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The Beta distribution is a PDF for continuous random variables defined in the interval [0, 1]. The PDF is uniquely defined by two positive parameters, namely α and β, that determine the shape of the distribution and its derivatives. The generalization to the multivariate domain is the Dirichlet distribution.

A random variable X is distributed according to a Beta distribution B(X; α, β) with parameters α > 0 and β > 0, in the interval [0, 1], if its PDF f_X(x) can be written as:

(1.41)

where the denominator is a normalizing constant to ensure that Eq. (1.41) is a valid PDF (i.e. the integral of the PDF is equal to 1). The mean μ_X of a Beta distribution is:

(1.42)

whereas the variance is:

(1.43)

The Beta distribution is a suitable model for random variables describing volumetric fractions, since it is defined on the interval [0, 1]; however, it is not commonly used in practical applications owing to the limited analytical tractability. In geoscience applications, it is often convenient to use the Beta PDF as a prior distribution in Monte Carlo sampling algorithms to sample bounded random variables such as fluid saturations. Figure 1.12 shows two examples of Beta distributions with parameters α = β = 0.1 (convex curve, solid line) and α = β = 2 (concave curve, dashed line) that could be used to describe the distribution of fluid saturations and mineral fractions in a hydrocarbon reservoir, respectively. The convex distribution in Figure 1.12 shows high likelihood for values near the boundaries of the interval [0, 1], and low likelihood in the middle of the interval. This behavior is often observed in fluid saturations in the subsurface, where partial saturations are unlikely owing to the gravity effect. The concave distribution in Figure 1.12 shows low likelihood for values near the boundaries of the interval [0, 1], and high likelihood in the middle of the interval. This behavior is often observed in mineral fractions in the subsurface, where rocks are generally made of mixtures of minerals rather than pure minerals.