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P left-parenthesis upper Y less-than-or-equal-to y overbar right-parenthesis equals upper P left-parenthesis mu Subscript upper Y Baseline plus sigma Subscript upper Y Baseline upper X less-than-or-equal-to y overbar right-parenthesis period"/> For example, if Y has a Gaussian distribution with mean μ_Y = 1 and variance , then we define a new random variable X = (Y − 1)/2. The probability P(Y ≤ 2) is then equal to the probability P(X ≤ 0.5) ≅ 0.69. Similarly, P(−3 ≤ Y ≤ 3) = P(−2 ≤ X ≤ 1) ≅ 0.82.

      In general, the probability that a Gaussian random variable X takes values in the interval (μ_X − σ_X, μ_X + σ_X) is approximately 0.68; in the interval (μ_X − 2σ_X, μ_X + 2σ_X) is approximately 0.95; and in the interval (μ_X − 3σ_X, μ_X + 3σ_X) is approximately 0.99. Therefore, there is a high probability that the values of the Gaussian random variable are in the interval of length 6σ_X centered around the mean μ_X, even though the tails of the distributions have non‐zero probability for values greater than μ_X + 3σ_X or less than μ_X − 3σ_X.

      Because a Gaussian PDF takes positive values in the entire real domain, when using Gaussian distributions for bounded random variables, such as volumetric fractions, or positive random variables, such as velocities, one should truncate the distribution to avoid non‐physical outcomes or apply a suitable transformation (Papoulis and Pillai 2002), such as the normal score or logit transformations of the original variables.

      

      1.4.4 Log‐Gaussian Distribution

      We then introduce the log‐Gaussian distribution (or log‐normal distribution) that is strictly related to the Gaussian distribution. Log‐Gaussian distributions are commonly used for positive random variables. For example, suppose that we are interested in the probability distribution of P‐wave velocity. To avoid positive values of the PDF for negative (hence non‐physical) P‐wave velocity values, we can take the logarithm of P‐wave velocity and assume a Gaussian distribution in the logarithmic domain. In this case, the distribution of P‐wave velocity is said to be log‐Gaussian.

We say that a random variable Y is distributed according to a log‐Gaussian distribution with mean μ_Y and variance , if X = log(Y) is distributed according to a Gaussian distribution . The PDF f_Y(y) can be written as:

Figure 1.9 Log‐Gaussian probability density function associated with the standard Gaussian distribution in Figure 1.8.

      (1.34)

      where μ_X and are the mean and the variance of the random variable in the logarithmic domain.

      The mean μ_Y and the variance of the log‐Gaussian distribution are related to the mean μ_X and variance of the associated Gaussian distribution, according to the following transformations:

      (1.35)

      (1.36)

      (1.37)

      (1.38)

Figure 1.9 shows the log‐Gaussian distribution associated with the standard Gaussian distribution shown in Figure 1.8.

      A log‐Gaussian distributed random variable takes only positive real values. Its distribution is unimodal but it is not symmetric since the PDF is skewed toward 0. The skewness s of a log‐Gaussian distribution is always positive and is given by:

      (1.39) Скачать книгу