 
  =
󰇛󰇜 󰇛󰇜󰇛󰇜󰇛󰇜  
  ,    ₹0.5    
  
        
1.   
 X  Y     a, b   :
󰇛󰇜 󰇛󰇜󰇛󰇜
:        
      
   
2.    
 C    , :
2.
󰇛󰇜

  
,      
3.   
󰇛󰇜 󰇛󰇜󰇛󰇜
    
     

4.   
󰇛󰇜 󰇛󰇜󰇛󰇜
   
5.   
 X  Y  ,
󰇛󰇜 󰇛󰇜󰇛󰇜
  ,         X  Y   
6.      
 
              ""   
     
,       
 
            
2.
' 
             
             '   
               
  " "            
 
 
4.()            ()  
    ?           
      
:     
    

               
  '        ,        
,  :
󰇛
󰇜 󰇛󰇜 
󰇛
󰇜
󰇛
󰇜󰇛
󰇜
󰇛

󰇜󰇛
󰇜
 :  =  ×  /    
     
    ;           


             
    :
󰇛
󰇜
󰇛
󰇜
󰇛󰇜
       :
󰇛
󰇜 󰇛
󰇜󰇛
󰇜
2.
          :
 
,
󰇛󰇜 󰇛

󰇜 󰇛

󰇜󰇛
󰇜
 :
󰇛
󰇜
󰇛
󰇜󰇛
󰇜
󰇛

󰇜󰇛
󰇜
    
 ,  
     , ,  ,          
 () , , ,  
         ()    
       ?
󰇛defect
󰇜
󰇛defect
󰇜 󰇛defect
󰇜 
:.󰇛
󰇜 
:.󰇛
󰇜 
:
󰇛󰇜   
:
󰇛
󰇜



 ,       ,     
        
      :  
  
 
  
  (    )           
      '  
        :󰇛󰇜 󰇛 󰇜
2.
󰇟󰇠 󰇛󰇜
        :󰇛󰇜
󰇟󰇠 󰇛󰇜

 ,            
  (   )
 
  
 :        
:        
 
 
:   ( ,      ) 
 '   
  

  :      ,
󰇟󰇠 󰇟󰇠󰇟󰇠
           
    
  
 
   
   : - ,
󰇟󰇠 
 
    :     ( 1  , 
 0),

󰇟
󰇠 󰇛󰇜
         
  
2.
     :   ,
󰇟󰇠 󰇟󰇠󰇟󰇠
  ,      

     (LOTUS)
     :  ,
󰇟󰇛󰇜󰇠 󰇛󰇜󰇛󰇜
(discrete)󰇟󰇛󰇜󰇠 󰇛󰇜󰇛󰇜

(continuous)
      
󰇛󰇜󰇟󰇛󰇜󰇠
 
   :   (  )      
  
 󰇟 󰇠
  (  ):
󰇟󰇟 󰇠󰇠 󰇟󰇠
         
  
      :  ,
󰇛󰇟󰇠󰇜 󰇟󰇛󰇜󰇠
 ,       ()     
  
      :
󰇛󰇜 󰇟󰇛󰇟󰇠󰇜
󰇠 󰇟
󰇠󰇛󰇟󰇠󰇜
          
  
     0.3    ₹10, 0.6    ₹2,  
 ₹0  
 
2.
 :
󰇟󰇠   
    ₹4 ,     .    , 
   
 ,      
    
          ()  
      
           '
 (        )     
 
 ,            "   ", 
          "   "

            : 
-
    ,   ()      '  ,
      
        
     ,     
  -   ,
LOTUS,    -   , ,     

 
-
5.    ?            
   
:🌟   ?
   
    100        
      ?  
!       ,
  ,           
 
    '   :
  (X-) ' 
2.
  (Y-) '   
  ...
     ,    ' ,     
 -   ,     '   (  '),   -
   
 , , -     :
 

 
 
 (
     )🔔)
     :
👉           -  ,  
      
          
   - 
   '    ,  :
  
 ( )
   
 
  
     
           
🎨      (  '   !)

    
 ,      :
2.
:
    
   
   
       X- 

        (PDF) 
 
📏  (  )
        
,   μ    σ 
        :
󰇛󰇜

󰇡󰇜

  😄
  :     '         
    (    )
     '       
 
   
      
         
  :
2.
  =  
             
 
   ,      !
        
 
 =  = 
  , , ,     -      
    :
👉 ()
👉 ( )
👉 (   -   )
          
     '     
 
     
     ,      () 
 :
       
       
     '   )
  (σ)        
 σ →     (         )
 σ →     (  )
  σ   ""   😄
2.
      = 1
   :
👉     = 100%
          
 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- 

          

  :
👉      

 : 7              
2.
    σ)
          :
μ →  (  )
σ →   ()
 
   ,  
     !
 
      , :
   

       
     : 170.2 cm170.25 cm170.253 cm
   
🎯    
?

       -     ' 
 
 :
        
     
     
      
     
        
   "  '  "        ,   
   
2.
6.           ?  
                
:🌟 : ,  , , 
🌟-
   -       -  
       '         
"   "    
        
             '  
👉  :    "        
?"  "  '     ?"    
    
🌟   
    ()          
       (PMF)     :
󰇛 󰇜


:
         
  ( 2.718) 
  
👉:       4    ,    
 2    :
󰇛 󰇜






🌟   
        :
1.     :    👉:    
 '   
2.    '  :         
👉:    -  '   
2.
3.        :    
   👉:       
4.    :        
 
👉 :          ,    
    
🌟        
           
   PMF :
󰇛 󰇜󰇡
󰇢
󰇛󰇜


,           :
 1: 
      (  )
      ( )
   (   )
 2:  PMF  
󰇛 󰇜

󰇛󰇜
󰇛󰇜

 3:   
, ,  .  
 .󰇜


        
 4: 
󰇛 󰇜

     PMF 
👉:          ,  
    ""   
🌟  
2.
1.   
 () =
 () =
👉 :   ,     
2.  
  , :
Poisson󰇛
󰇜
Poisson󰇛
󰇜
Poisson󰇛
󰇜
👉:        3       5 
 ,     8      ,   (8) 
   
3.    ()
           ,    
  :        '  

4.   
 =
 =
👉 ,            
5.    
 ,         :
󰇛󰇜
👉: ,            
 
6.   
             :
     
       
       
2.
🌟    
1.   :         
2.  :        
3.  :     
4. :        
5. :    '     
👉  
           
📖  
  '   
        '  , 
  
       
    ,  

    -   10            
    8, 10,  12          

🌟
       ,   ,     
   '              
          -    ,  
,       -     
 
-
7.()             
 
() 
         ?
: ():           
               
         :
    '         
    '  
      
 
2.
  ?
               
  :
 =    
 :
   (        )
        
          
   ,   
       
   ?
            ,    

 , 
  (   )         
 
  :
 =       
 :
         ,    
             
              
     ,       
     
        '  :
1. 
:      
2.
:      
2. 
 
:      10       

     
:             
3. 
:    , , ,  ,   
:        
4. 
:  '   
        ,   
 ,    
:   '           
    '   ,      
5. 
:           :
  ,  ,    
:     /     : 
,  ,  
6.   
        (     )
:, 
    
:, 
         
2.
 
  
            
      (   )
   
   -   
   -     
      '    
   
 ,  :
           
              
  ,   ,    '   
 ():            ?
        
      -  :
  
,     
 ,  ,    
 
           ,  
    :
   
          
      
       


          
2.
     ?
   , 
    '  ''  -
     
      
  :         
     
     
     -- :
 1:    
,     
     :   

 2:    
   '  
   '   :

 / 
 
 (/)
 
      :

-
   
 :  
    :
 
 
 
2.
       
 3:     
  
 300     
 4:        
  :
       
          :     
    ,      
    
 
 5:      
,   ,      
     :
 
  
  
  
 
 6:     
 ,               
   
 ?

 :
   
 
 
 
2.
    
     
 - 
      :
60%  
40%  
 100         :

2    :   

 60   40   
    
 
 
       ,  
  80   
20       

             
                   
  ,              
             
               
 
8.()         
  
 (      )        
?
()         
:🌟-
 ,               
  
         ,   

2.
      
   
       
  -                
     ,         
        -,    
                  -
   
👉  :        ,       
    
🌟(a)     
1.  
:          
:, ,   ' 
:          
:
o      
o      
o          
👉          
,   
2.  
:   ,      '     
:-, ,   '  
:          " 
"     
:
o      
o  -      
:
o     
o           
 
👉           -    

 ,  
 
 
 

  
2.
-
     
 -
   
 
🌟   
         '     
   
              
  :       
1.    (SRSWR)
:
1.        
2.   ,           
3.              
:  
10
  3       

SRSWR    ,        , 
 
      ,       
:
o     
o    
👉             - 
   
 
2.    (SRSWOR)
:
1.        
2.      ,    
 
3.            
  
:10
  3          
     ,   
  
:
o     
 
o    
👉             - 
    
 

SRSWR  SRSWOR  

   (SRSWR)
  (SRSWOR)
2.

  
  

   
  
 



  
   
🌟()    

  (SE)        (   )  
       
👉  :             "" 
 
     
        :

    SE ( )
    SE ( )
   
1.  :SE 
             
2.    :SE      -
     
3.  :SE       t-  z-   
   
4.   :SE            
   
  
1.    '   :  SE 
 
2.   '   :  SE 


3.  :SE    ,    
4.    
:SE        
🌟    
2.
   
           
30       160  
   
  
               
         
 SE = 2 cm,           ±2 cm    
 
 SE = 10 ,        ,    
 

👉 , SE      - "  "    
📖  
         
 :
      -     

 :
      - ,     
   
 :             

 
🌟
     
 ,   
       '   
          ,    

         -  
  
"              
 
       ,       
 "