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Transcript
The Moments of Student’s t Distribution

The kurtosis of Student’s t is infinite when df ≤ 4. Otherwise it is 6/(df – 4)

The skewness of Student t is 0, unless df ≤ 3, in which case it is not defined.

The standard deviation of Student’s t distribution is the square root of df / (df – 2),
unless df ≤ 2, in which case it is not defined.

The mean of Student’s t distribution is 0, unless df = 1, in which case it is not
defined.
I have my students run a little Monte Carlo to create the sampling distribution of t
and then fiddle with sample size and the shape of the population from which the scores
are sampled. This allows them to demonstrate the consistency of the mean and the
variance, the central limit theorem, and so on. When I ran it with sample sizes of 4, I
found that the obtained distributions of sample means (100,000 to 1,000,000 samples in
each distribution) had outrageous values for g1 and g2, and these values changed wildly
from one replication to the next. For example, on one run, g1 = .04 and g2 = 102. On
the next run g1 = .4.66 and g2 = 707. On a third run g1 = -26 and g2 = 8,063. Now I
know why – neither skewness nor kurtosis are defined for Student’s t on 3 degrees of
freedom.
It was Robert J. MacG. Dawson who explained this to me, answering a query
about it on Edstat-L:
The skewness (in the third-moment sense) is only defined
for df >= 4. It's a matter of convergence of improper integrals.
T_1 (two data, one degree of freedom) is the Cauchy
distribution, and has no mean. This is because in computing it
you are dividing one normally distributed RV by another - hence
[in polar coordinates] the CDF is basically arctangent - hence
the pdf is 1/(1+x^2)and the mean is the integral from -infinity
to infinity of x/(1+x^2)which is easily seen to diverge. Its
Cauchy principle value is 0 as befits a symmetric function, but
the limits at -infinity and infinity cannot be evaluated
independently. If you try to compute a sample mean for a T_1
simulation it will not converge as the sample size increases.
By a similar argument T,_2 (3 data, pdf 1/{(2+x^2)^(3/2)}
is of order x^-3 at infinity and has a mean but no standard
deviation; and T_3 (4 data) has mean, SD, but no third moment.
There are also measures of skewness (eg, Q1+Q3-2M, which is
completely order based) that are defined even for T_1. They are
also zero.
Karl L. Wuensch, East Carolina University, November, 2009