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of the favorable event, generally called the probability of success, is a real number p such that 0 ≤ p ≤ 1. The probability mass function p_X(x) is then:

(1.29) p Subscript upper X Baseline left-parenthesis x right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column p 2nd Column x equals 1 2nd Row 1st Column 1 minus p 2nd Column x equals 0 period EndLayout

The mean of the Bernoulli distribution is then μ_X = p and the variance is sigma Subscript upper X Superscript 2 Baseline equals p left-parenthesis 1 minus p right-parenthesis .

The Bernoulli distribution has several applications in earth sciences. In reservoir modeling, for example, we can use the Bernoulli distribution for the occurrence of a given facies or rock type. For instance, we define a successful event as finding a high‐porosity sand rather than impermeable shale. The probability of success is generally unknown and it depends on the overall proportions of the two facies.

1.4.2 Uniform Distribution

A common distribution for discrete and continuous properties is the uniform distribution on a given interval. According to a uniform distribution, a random variable is equally likely to take any value in the assigned interval. Hence, the PDF is constant within the interval, and 0 elsewhere. If a random variable X is distributed according to a uniform distribution U([a, b]) in the interval [a, b], then its PDF f_X(x) can be written as:

(1.30) f Subscript upper X Baseline left-parenthesis x right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column x less-than a 2nd Row 1st Column StartFraction 1 Over b minus a EndFraction 2nd Column a less-than-or-equal-to x less-than-or-equal-to b 3rd Row 1st Column 0 2nd Column x greater-than b period EndLayout

The mean μ_X of a uniform distribution in the interval [a, b] is:

(1.31) mu Subscript upper X Baseline equals StartFraction a plus b Over 2 EndFraction

and it coincides with the median, whereas the variance sigma Subscript upper X Superscript 2 is:

(1.32) sigma Subscript upper X Superscript 2 Baseline equals StartFraction b minus a Over 12 EndFraction period

An example of uniform distribution in the interval [1, 3] is shown in Figure 1.7. The uniform distribution is sometimes called non‐informative because it does not provide any additional knowledge other than the interval boundaries.

1.4.3 Gaussian Distribution

Most of the random variables of interest in reservoir modeling are continuous. The most common PDF for continuous variables is the Gaussian distribution, commonly called normal distribution. We say that a random variable X is distributed according to a Gaussian distribution script upper N left-parenthesis upper X semicolon mu Subscript upper X Baseline comma sigma Subscript upper X Superscript 2 Baseline right-parenthesis with mean μ_X and variance , if its PDF f_X(x) can be written as:

(1.33)

A Gaussian distribution script upper N left-parenthesis upper X semicolon mu Subscript upper X Baseline equals 0 comma sigma Subscript upper X Superscript 2 Baseline equals 1 right-parenthesis with 0 mean and variance equal to 1 is also called standard Gaussian distribution (or normal distribution). The Gaussian distribution is symmetric and unimodal (Figure 1.8) and can be used to describe a number of phenomena in nature. The Gaussian distribution is defined on the entire set ℝ of real numbers and it is always positive. For example, the standard Gaussian distribution in Figure 1.8 is always greater than 0, but the probability of the random variable being greater than 3 or less than 3 is close to 0, hence negligible.

Graph depicts uniform probability density function in the interval [1, 3].

Figure 1.7 Uniform probability density function in the interval [1, 3].

Graph depicts standard Gaussian probability density function with 0 mean and variance equal to 1.

Figure 1.8 Standard Gaussian probability density function with 0 mean and variance equal to 1.

As shown in Eq. (1.15), in order to compute any probability associated with the random variable X, we must compute an integral. For a Gaussian distribution, the integral of Eq. (1.33) has no analytical form; therefore, we must use numerical tables of the cumulative density function of the standard Gaussian distribution (Papoulis and Pillai 2002). For example, if X is a random variable distributed according to a standard Gaussian distribution script upper N left-parenthesis upper X semicolon mu Subscript upper X Baseline equals 0 comma sigma Subscript upper X Superscript 2 Baseline equals 1 right-parenthesis , then we can compute the following probabilities P(X ≤ 1) ≅ 0.84, P(X ≤ 1.5) ≅ 0.932, P(X ≤ 2) ≅ 0.977, using the numerical tables. Similarly, we obtain that P(−1 ≤ X ≤ 1) ≅ 0.68, P(−2 ≤ X ≤ 2) ≅ 0.954, and P(−3 ≤ X ≤ 3) ≅ 0.997.

To compute a probability associated with a non‐standard Gaussian distribution script upper N left-parenthesis upper Y semicolon mu Subscript upper Y Baseline comma sigma Subscript upper Y Superscript 2 Baseline right-parenthesis with mean μ_Y and variance , we apply the transformation X = (Y − μ_Y)/σ_Y and we use the numerical tables of the standard Gaussian distribution. Indeed, the random variable X is a standard Gaussian distribution with 0 mean and variance equal to 1, and Скачать книгу