Obtaining a Beta Distribution through specification of mean and variance

In this section, we provide the details on how to establish a Beta Distribution on the interval (where a, b are real numbers), given a mean and variance for a Beta Distribution on the unit interval [0,1]. To begin, we note that the mean and variance for the Beta Distribution on the unit interval satisfy the equations:

where and are the parameters for the Beta Distribution (See, for example, (Evans et al., 2000)).

Through some straightforward algebra, we obtain:

In this way, we obtain a Beta Distribution on the interval [0,1]. We then use the linear transformation to transform this distribution to a Beta Distribution on the interval [a, b].

References

1. Evans, M., Hastings, N. and Peacock, B. (2000) Statistical Distributions. 3rd ed. J. Wiley and Sons, Inc., New York.