Noncentral beta distribution
Encyclopedia
In probability theory
and statistics
, the noncentral beta distribution is a continuous probability distribution that is a generalization of the (central) beta distribution.
for the noncentral beta distribution is:
where
is the beta function, 
and 
are the shape parameters, and 
is the noncentrality parameter
.
for the noncentral beta distribution is:
where
is the regularized incomplete beta function, 
and 
are the shape parameters, and 
is the noncentrality parameter
.
, the noncentral beta distribution is equivalent to the (central) beta distribution.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
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 noncentral beta distribution is a continuous probability distribution that is a generalization of the (central) beta distribution.
Probability density function
The probability density functionProbability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
for the noncentral beta distribution is:
where
is the beta function, and are the shape parameters, and is the noncentrality parameterNoncentrality parameter
Noncentrality parameters are parameters of families of probability distributions which are related to other "central" families of distributions. If the noncentrality parameter of a distribution is zero, the distribution is identical to a distribution in the central family...
.
Cumulative distribution function
The cumulative distribution functionCumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...
for the noncentral beta distribution is:
where
is the regularized incomplete beta function, and are the shape parameters, and is the noncentrality parameterNoncentrality parameter
Noncentrality parameters are parameters of families of probability distributions which are related to other "central" families of distributions. If the noncentrality parameter of a distribution is zero, the distribution is identical to a distribution in the central family...
.
Special cases
When, the noncentral beta distribution is equivalent to the (central) beta distribution.