Matvec Users’ Guide

98 CHAPTER 13. STATISTICAL DISTRIBUTIONS
> D.mgf(2)
***ERROR***
FDist:mgf(): not available yet
> D.nonct(3,0.95)
1.626
13.2.6 Gamma distribution
Definition
The random variable X has a gamma distribution if its probability density function is defined by
f(x) =
1
Γ(α)θ α xα−1 e −x/θ , 0 ≤ x < ∞. (13.8)
where α (real) and θ (real) are the parameters with their ranges α, θ > 0. In short, we say X ∼ Gamma(α, θ).
Properties
1. moment generating function
M(t) =
1
(1 − θt) α , t < 1/θ
2. E(X) = αθ, Var(X) = αθ 2
Matvec interface
An object of Gamma(α, θ) can be created by
D = StatDist("Gamma",alpha,theta);
Matvec provided several standard member functions to allow user to access most of properties and
functions of Gamma(α, θ):
pdf D.pdf(x) returns the probability density function (pdf) values of x which could be a vector or matrix.
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 α, D.parameter(2) returns θ.
mean D.mean() returns the expected value.
variance D.variance() returns the variance.