is a standard normal random variable,
then the distribution of 
is a Chi-Squared distribution
(see Figure 3.6). A Chi-Squared random variable
is represented by 
. Thus:
(6)If is a standard normal random variable,
then the distribution of is a Chi-Squared distribution
(see Figure 3.6). A Chi-Squared random variable
is represented by . Thus:(6)
|
Figure 3.6: Chi-Squared distribution. |
The distribution of the variable 
mentioned in the previous equation
is also referred to as centrally distributed Chi-Squared with
one degree of freedom. The degree of freedom is one here because
here the Chi-Squared random variable is obtained from a single standard
normal random variable 
. The previous equation may also be
represented by including the degree of freedom into the equation as: 
If 
, 
, 
...
are 
independent standard normal random
variables then: 
is also a Chi-Squared random variable.
The distribution of 
is said to be centrally Chi-Squared
with 
degrees of freedom, as the Chi-Squared
random variable is obtained from 
independent standard normal random
variables.
If 
is a normal random variable then the
distribution of 
is said to be non-centrally distributed
Chi-Squared with one degree of freedom. Therefore, 
is a Chi-Squared random variable and
can be represented as:
If 
, 
, 
...
are 
independent normal random variables
then: 
is a non-centrally distributed Chi-Squared random variable with 
degrees of freedom.
See Also:
