Beta Distribution - Characterization - Probability Density Function

Probability Density Function

The probability density function of the beta distribution, for 0 ≤ x ≤ 1, and shape parameters α > 0 and β > 0, is a power function of the variable x and of its reflection (1 − x) as follows:

\begin{align} f(x;\alpha,\beta) & = \mathrm{constant}\cdot x^{\alpha-1}(1-x)^{\beta-1} \\ & = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{\int_0^1 u^{\alpha-1} (1-u)^{\beta-1}\, du} \\ & = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1} \\ & = \frac{1}{\Beta(\alpha,\beta)}\, x ^{\alpha-1}(1-x)^{\beta-1} \end{align}

where is the gamma function. The beta function, appears as a normalization constant to ensure that the total probability integrates to unity. In the above equations x is a realization, -an observed value that actually occurred-, of a random process X.

This definition includes both ends x = 0 and x = 1, which is consistent with definitions for other continuous distributions supported on a bounded interval which are special cases of the beta distribution, for example the arcsine distribution, and consistent with several authors, such as N. L. Johnson and S. Kotz. However, several other authors, including W. Feller, choose to exclude the ends x = 0 and x = 1, (such that the two ends are not actually part of the density function) and consider instead 0 < x < 1.

Several authors, including N. L. Johnson and S. Kotz, use the nomenclature p instead of α and q instead of β for the shape parameters of the beta distribution, reminiscent of the nomenclature traditionally used for the parameters of the Bernoulli distribution, because the beta distribution approaches the Bernoulli distribution in the limit as both shape parameters α and β approach the value of zero.

In the following, that a random variable X is Beta-distributed with parameters α and β will be denoted by:

Other notations for Beta-distributed random variables used in the statistical literature are and .

Read more about this topic:  Beta Distribution, Characterization

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