This will help the people alot in making their cocepts clear about the NORMAL DIsTRIBUTION...... And it is my passion to aware people..... to remove difficulyies from the way of people

1. Muhammad Umer
15103122-017

2. Normal Distribution
1. Probability Density Funtion
2. Mean for normal distribution
function

3. Probability Density Funtion
2
2
( )
2
1
( )
2
x
f x e


 




4. • Assuption
( ) 1f x dx



2
x
e dx 





5. 2
2
( )
2
1
( )
2
x
f x dx e dx


  

 

 
2
21
2
1
2
x
y
x
y


 
 
 
 
 
 
2
2
dx
dy
dx dy


 

6. 21
( ) 2
2
y
f x dx e dy

 

 
 
 
22
( )
2
y
f x dx e dy

 

 



 

7. 1
( ) ( 2 )
2
f x dx 



 

( ) 1f x dx




8. Mean for Normal Distribution
( )
( ) ( )
E X
E X xf x dx




 

9. • Now let suppose that
Y= x-
dy = dx
2
2
( )
2
1
( )
2
x
E X x e dx
 



 


10. 2
2
( )
2
1
( ) ( )
2
y
E X y e dy



 
  
2
2
( )
2
1
( ) ( ) ( )
2
y
E X y e dy



 
  

11. • If we plot the function then we will got an odd
signal. Ie( )
• As the sum of the negative and positive side
values on both sides of the real axis is zero.
2
2
( )
2
1
( ) 0
2
y
y e dy




  
2
t
te

12. Hence we can calculate the mean of the
normal distribution funtion.
( ) 0E X  
( )E X 