Matvec Users’ Guide

13.2. CONTINUOUS DISTRIBUTION 91
4. the cumulative distribution function (cdf) for X ∼ N(0, 1) is
Φ(x) =
=
∫ x
−∞
1
√
2π
exp(− w2
2 ) dw
{
1+erf(x/ √ 2)
2
x ≥ 0
1−erf(−x/ √ 2)
2
x < 0
5. E(X) = µ, Var(X) = σ 2 .
6. the median = µ.
7. if X 1 , X 2 , . . . , X n is a sample from N(µ, σ 2 ), let
∑ n
i=1 ¯X = X ∑ n
i
, and s 2 i=1
=
(X i − ¯X) 2
n
n − 1
then ¯X ∼ N(µ, σ2
s2
n
), (n − 1)
σ
∼ χ 2 (n − 1), and both are independent.
2
Matvec interface
An object of N(µ, σ 2 ) can be created by
D = StatDist("Normal",mu,sigma2);
D = StatDist("Normal");
// standard normal N(0,1)
Matvec provided several standard member functions to allow user to access most of properties and
functions of N(µ, σ 2 ):
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 µ and D.parameter(2) returns σ 2 .
mean D.mean() returns the expected value of X ∼ N(µ, σ 2 ).
variance D.variance() returns the variance of X ∼ N(µ, σ 2 ).
Examples
> D = StatDist("Normal",2, 10)
NormalDist(2,10)
> D.mean()
2
> D.pdf([-2,0,2])
Col 1 Col 2 Col 3
Row 1 0.0566858 0.103288 0.126157
> D.cdf([-3.20148,2,7.20148])