8.2. The Normal Probability Density
The normal probability density, also called the Gaussian density, might be the most
commonly used probability density function in statistics.
Normal Probability Density Function:
A random variable X taking values in [, ] has the normal
probability density function f (x) if
f ( x) 
1
e
2 
  x 2
2 2
, -  x   ,
where
  E ( X ),  2  Var( X ),   3.14159
f(x)
The graph of f (x) is
u
x
Properties of Normal Density Function:
(a)


1
  E ( X )   xf ( x)dx   x 
e
2 


1
  x   2
2 2
dx
and
  Var ( X ) 
2

 x   
2


1
f ( x)dx   x    
e
2



2
 x   2
2 2
dx
(b)
  the mean of the normal random variable X
 the median of the normal random variable X ( P( X   )  P( X  u ))
 the mode of the normal probabilit y density ( f (u )  f ( x), x   )
(c) X is a random variable with the normal density function. X is
denoted by
X ~ N ( , 2 )
f(x)
(d) The standard deviation determine the width of the curve. The
normal density with larger standard deviation would be more
dispersed than the one with smaller standard deviation. In the
following graph, two normal density functions have the same
means but different standard deviations, one is 1 (the solid line)
and the other is 2 (the dotted line):
u
x
(e) The normal density is symmetric with respect to mean. That is,
f (   c)  f (   c), where c is any number
Also,
2
.
P  X    c  P  c  X    .
(f) The probability of a normal random variable follows the
empirical rule introduce previously. That is,
P(u    X     )  0.6826  68%
P(   2  X    2 )  0.9544  95%
P(   3  X    3 )  0.9973  100%
i.e., the probability of X taking values within one standard
deviation is about 0.68, within two standard deviations about
0.95, and within three standard deviation about 1.
Standard Normal Probability Density Function:
A random variable Z, taking values in [, ] has the standard
normal probability density function f (x) if
f ( x) 
1
e
2
x2
2
, -  x  ,
where
  E (Z )  0,  2  Var(Z )  1 .
Note: we denote Z as Z ~ N (0,1)
The probability of Z taking values in some interval can be found by the normal table.
The probability of Z taking values in [0,z], z  0, can be obtained by the normal
table. That is,
P(0  Z  z )  the area of the region between tw o vertical lines
z

0
1
2
e
 x2
2
dx
3
The graph is given below:
-z
0
z
Example:
P(0  Z  1.0)  0.3413
P(0  Z  1.03)  0.3485
P(1.0  Z  1.0)  P(1  Z  0)  P(0  Z  1)
 2 P(0  Z  1)  0.6826 (symmetry of Z )
P( Z  1.5)  1  P( Z  1.5)  1  P( Z  0)  P(0  Z  1.5)
1
 1  (  0.4332)  0.0668
2
P( Z  1.5)  P(1.5  Z  0)  P( Z  0)
1
 P(0  Z  1.5)  (symmetry of Z )
2
 0.4332  0.5  0.9332
4
P(1  Z  1.5)  P(0  Z  1.5)  P(0  Z  1)
 0.4332  0.3413  0.0919
Example:
P( Z  x)  0.0099 . What is x?
[solutions:]
P( Z  x)  1  P( Z  x)  1  0.0099  0.9901
1
 x  0 (if x  0, then P( Z  x)  )
2
 P( Z  x)  P( Z  0)  P(0  Z  x) 
 P(0  Z  x)  0.9901 
1
 P(0  Z  x)  0.9901
2
1
 0.4901  x  2.33
2
Computing Probabilities for any Normal Random Variable:
Once the probability of the standard normal random variable can be
obtained, the probability of any normal random variable (not
standard) can be found via the following important rescaling:
X ~ N ( , 2 ) 
X 

 Z ~ N (0,1)
Example:
Let
X ~ N (1,4) . Please find P(1  X  3) .
[solutions:]
  1 and   2. Then,
5
0 X 1 2
P(1  X  3)  P(1  1  X  1  3  1)  P( 
 )  P(0  Z  1)  0.3413
2
2
2
Online Exercise:
Exercise 8.2.1
Exercise 8.2.2
6