Noncentral beta distribution

In probability theory and statistics, the noncentral beta distribution is a continuous probability distribution that is a generalization of the (central) beta distribution.

Probability density function

The probability density function for the noncentral beta distribution is:

$f(x) = \sum_{j=0}^\infin \frac{1}{j!}\left(\frac{\lambda}{2}\right)^je^{-\lambda/2}\frac{x^{a+j-1}(1-x)^{b-1}}{B(a+j,b)}$

where B is the beta function, a and b are the shape parameters, and λ is the noncentrality parameter.

Cumulative distribution function

The cumulative distribution function for the noncentral beta distribution is:

$F(x) = \sum_{j=0}^\infin \frac{1}{j!}\left(\frac{\lambda}{2}\right)^je^{-\lambda/2}I_x(a+j,b)$

where I_x is the regularized incomplete beta function, a and b are the shape parameters, and λ is the noncentrality parameter.

Special cases

When λ = 0, the noncentral beta distribution is equivalent to the (central) beta distribution.

References

• M. Abramowitz and I. Stegun, editors (1965) "Handbook of Mathematical Functions", Dover: New York, NY.

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