The document discusses the properties of the normal distribution, focusing on the probability density function and the calculation of its mean. It includes mathematical expressions and integrals related to these concepts, illustrating how the probabilities and means are derived. The analysis concludes that the mean of the normal distribution function can be calculated through specific integrals.

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 