Bates distribution
Encyclopedia
In probability
and statistics
, the Bates distribution, is a probability distribution
of the mean
of a number of statistically independent uniformly distributed random variables on the unit interval
. This distribution is sometimes confused with the Irwin–Hall distribution, which is the distribution of the sum (not mean) of n independent random variables uniformly distributed from 0 to 1.
of the mean
, X, of n independent uniformly distributed random variables on the unit interval
, Ui:
The equation defining the probability density function of a Bates distribution random variable x is
for x in the interval (0,1), and zero elsewhere. Here sgn(x − k) denotes the sign function
:
More generally, the mean of n independent uniformly distributed random variables on the interval [a,b]
would have the probability density function of
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, the Bates distribution, is a probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
of the mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
of a number of statistically independent uniformly distributed random variables on the unit interval
Unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...
. This distribution is sometimes confused with the Irwin–Hall distribution, which is the distribution of the sum (not mean) of n independent random variables uniformly distributed from 0 to 1.
Definition
The Bates distribution is the continuous probability distributionProbability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
of the mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
, X, of n independent uniformly distributed random variables on the unit interval
Unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...
, Ui:
The equation defining the probability density function of a Bates distribution random variable x is
for x in the interval (0,1), and zero elsewhere. Here sgn(x − k) denotes the sign function
Sign function
In mathematics, the sign function is an odd mathematical function that extracts the sign of a real number. To avoid confusion with the sine function, this function is often called the signum function ....
:
More generally, the mean of n independent uniformly distributed random variables on the interval [a,b]
would have the probability density function of