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:
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
Famous quotes containing the words probability and/or function:
“The probability of learning something unusual from a newspaper is far greater than that of experiencing it; in other words, it is in the realm of the abstract that the more important things happen in these times, and it is the unimportant that happens in real life.”
—Robert Musil (18801942)
“The fact remains that the human being in early childhood learns to consider one or the other aspect of bodily function as evil, shameful, or unsafe. There is not a culture which does not use a combination of these devils to develop, by way of counterpoint, its own style of faith, pride, certainty, and initiative.”
—Erik H. Erikson (19041994)