692 B. PROBABILITY DISTRIBUTIONS
of Gaussians having the same mean but different variances.
St(x|μ, λ, ν)=
Γ(ν/2+1/2)
Γ(ν/2)
(
λ
πν
) 1 / 2 [
1+
λ(x−μ)^2
ν
]−ν/ 2 − 1 / 2
(B.64)
E[x]=μ forν> 1 (B.65)
var[x]=
1
λ
ν
ν− 2
forν> 2 (B.66)
mode[x]=μ. (B.67)
Hereν> 0 is called the number of degrees of freedom of the distribution. The
particular case ofν=1is called theCauchydistribution.
For aD-dimensional variablex, Student’s t-distribution corresponds to marginal-
izing the precision matrix of a multivariate Gaussian with respect to a conjugate
Wishart prior and takes the form
St(x|μ,Λ,ν)=
Γ(ν/2+D/2)
Γ(ν/2)
|Λ|^1 /^2
(νπ)D/^2
[
1+
∆^2
ν
]−ν/ 2 −D/ 2
(B.68)
E[x]=μ forν> 1 (B.69)
cov[x]=
ν
ν− 2
Λ−^1 forν> 2 (B.70)
mode[x]=μ (B.71)
where∆^2 is the squared Mahalanobis distance defined by
∆^2 =(x−μ)TΛ(x−μ). (B.72)
In the limitν→∞, the t-distribution reduces to a Gaussian with meanμand pre-
cisionΛ. Student’s t-distribution provides a generalization of the Gaussian whose
maximum likelihood parameter values are robust to outliers.
Uniform
This is a simple distribution for a continuous variablexdefined over a finite interval
x∈[a, b]whereb>a.
U(x|a, b)=
1
b−a
(B.73)
E[x]=
(b+a)
2
(B.74)
var[x]=
(b−a)^2
12
(B.75)
H[x]=ln(b−a). (B.76)
Ifxhas distributionU(x| 0 ,1), thena+(b−a)xwill have distributionU(x|a, b).
