alt="mu overbar Subscript upper X"/> is an estimate of the mean and it is computed as the average of the data:
(1.19)
Figure 1.3 Comparison of statistical estimators: mean (square), median (circle), and mode (diamond) for different distributions (symmetric unimodal, skewed unimodal, and multimodal distributions).
whereas the sample variance
(1.20)
where the constant 1/(n − 1) makes the estimator unbiased (Papoulis and Pillai 2002).
1.3.2 Multivariate Distributions
In many practical applications, we are interested in multiple random variables. For example, in reservoir modeling, we are often interested in porosity and fluid saturation, or P‐wave and S‐wave velocity. To represent multiple random variables and measure their interdependent behavior, we introduce the concept of joint probability distribution. The joint PDF of two random variables X and Y is a function fX,Y : ℝ × ℝ → [0, +∞] such that 0 ≤ fX,Y(x, y) ≤ +∞ and
The probability P(a < X ≤ b, c < Y ≤ d) of X and Y being in the domain (a, b] × (c, d] is defined as the double integral of the joint PDF:
(1.21)
Given the joint distribution of X and Y, we can compute the marginal distributions of X and Y, respectively, as:
(1.22)
(1.23)
In the multivariate setting, we can also introduce the definition of conditional probability distribution. For continuous random variables, the conditional PDF of X∣Y is:
where the joint distribution fX,Y(x, y) is normalized by the marginal distribution fY(y) of the conditioning variable. An analogous definition can be derived for the conditional distribution fY∣X(y) of Y∣X. All the definitions in this section can be extended to any finite number of random variables.
An example of joint and conditional distributions in a bivariate domain is shown in Figure 1.4. The surface plot in Figure 1.4 shows the bivariate joint distribution fX,Y(x, y) of two random variables X and Y centered at (1, −1). The contour plot in Figure 1.4 shows the probability density contours of the bivariate joint distribution as well as the conditional distribution fY∣X(y) for the conditioning value x = 1 and the marginal distributions fX(x) and fY(y).
The conditional probability distribution in Eq. (1.24) can also be computed using Bayes' theorem (Eq. 1.8) as:
Figure 1.4 Multivariate probability density functions: bivariate joint distribution (surface and contour plots), conditional distribution for x = 1, and marginal distributions.
Figure 1.5 shows an example where the uncertainty in the prior probability distribution of a property X is relatively large, and it is reduced in the posterior probability distribution of the property X conditioned on the property Y, by integrating the information from the data contained in the likelihood function. In seismic reservoir characterization, the variable X could represent S‐wave velocity and the variable Y could represent P‐wave velocity. If a direct measurement of P‐wave velocity is available, we can compute the posterior probability distribution of S‐wave velocity conditioned on the P‐wave velocity measurement. The prior distribution is assumed to be unimodal with relatively large variance. By integrating the likelihood function, we reduce the uncertainty in the
