Matvec Users’ Guide

102 CHAPTER 13. STATISTICAL DISTRIBUTIONS
where µ (real), σ 2 (real) and θ (real) are the parameters with their ranges −∞ < µ, θ < ∞ and σ 2 > 0. In
short, we say X ∼ lognormal(µ, σ 2 , θ).
Alternatively, if log(X − θ) ∼ N(µ, σ 2 ), then we say X has a lognormal(µ, σ 2 , θ) distribution.
The lognormal distribution with θ = 0 is known as two-parameter lognormal distribution lognormal(µ, σ 2 ).
Properties
1. E(X) = exp(µ + σ 2 /2) + θ, Var(X) = exp(2µ + 2σ 2 ) − exp(2µ + σ 2 ) + 2θ 2 .
2. mode(X) = exp(µ − σ 2 ) + θ
3. median(X) = exp(µ) + θ
Matvec interface
An object of lognormal(µ, σ 2 , θ) can be created by
D = StatDist("LogNormal",mu,sigma2,theta);
D = StatDist("LogNormal",mu,sigma2);
Matvec provided several standard member functions to allow user to access most of properties and
functions of lognormal(µ, σ 2 , 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 µ, D.parameter(2) returns σ 2 , and D.parameter(3) returns θ.
mean D.mean() returns the expected value.
variance D.variance() returns the variance.
Examples
> D = StatDist("LogNormal",1,3,2);
LogNormalDist(1,3,2)
> D.mean()
14.1825
> D.sample(1000).mean()
14.087
> D.pdf([2.01,10,30])
Col 1 Col 2 Col 3
Row 1 0.122530 0.0237094 0.00332270
> D.cdf([2.01,10,30])
Col 1 Col 2 Col 3
Row 1 0.000605776 0.733429 0.910929
> D.inv([0.000605776,0.733429,0.910929])
Col 1 Col 2 Col 3
Row 1 2.01000 10.0000 30.0000