The probability density function for Student's distribution is given by
(1)

Now define
(2)

then the cumulative distribution function is given by
(3)

The mean is 0, so the moments are
(4)
 
(5)
 
(6)
 
(7)

The mean, variance, skewness, and kurtosis excess are
(8)
 
(9)
 
(10)
 
(11)

The characteristic function is
(12)

where is a modified Bessel function of the second kind.
Letting
(13)

where is the sample mean and is the population mean gives Student's tdistribution.