Matvec Users’ Guide

13.3. DISCRETE DISTRIBUTION 107
13.3.4 Negative binomial distribution
Definition
The random variable X has a negative binomial distribution if its probability density function (pdf) is defined
by
f(x) = x + n − 1)!
x!(n − 1)! pn (1 − p) x , x = 0, 1, 2, . . . , (13.17)
where p (real) and n (integer) are the parameters with their ranges 0 ≤ p ≤ 1 and n ≥ 1.
For example, let X be the number of failures before the n’th success in a sequence of Bernoulli trials with
probability of success p, then X is distributed as negative binomial distribution with parameters n and p.
Properties
1. its moment generation function is
M(t) =
2. E(X) = r(1−p)
p
, Var(X) = r(1−p)
p
. 2
Matvec interface
p r
[1 − (1 − p)e t , t < −ln(1 − p)
]
r
An object of the negative binomial distribution can be created by
D = StatDist("NegativeBinomial",n,p);
Matvec provided several standard member functions to allow user to access most of properties and
functions of the negative binomial distribution:
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 n, number of Bernoulli trials; whereas D.parameter(2) returns p, the probability
of success in a Bernoulli trial.
mean D.mean() returns the expected value.
variance D.variance() returns the variance.
Examples
> D = StatDist("NegativeBinomial",4,0.1)
NegativeBinomialDist(4,0.1)
> D.mean()
36
> D.sample(1000).mean()
36.256
> D.pdf([0,10,100])