Let and be random variables such that
and are independent
Define a new random variable by
Then the distribution of is called the distribution with degrees of freedom, denoted by .
where is the beta function.
Below are graphs of some probability density functions for various degrees () of freedom.
, , ,
the distribution is also known as the Student’s distribution. The name Student came from the 19th Century research chemist William Sealy Gossett, who was employed by the brewing company Guinness to improve the yield of crops used to produce its beer. Gossett conducted agricultural experiments and used random numbers to help determine the sampling distribution of the data he collected. Because the brewing company wanted to keep the research results confidential for competitve reasons, Gossett had to use a pen name to publish his findings. Student was his pen name and the distribution he found turned out to be the distribution.
is asymptotically (as ) , the standard normal distribution with mean and variance .
If , exists iff . Therefore, a distribution has no mean if it has one degree of freedom. For , . For , .
Please note that the statistic does not depend on , and thus is used often in testing hypotheses involving comparison of the sample mean to the true mean, given a set of random samples that are normally distributed with an unknown mean and an unknown variance. This is an example of a test.
|Date of creation||2013-03-22 14:26:50|
|Last modified on||2013-03-22 14:26:50|
|Last modified by||CWoo (3771)|