- If is a standard normal r.v., the distribution of is called the chi-square distribution with 1 degree of freedom.
- If are independent chi-square r.v. with 1 degree of freedom, the distribution of is called the chi-square distribution with n degrees of freedom and is denoted by
- has a distribution
If and and are independent, then the distribution of is called the distribution with n degrees of freedom.
- if are a random sample from a , we know that
- but if is unknown, we can use to replace , and
t-distribution converges to a normal distribution if the freedom degree approaches to infinity.
Let and be independent r.v. with and degrees of freedom, respectively. The distribution of
is call the distribution with and degrees of freedom and is denoted by
A variance ratio may have an F distribution even if the parent populations are not normal. It is enough that their pdf are symmetry functions.
if , then