Let and be random variables such that
and are independent
, the chi-squared distribution with degrees of freedom
Define a new random variable by
By transformation of the random variables and , one can show that the probability density function of the F distribution of has the form:
for , where is the beta function. for .
For a fixed , say 10, below are some graphs for the probability density functions of the F distribution with degrees of freedom.
The next set of graphs shows the density functions with degrees of freedom when is fixed. In this example, .
If , the non-central chi-square distribution with m degrees of freedom and non-centrality parameter , with and defined as above, then the distribution of is called the non-central F distribution with m and n degrees of freedom and non-centrality parameter .
the “F” in the F distribution is given in honor of statistician R. A. Fisher.
If , then .
If , the t distribution with degrees of freedom, then .
If , then
Suppose are random samples from a normal distribution with mean and variance . Furthermore, suppose are random samples from another normal distribution with mean and variance . Then the statistic defined by
where and are sample variances of the and the , respectively, has an F distribution with m and n degrees of freedom. can be used to test whether . is an example of an F test.
|Date of creation||2013-03-22 14:26:56|
|Last modified on||2013-03-22 14:26:56|
|Last modified by||CWoo (3771)|
|Synonym||Fisher F distribution|
|Synonym||central F distribution|
|Defines||non-central F distribution|