Matvec Users’ Guide

13.2. CONTINUOUS DISTRIBUTION 93
Examples
> D = StatDist("Uniform",1, 4)
UniformDist(1,4)
> D.variance()
0.75
> D.pdf([1,2,3,4])
Col 1 Col 2 Col 3 Col 4
Row 1 0.333333 0.333333 0.333333 0.333333
> D.cdf([1,2,3,4])
Col 1 Col 2 Col 3 Col 4
Row 1 0.00000 0.333333 0.666667 1.00000
> D.inv([0,1/3,2/3,1])
Col 1 Col 2 Col 3 Col 4
Row 1 1.00000 2.00000 3.00000 4.00000
> U.sample(1,3)
Col 1 Col 2 Col 3
Row 1 3.59627 1.36787 2.17816
13.2.3 χ 2 distribution
Definition
The random variable X has a non-central χ 2 distribution if its probability density function (pdf) is defined
by
exp(−(x + λ)/2) ∑ ∞
x r/2+j−1 λ j
f(x) =
2 1 2 r Γ(r/2 + j)2 2j j! , 0 ≤ x < ∞ (13.3)
j=0
where r (integer) and λ (real) are the parameters with their ranges r ≥ 1, λ ≥ 0. The parameter r is
commonly called the degrees of freedom and λ non-centrality parameter. In short, we say X ∼ χ 2 (r, λ).
Alternatively, If Z 1 , Z 2 , . . . , Z r are independent and distributed as N(δ i , 1) then the random variable
r∑
X = Z 2
i=1
is called the non-central χ 2 distribution with r degrees of freedom and non-centrality parameter λ = ∑ r
i=1 δ i.
When λ = 0, the p.d.f of χ 2 (r, λ) reduces to
f(x) =
1
Γ(r/2)2 r/2 xr/2−1 e −x/2 , (13.4)
we say X ∼ χ 2 (r) which is commonly called (central) χ 2 distribution.
The central χ 2 distribution is the special case of Gamma distribution: χ 2 (r) = Γ(r/2, 2)
For example, if X 1 , X 2 , . . . , X n is a sample from N(µ, σ 2 ), then
(n − 1) s2
σ 2 ∼ χ2 (n − 1)
and is independent of the sample mean ¯X.
Properties
1. its moment generation function is
2. E(X) = r + λ, Var(X) = 2(r + 2λ).
M(t) = (1 − 2t) −r/2 λt
exp(
1 − 2t ), t < 1 2
3. If χ 2 (r 1 , λ 1 ) and χ 2 (r 2 , λ 2 ) are independent, then χ 2 (r 1 , λ 1 ) + χ 2 (r 2 , λ 2 ) = χ 2 (r 1 + r 2 , λ 1 + λ 2 )