Matvec Users’ Guide

96 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).
nonct D.nonct(cv,p) returns non-centrality value given the critical value (cv) and p value (cdf). Both cv and
p could be either vector or matrix as long as the sizes are the same.
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 r, degree of freedom; whereas D.parameter(2) returns δ, non-centrality parameter.
mean D.mean() returns the expected value of X ∼ t(r, δ).
variance D.variance() returns the variance of X ∼ t(r, δ).
Examples
> D = StatDist("t",15,2)
tDist(15,2)
> D.mean()
2.10746
> D.variance()
1.32782
> D.pdf(2)
***ERROR***
tDist::pdf(): non-centrality: not available
> D.cdf([-1,0,1,5])
Col 1 Col 2 Col 3 Col 4
Row 1 0.00165624 0.0227501 0.158591 0.986127
> D.inv([0.00165624,0.0227501, 0.158591,0.986127])
Col 1 Col 2 Col 3 Col 4
Row 1 -1.00000 -0.000122502 0.999999 5.00002
> D.nonct(5,0.986127)
1.99998
> D.sample(3)
i = 1 i = 2 i = 3
0.952105 2.99387 0.707365
13.2.5 F distribution
Definition
If U 1 ∼ χ 2 (r 1 , δ) and U 2 ∼ χ 2 (r 2 ) are independent then the random variable
X = U 1/r 1
U 2 /r 2
is called the non-central F distribution with r1 (integer) and r2 (integer) degrees of freedom and noncentrality
parameter δ (real), denoted by F (r 1 , r 2 , δ) where r 1 , r 2 ≥ 1 and δ ≥ 0.