Matvec Users’ Guide

13.2. CONTINUOUS DISTRIBUTION 95
13.2.4 t distribution
Definition
The random variable X has a non-central t distribution if its probability density function (pdf) is defined by
f(x) = exp(δ2 /2)Γ((r + 1)/2) r
√ (
πrΓ(r/2) r + x 2 )(r+1)/2 (13.5)
[
∞∑ Γ((r + j + 1)/2) xδ √ ] j
2
×
√ , −∞ < x < ∞ (13.6)
j!Γ((r + 1)/2) r + x
2
j=0
where r (integer) and δ (real) are the parameters with their ranges r ≥ 1, −∞ < δ < ∞; r is commonly
called the degrees of freedom and δ non-centrality parameter. In short, we say X ∼ t(r, δ).
Alternatively, If Z ∼ N(δ, 1) and U ∼ χ 2 (r) are independent then the random variable
X =
Z √
U/r
is called the non-central t distribution with r degrees of freedom and non-centrality parameter δ.
When δ = 0, the p.d.f of t(r, δ) reduces to
Γ((r + 1)/2) r
f(x) = √ ( πrΓ(r/2) r + x 2 )(r+1)/2 (13.7)
we say X ∼ t(r) which is commonly called (central) t distribution.
For example, if X 1 , X 2 , . . . , X n is a random sample from N(µ, σ 2 ), then
¯X − µ
σ √ n
∼ N(0, 1), (n − 1)
s2
σ 2 ∼ χ2 (n − 1),
and both are independent. Thus,
Because
Thus,
¯X − µ
s/ √ n
∼ t(n − 1)
¯X
σ √ ∼ N(µ, 1)
n
¯X
s/ √ n ∼ t(n − 1, µ
σ √ n )
Properties
1. E(X) = (r/2) 1/2 Γ((r−1)/2)
Γ(r/2)
δ, Var(X) = r
r−2 (1 + δ2 ) − E 2 (X).
Matvec interface
An object of t(r, δ) can be created by
> D = StatDist("t",r,delta)
> D = StatDist("t",r)
Matvec provided several standard member functions to allow user to access most of properties and
functions of t(r, δ):
pdf D.pdf(x) returns the probability density function (pdf) values of x which could be a vector or matrix.