
Probability and Statistics Overview 43
probability density function is
p(x) =
Γ((k + 1)/2)
Γ(k/2)
1
√
kπ
1
(1 + x
2
/k)
(k+1)/2
, x ∈ R,
where Γ is the gamma function defined by (2.103). The mean and variance
are given by
E(X) = 0 if k > 1 (otherwise undefined),
and
Var(X) = k/(k −2) if k > 2 (otherwise undefined).
Figure 2.10 compares the pr obability density function of a Student’s t dis-
tribution with various deg rees of freedom to the probability density function
of a standard normal distribution. From this figure we see that the proba-
bility density function of a Student’s t distribution is symmetric like that of
the no rmal distribution, with “heavier tails,” ...