probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

and statistics, the noncentral beta distribution is a

continuous probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...

that is a noncentral generalization of the (central)

beta distribution In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...

. The noncentral beta distribution (Type I) is the distribution of the ratio :

X = \frac,

where

\chi^2_m(\lambda)

is a noncentral chi-squared random variable with degrees of freedom ''m'' and noncentrality parameter

\lambda

, and

\chi^2_n

is a central chi-squared random variable with degrees of freedom ''n'', independent of

\chi^2_m(\lambda)

. In this case,

X \sim \mbox\left(\frac,\frac,\lambda\right)

A Type II noncentral beta distribution is the distribution of the ratio :

Y = \frac,

where the noncentral chi-squared variable is in the denominator only. If

Y

follows the type II distribution, then

X = 1 - Y

follows a type I distribution.

Cumulative distribution function

The Type I

cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...

is usually represented as a Poisson mixture of central

beta Beta (, ; uppercase , lowercase , or cursive ; grc, βῆτα, bē̂ta or ell, βήτα, víta) is the second letter of the Greek alphabet. In the system of Greek numerals, it has a value of 2. In Modern Greek, it represents the voiced labi ...

random variables: :

F(x) = \sum_^\infty P(j) I_x(\alpha+j,\beta),

where λ is the noncentrality parameter, ''P''(.) is the Poisson(λ/2) probability mass function, ''\alpha=m/2'' and ''\beta=n/2'' are shape parameters, and

I_x(a,b)

is the

incomplete beta function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t ...

. That is, :

F(x) = \sum_^\infty \frac\left(\frac\right)^je^I_x(\alpha+j,\beta).

The Type II

in mixture form is :

F(x) = \sum_^\infty P(j) I_x(\alpha,\beta+j).

Algorithms for evaluating the noncentral beta distribution functions are given by Posten and Chattamvelli.

Probability density function

The (Type I)

probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...

for the noncentral beta distribution is: :

f(x) = \sum_^\infin \frac\left(\frac\right)^je^\frac.

where

B

is the

beta function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^ ...

\alpha

and

\beta

are the shape parameters, and

\lambda

is the noncentrality parameter. The density of ''Y'' is the same as that of ''1-X'' with the degrees of freedom reversed.

Related distributions

Transformations

X\sim\mbox\left(\alpha,\beta,\lambda\right)

, then

\frac

follows a

noncentral F-distribution In probability theory and statistics, the noncentral ''F''-distribution is a continuous probability distribution that is a noncentral generalization of the (ordinary) ''F''-distribution. It describes the distribution of the quotient (''X''/''n' ...

with

2\alpha, 2\beta

degrees of freedom, and non-centrality parameter

\lambda

. If

X

follows a

F_\left( \lambda \right)

with

\mu_

numerator degrees of freedom and

\mu_

denominator degrees of freedom, then :

Z = \cfrac

follows a noncentral Beta distribution: :

Z \sim \mbox\left(\frac\mu_,\frac\mu_,\lambda\right)

. This is derived from making a straightforward transformation.

Special cases

When

\lambda = 0

, the noncentral beta distribution is equivalent to the (central)

References

Citations

Sources

* M. Abramowitz and I. Stegun, editors (1965) " Handbook of Mathematical Functions", Dover: New York, NY. * * * Christian Walck, "Hand-book on Statistical Distributions for experimentalists." {{ProbDistributions, continuous-bounded Continuous distributions b