Matvec Users’ Guide

92 CHAPTER 13. STATISTICAL DISTRIBUTIONS
Col 1 Col 2 Col 3
Row 1 0.0500001 0.500000 0.950000
> D.inv([0.05,0.5,0.95])
Col 1 Col 2 Col 3
Row 1 -3.20148 2.00000 7.20148
> D.sample(2,3)
Col 1 Col 2 Col 3
Row 1 -1.26660 2.85846 5.64868
Row 2 3.51262 -1.97644 3.44860
13.2.2 Uniform distribution
Definition
The random variable X has a uniform distribution if its probability density function (pdf) is defined by
f(x) = 1
b − a , a ≤ x ≥ b (13.2)
where a (real) and b (real) are the parameters with their range −∞ < a < b < ∞. In short, we say
X ∼ U(a, b). The uniform distribution with a = 0 and b = 1 is known as the standard uniform distribution
U(0, 1). One of interesting example is that if D is an object of StatDist, then D.cdf(x) is distributed as
U(0, 1).
Properties
1. E(X) = a+b
(b−a)2
2
, Var(X) =
12
.
2. its moment generation function is
Matvec interface
An object of U(a, b) can be created by
D = StatDist("Uniform",a,b);
D = StatDist("Uniform");
M(t) = etb − e ta
, t ≠ 0; M(0) = 1
t(b − a)
// standard uniform U(0,1)
Matvec provided several standard member functions to allow user to access most of properties and
functions of U(a, b):
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 a and D.parameter(2) returns b.
mean D.mean() returns the expected value of X ∼ U(a, b).
variance D.variance() returns the variance of X ∼ U(a, b).