Matvec Users’ Guide

13.2. CONTINUOUS DISTRIBUTION 99
Examples
> D = StatDist("Gamma",2,20)
GammaDist(2,20)
> D.mean()
40
> D.sample(1000).mean()
40.3099
> D.pdf([1,10,100])
Col 1 Col 2 Col 3
Row 1 0.00237807 0.0151633 0.00168449
> D.cdf([1,10,100])
Col 1 Col 2 Col 3
Row 1 0.00120910 0.0902040 0.959572
> D.inv([0.00120910,0.0902040,0.959572])
Col 1 Col 2 Col 3
Row 1 0.999998 10.0000 99.9998
13.2.7 Exponential distribution
Definition
The random variable X has a exponential distribution if its probability density function is defined by
f(x) = 1 θ e−x/θ , 0 ≤ x < ∞. (13.9)
where θ (real) is the parameter with its range θ > 0. In short, we say X ∼ E(θ). The exponential distribution
with θ = 1 is known as the standard exponential distribution.
Properties
1. moment generating function
M(t) = 1
1 − θt , t < 1 θ
2. E(X) = θ, Var(X) = θ 2
3. the cumulative distribution function (cdf) is defined
{ 0 −∞ < x < 0
F (x) =
1 − e −x/θ 0 ≤ x < ∞
4. the median m is found by solving F (m) = 0.5. That is m = θ log(2). because log(2) < 1. Thus the
median is always less than the mean (θ).
Matvec interface
An object of E(θ) can be created by
D = StatDist("Exponential",theta);
D = StatDist("Exponential");
Matvec provided several standard member functions to allow user to access most of properties and
functions of E(θ):
pdf D.pdf(x) returns the probability density function (pdf) values of x which could be a vector or matrix.