## 3 Random Variables
### Distribution of Random Variables  
To appropriately represent the variability in each of the random variables, a suitable probability distribution is required, along with their respective means and standard deviations. The most frequently utilized distributions for expressing variabilities in soil properties are the normal distribution and the lognormal distribution [43; 44] due to theoretical principles like the central limit theorem, empirical observations, and mathematical convenience. These distributions facilitate statistical analysis, parameter estimation, and compatibility with geotechnical models while effectively capturing soil variability. Additionally, they aid in data interpolation and ensure physically meaningful representations of soil properties. The implementation of these distributions within a Monte Carlo framework aims to select the optimal distributional choices for various soil types and stochastic variables, especially since the uncertainties of the studied empirical parameters are not well studied in the literature. The normal distribution is the most widely used distribution [45]. It relies on the fact that the sums of random variables tend to follow a Gaussian distribution, as demonstrated by the central limit theorem. Numerous natural phenomena that involve the accumulation of factors often exhibit a normal distribution [33]. However, a main drawback of the normal distribution is allowing negative values, which is problematic when attempting to model physical quantities that cannot be negative in practice, such as density. For example, the probability of getting a negative drag coefficient in sand is nontrivial for the typical mean and COV involved in this work. This is demonstrated in Figure 2, where the normal mean (\(\mu\)) and COV of \(R_{sand}\) were 0.94 MPa and 58%, resulting in an approximately 17% chance of getting a negative \(R_{sand}\). It is noteworthy that the difference in the peaks of the two distributions returns to the nature of these distributions. First, in the case of normal distribution, the mode, \(M\), coincides with the mean value; thus, this distribution is symmetric. However, in the case of lognormal distribution, the mode is given by:  
\[M=e^{\mu-\sigma^{2}} \tag{9}\]  
where \(\mu\) and \(\sigma^{2}\) are the mean and variance of the underlying normal distribution before exponentiation. Since \(\sigma^{2}\) is positive, the mode is always smaller than the mean. To address this issue, it is preferable to use a non-negative distribution as long as it accurately represents the population concerned. Several non-negative distributions, such as the exponential, gamma, and Weibull distributions, are available [46]. However, the simplest and most commonly used positive distribution for describing soil properties is the lognormal distribution [47]. Since the effect of employing a normal versus lognormal distribution of properties on the predicted DoB of UXOs is unknown, and to identify the proper distribution to use for each random variable in natural soils \((\rho,C,R)\), four approaches, involving 27 different combinations of distributions, were explored in the context of a Monte Carlo simulation (Table 2).