ACST828 LECTURE 6
Part 1:
Normal distribution:

X ~ N   , 2 

mean (average)



Variance

2

probability density function

 1  x   2 
1
exp    f  x 
 2   
 2




cumulative density function

 1  t   2 
1
F  x   dt exp   
 2   

2

 


Standard Normal Density

X ~ N  0,1

probability density function

n x 

cumulative density function

x

N  x 

1
1  exp   x 2 
2
2  x 



important result: standardization 1 exp   t 2 dt
2
2
1

if X~N   , 2  and Z= then Z~N  0,1

X-



1

Mathematical Expectation:
Given a random variable X and its pdf f  x  we define the expectation of the function g  X  to be the integral E  g  X   



 g  x  f  x  dx



Note that g  X  is also a random variable

The Moment Generating Function (MGF)
The MGF of a random variable X is a function of t denoted by
M X  t   E  e xt  which is an expectation

MGF of normal
If X ~ N   ,  2 
1  x 



1
Xt
xt
Then M X  t   E  e     e e 2  
  2
 
Lognormal Distribution:


2

1

 t   2t 2
 dx  e 2



Y has the lognormal distribution with parameters   , 2  if:





its logarithm is normally distributed X  log e Y  ~ N  , 2 .

This in turn means that Y  e X

2

The cumulative density function of Y is
 log e  y    
FY  y   Pr Y  y   N 



 x 12
1  2t where N  x    e dt the cdf of the standard normal distribution
2


The probability density function of Y is
 1  log  y    2 
1
e fY  y   exp   

 2

y 2




for y > 0

The mean and variance of the Lognormal distribution:
 The mean of Y is E Y   e

1
2

  2

e

1
E  X