Matvec Users’ Guide

100 CHAPTER 13. STATISTICAL DISTRIBUTIONS
cdf D.cdf(x) returns the cumulative distribution function (cdf) values of x which could be a vector or
matrix.
mgf D.mgf(t) returns the moment-generating function (mgf) values of t which could be a vector or matrix.
inv D.inv(p) is the inverse function of D.cdf(x), where p could be a vector or matrix. That is if p =
D.cdf(x), then x = D.inv(p).
sample D.sample(), D.sample(n), and D.sample(m,n) return a random scalar or a vector of size n or a matrix
of size m by n.
parameter D.parameter(1) returns θ.
mean D.mean() returns the expected value.
variance D.variance() returns the variance.
Examples
> D = StatDist("Exponential",10)
ExponentialDist(10)
> D.sample(1000).mean()
9.69597
> D.pdf([0,2,10])
Col 1 Col 2 Col 3
Row 1 0.100000 0.0818731 0.0367879
> D.cdf([0,2,10])
Col 1 Col 2 Col 3
Row 1 0.00000 0.181269 0.632121
> D.inv([0,0.181269,0.632121])
Col 1 Col 2 Col 3
Row 1 5.63450e-07 2.00000 10.0000
13.2.8 Beta distribution
Definition
The random variable X has a non-central beta distribution if its probability density function is defined by
f(x) = .... (13.10)
where α (real), β (real) and λ (real) are the parameters with their ranges α, β > 0 and λ ≥ 0. In short, we
say X ∼ Beta(α, β, λ).
If λ = 0, then its pdf reduces to
f(x) =
Γ(α + β)
Γ(α)Γ(β) xα−1 (1 − x) β−1 , 0 < x < 1 (13.11)
we say X ∼ Beta(α, β) which is commonly called (central) Beta distribution.
Alternatively, if X 1 ∼ χ 2 (r 1 , λ) and X 2 ∼ χ 2 (r 2 ) are independent, then the ratio
X =
is defined as a non-central beta distribution Beta(r 1 /2, r 2 /2, λ)
Properties
1. for X ∼ Beta(α, β), E(X) = α
α+β , Var(X) =
X 1
X 1 + X 2
(13.12)
αβ
(α+β+1)(α+β) 2 .