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:

$f(x)=\sum _{j=0}^{\infty }{\frac {1}{j!}}\left({\frac {\lambda }{2}}\right)^{j}e^{-\lambda /2}{\frac {x^{a+j-1}(1-x)^{b-1}}{B(a+j,b)}}$

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

Cumulative distribution function

The cumulative distribution function for the noncentral beta distribution is:

$F(x)=\sum _{j=0}^{\infty }{\frac {1}{j!}}\left({\frac {\lambda }{2}}\right)^{j}e^{-\lambda /2}I_{x}(a+j,b)$

where $I_{x}$ is the regularized incomplete beta function, $a$ and $b$ are the shape parameters, and $\lambda$ is the noncentrality parameter.

Special cases

When $\lambda =0$, 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.
• Hodges, J.L. Jr (1955). "On the noncentral beta-distribution". Annals of Mathematical Statistics. 26: 648–653.
• Seber, G.A.F. (1963). "The non-central chi-squared and beta distributions". Biometrika. 50: 542–544.