A.8 Student’s t distribution

X has a Student’s t distribution on ν degrees of freedom, denoted

if it has the same distribution as

where and are independent, or equivalently if X has density

It follows that if are independently and

then

The mean and variance are

Because the distribution is symmetrical and unimodal, the median and mode both equal the mean, that is

As the distribution approaches the standard normal form.

It may be noted that Student’s t distribution on one degree of freedom is the standard Cauchy distribution C(0, 1).