Database Reference
In-Depth Information
number of balls. When p is known, the calculation is easy to do. However,
most of the time the various draws of X will be observed, and the name of
the game will be to figure out p.
Statistical Distributions
Determining a range of likely values for p given some observations of X,
called
data
, makes use of a mathematical model called a
distribution
. As
the name implies, a distribution describes the way probability is distributed
among the possible values of X.
There are some exceptions, such as non-parametric methods, but nearly
every familiar calculation is based on either an implicit or explicit statement
of the particular mathematical model that underlies the data. Some of these
underlying models are so well known that they have been given names and
standard interpretations. The next two sections introduce some of these
famous models. They come in two flavors: discrete and continuous.
Discrete Distributions
Discrete distributions are described by a
probability mass function
, which
is the probability that a random variable X will take on a particular value k.
This is usually written as a
p(k)
. The result of adding p(k) for all possible
values of k will always be equal to one. There are a number of discrete
distributions with well-studied interpretations, but the five described in the
next three sections are particularly useful to know in this setting.
Binomial and Hypergeometric Distributions
In general, the probability distribution from the last section is known as the
binomial distribution
. Its probability mass function (PMF) is implemented
as follows:
public static double dbinom(long k,double n,double p) {
return Arithmetic.
binomial
(n, k)
*Math.
pow
(p, k)
*Math.
pow
(1-p, n-k);
}
