Implementing a random variable generator based on a d-ary tree

In the past, residual factor R in strain-softening soil slopes has been included, either directly or indirectly, as a deterministic variable in both deterministic and probabilistic studies. This paper discusses the uncertainties associated with R and outlines a systematic approach for the reliability analysis of a natural slope in which shear strength parameters and pore pressure ratio are random variables, each assumed with a lognormal probability distribution. For the residual factor R, seven probability distribution options under the generalized beta-distribution system are considered. Slope reliability is computed based on the first order reliability method (FORM) and validated against Monte-Carlo simulation (MCS). Results obtained from two illustrative examples indicate that the probability of failure, with R as one of six random variables, can be orders of magnitude higher than that based on five random variables with R considered as a deterministic parameter. The magnitude of influence of R as a random variable is, however, highly dependent on its probability distribution, the left-skewed triangular distribution having the most significant influence in both the examples. Results of sensitivity analyses reveal that, for almost all of its assumed probability distributions, R is the most dominant among the six random variables. Effects of variation of some of the statistical and correlation properties of the other random variables, viz. the shear strength parameters and the pore pressure ratio, on the results of reliability analyses are also studied.

This chapter introduces probability theory as a system of models, based on measure theory, of some real-world phenomena. The models are measure spaces of total measure 1 and usually have certain distinguished measurable functions defined on them. Section 1 begins by establishing the measure-theoretic framework and a short dictionary for passing back and forth between terminology in measure theory and terminology in probability theory. The latter terminology includes events, random variables, mean, probability distribution of a random variable, and joint probability distribution of several random variables. An important feature of probability is that it is possible to work with random variables without any explicit knowledge of the underlying measure space, the joint probability distributions of random variables being the objects of importance. Section 2 introduces conditional probability and uses that to motivate the mathematical definition of independence of events. In turn, independence of events leads naturally to a definition of independent random variables. Independent random variables are of great importance in the subject and play a much larger role than their counterparts in abstract measure theory. Examples at the end of the section indicate the extent to which functions of independent random variables can remain independent. The techniques in the examples are of use in the subject of statistical inference, which is introduced in Section 10. Section 3 states and proves the Kolmogorov Extension Theorem, a foundational result allowing one to create stochastic processes involving infinite sets of times out of data corresponding to finite subsets of those times. A special case of the theorem provides the existence of infinite sets of independent random variables with specified probability distributions. Section 4 establishes the celebrated Strong Law of Large Numbers, which says that the Cesaro sums of a sequence of identically distributed independent random variables with finite mean converge almost everywhere to a constant random variable, the constant being the mean. This is a theorem that is vaguely known to the general public and is widely misunderstood. The proof is based on Kolmogorov’s inequality. Sections 5–8 provide background for the Central Limit Theorem, whose statement and proof are in Section 9. Section 5 discusses three successively weaker kinds of convergence for random variables—almost sure convergence, convergence in probability, and convergence in distribution. Convergence in distribution will be the appropriate kind for the Central Limit Theorem. Section 6 contains the Portmanteau Lemma, which gives some equivalent formulations of convergence in distribution, Section 7 introduces characteristic functions as Fourier transforms of probability distributions, and Section 8 proves the Levy Continuity Theorem, which formulates convergence in distribution in terms of characteristic functions. Section 9 contains the statement and proof of the Central Limit Theorem, followed by some simple examples. This theorem is the most celebrated result in probability theory and has many applications in mathematics and other fields. Section 10 is a brief introduction to the subject of statistical inference, showing how the Central Limit theorem plays a role in practice through the $t$ test of W. S. Gosset.

In mathematics and statistics, the desire to eliminate mathematical tedium and facilitate exploration has lead to computer algebra systems. These computer algebra systems allow students and researchers to perform more of their work at a conceptual level. The design of generic algorithms for tedious computations allows modelers to push current modeling boundaries outward more quickly. Probability theory, with its many theorems and symbolic manipulations of random variables is a discipline in which automation of certain processes is highly practical, functional, and efficient. There are many existing statistical software packages, such as SPSS, SAS, and S-Plus, that have numeric tools for statistical applications. There is a potential for a probability package analogous to these statistical packages for manipulation of random variables. The software package being developed as part of this dissertation, referred to as “A Probability Programming Language” (APPL) is a random variable manipulator and is proposed to fill a technology gap that exists in probability theory. My research involves developing algorithms for the manipulation of discrete random variables. By defining data structures for random variables and writing algorithms for implementing common operations, more interesting and mathematically intractable probability problems can be solved, including those not attempted in undergraduate statistics courses because they were deemed too mechanically arduous. Algorithms for calculating the probability density function of order statistics, transformations, convolutions, products, and minimums/maximums of independent discrete random variables are included in this dissertation.

An important component in the assessment of the economic risk of a prospect prior to development is the calculation of in-place and recoverable reserves and their associated uncertainty. This calculation requires the multiplication of the probability distributions of volume, porosity, net-pay to gross etc. to form the probability distribution of reserves. Two methods have been advocated for this calculation; Parametric methods and Monte-Carlo methods In (i) component distributions are combined analytically to give measures of average and confidence levels in reserves. whereas in (ii) the distribution of the components are randomly sampled to build-up the distribution of reserves. Although at present the Monte-Carlo approach has been widely advocated in the petroleum industry, due to their general applicability, the parametric methods have many adherents citing their ease and speed of use as principle reasons. However, there is a seeming lack of rigorous analysis in either method which would make the selection of the most appropriate method for a particular problem more objective. In this paper we address this area and show how the errors in both methods can. be reduced. Additionally, we suggest a new approach which combines the strengths of both methods into a flexible algorithm for reserves calculations.

An investment risk analysis by the probability distribution technique was proposed and published by Hillier in a 1963 issue of Management Science. His work was quite theoretical. A simplified form of the method was presented in petroleum literature by Davidson and Cooper in 1975. However, their work did not extend to using the method in practical decision making. Both of these works critically discussed why this approach is preferable to earlier methods. In as much as the main value of such an innovation and the ultimate need for its publication is its application, discussion of the practical aspects of the method is important. The purpose of this paper is to fulfill this need. Introduction Regardless of what criterion an investor uses to determine the profitability of his venture, the amount of risk involved must be considered for a sound, final decision. As such, many techniques have evolved over the years for quantifying risk. Among the older techniques which have been used are:Payback Period - The length of time required to Payback Period - The length of time required to recover an investment through the net cash flows from the project.Expected Monetary Value - Sum of the products of investment outcomes and their respective products of investment outcomes and their respective chances of occurrence.The Certainity Equivalent - A method of discounting cash flow by a risk-free interest rate.Sensitivity Analysis - Determining the profitability measure by varying an uncertain parameter. For reasons well discussed in parameter. For reasons well discussed in literature, all of these techniques are weak and are inadequate for proper risk portrayal. The more recent and certainly more sophisticated method of risk analysis is the probability distribution of the profitability index. One form of this is the Monte profitability index. One form of this is the Monte Carlo simulation which uses randomly selected parameters from appropriate ranges of values to generate parameters from appropriate ranges of values to generate a series of possible outcomes. While this technique is a quite reliable tool, it does not lend itself to routine application because of the computer work involved. Another form is the, analytical technique proposed by Fredrick Hillier. This approach adopts proposed by Fredrick Hillier. This approach adopts well know statistical theories and assumes the distribution of the profitability index to derive the probability distribution. Hillier's work, though probability distribution. Hillier's work, though amenable to hand calculators, was quite theoretical. A simplified form of the method was presented in petroleum literature by Davidson and presented in petroleum literature by Davidson and Cooper in 1975. However, their work did not extend to using the method in practical decision making. This paper discusses the practical aspects of Hillier's approach. Although Davidson and Cooper suggested log-normal distribution assumption for analysis of petroleum investments, this work is based on the validity of normal distribution when the random variable is the cash flow. Both assumptions are compared in the discussion. Arriving at a decision to accept or reject an investment proposal on the basis of the method requires the use of utility theory. Therefore, the concept of the theory is reviewed. Its use in conjunction with the probability distribution of net present value is then shown. Finally, a decision present value is then shown. Finally, a decision concerning two or more risky investments may be influenced by the skewness of normal distribution. Hence, the semi-variance equation for testing the normality or magnitude and direction of skewness is presented and its application discussed. DERIVATION OF PROBABILITY DISTRIBUTION In his work, Hillier assumed that under conditions of uncertainity, cash flows from an investment are random variables which can be characterized by their means and variances. Therefore, by the central limit theorem, the distribution of these means will be approximately normal. Moreover, their sum will approach a normal distribution. Based on these, he derived the probability distributions of net present values and internal rate of return for three possible cases of cash flows from an investment. These cases include (1) statistically independent cash flows (2) perfectly correlated cash flows and, (3) combined independent and correlated cash flows. This paper considers the second case only.

This chapter reviews some subjects in mathematics and physics that are used in different contexts throughout this book. The selection of subjects and the level of their coverage reflect the author’s perception of what potential users of this text were exposed to in their earlier studies. Therefore, only brief overview is given of some subjects while somewhat more comprehensive discussion is given of others. In neither case can the coverage provided substitute for the actual learning of these subjects that are covered in detail by many textbooks. A random variable is an observable whose repeated determination yields a series of numerical values (“realizations” of the random variable) that vary from trial to trial in a way characteristic of the observable. The outcomes of tossing a coin or throwing a die are familiar examples of discrete random variables. The position of a dust particle in air and the lifetime of a light bulb are continuous random variables. Discrete random variables are characterized by probability distributions; Pn denotes the probability that a realization of the given random variable is n. Continuous random variables are associated with probability density functions P(x): P(x1)dx denotes the probability that the realization of the variable x will be in the interval x1 . . . x1+dx.

Chapter Three - PROBABILITY DISTRIBUTIONS

This chapter is devoted to illustrating some widely used techniques for generating random variables. In this chapter, the author will first introduce inverse transform technique in Section 14.1. Specifically, examples of applying the inverse transform technique to generate both discrete and continuous random variables will be provided. Based on the developed random generators, Section 14.2 will present transformation technique to generate related random variables in the same family.

Input modeling is the selection of a probability distribution to capture the uncertainty in the input environment of a stochastic system. Example applications of input modeling include the representation of the randomness in the time to failure for a machining process, the time between arrivals of calls to a call center, and the demand received for a product of an inventory system. Building simulations of stochastic systems requires the development of input models that adequately represent the uncertainty in such random variables. Since there are an abundance of probability distributions that can be used for this purpose, a natural question to ask is how to identify the probability distribution that best represents the particular situation under study. For example, is the exponential distribution a reasonable choice to represent the time to failure for a machining process, or is it better to use an empirical distribution function obtained from the historical time-to-failure data? Recognizing the fact that there is no true input model waiting to be found, the goal of stochastic input modeling is to obtain an approximation that captures the key characteristics of the system inputs. The development of a good input model requires the collection of as much information as possible about the relevant randomness in the system as well as the historical data consisting of the past realizations of the random variables of interest. In the presence of a data set, the input model can be identified by fitting a probability distribution to the historical data. However, it may be difficult and/or costly to collect data for the stochastic system under study; it can also be impossible to properly collect any data at all such as when the proposed system does not exist. In the absence of historical data, any relevant information (e.g., expert opinion and the conventional bounds suggested by the underlying physical situation) can be used for input modeling. This article addresses the key issues that arise in stochastic input modeling both in the presence and in the absence of historical data. The first step in input modeling is to identify the sources of randomness in the input environment of the system under study. Many stochastic systems contain multiple sources of uncertainty, e.g., the completion time of an item on a particular machine, the potential breakdown of the machine, and the percentage of defective items produced by the machine might be among the sources of uncertainty in a manufacturing setting. Throughout, the random vector X = (X1, X2, …, X K )′ is used to represent the collection of K different inputs of a stochastic system, where X k is the random variable denoting the kth system input. The K components of this random vector might also be correlated with each other. Therefore, the stochastic properties of the random inputs X k , k = 1, 2, …, K, are captured in the joint probability

sumption in period j, such that either the risk aversion index -u(x)/u'(x), or the risk aversion index -xu'(x)/u'(x), is a positive constant for all x > 0. In a second paper [6], it was further shown that this model, developed with the individual in mind, also gives rise to an induced theory of the firm under risk for the same class of utility functions. In the foregoing model, it was assumed that the individual's horizon was infinite (or known with certainty). In this paper, we consider the same basic model with three modifications. First, we postulate that the individual's, lifetime is a random variable with a known probability distribution. Second, we introduce a utility function intended to represent the individual's bequest motive. Third, we offer the individual the opportunity to purchase insurance on his life. It is found that when some or all of these modifications are made, all of the more important properties possessed by the optimal consumption and investment strategies under a certain horizon are preserved, albeit only under special conditions. In Section 2, the various components of the decision process are constructed. In the earlier model, the individual's objective was assumed to be the maximization of expected utility from consumption over time. Here, we postulate, more generally, that his objective is to maximize expected utility from consumption as long as he lives and from the bequest left upon his death. As before, the individual's resources are assumed to consist of an initial capital position (which may be negative) and a non-capital income stream. The latter, which may possess any time-shape, is assumed to be known with certainty and to terminate upon his death. In addition to insurance available at a fair' rate, the individual faces both financial opportunities (borrowing and lending) and an arbitrary number of productive investment opportunities. The interest rate is presumed to be known but may have any time-shape. The returns from the productive opportunities are assumed to be random variables, whose probability distributions may differ from period to period but are assumed to satisfy the no-easy-money condition. While no limit is placed on borrowing, the individual is required to be solvent at the time of his death with

The mutation operator is the only source of variation in Evolutionary Programming. In the past these have been human nominated and included the Gaussian, Cauchy, and the Levy distributions. We automatically design mutation operators (probability distributions) using Genetic Programming. This is done by using a standard Gaussian random number generator as the terminal set and and basic arithmetic operators as the function set. In other words, an arbitrary random number generator is a function of a randomly (Gaussian) generated number passed through an arbitrary function generated by Genetic Programming. Rather than engaging in the futile attempt to develop mutation operators for arbitrary benchmark functions (which is a consequence of the No Free Lunch theorems), we consider tailoring mutation operators for particular function classes. We draw functions from a function class (a probability distribution over a set of functions). The mutation probability distribution is trained on a set of function instances drawn from a given function class. It is then tested on a separate independent test set of function instances to confirm that the evolved probability distribution has indeed generalized to the function class. Initial results are highly encouraging: on each of the ten function classes the probability distributions generated using Genetic Programming outperform both the Gaussian and Cauchy distributions.

Whether an experiment yields qualitative or quantitative outcomes, methods of statistical analysis require that we focus on certain numerical aspects of the data (such as a sample proportion x/n, mean \( \bar{x} \), or standard deviation s). The concept of a random variable allows us to pass from the experimental outcomes themselves to a numerical function of the outcomes. There are two fundamentally different types of random variables—discrete random variables and continuous random variables. In this chapter, we examine the basic properties and discuss the most important examples of discrete variables. Chapter 4 focuses on continuous random variables.KeywordsNegative BinomialSingle SpikeletBinomial ExperimentUnacceptable VoltageAcceptance Sampling PlansThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The random variables in two‐stage programming under uncertainty are generally treated in a passive manner in that no information regarding the random variables or the process generating the random variables may be obtained. This paper develops the economics of information for the case in which the probability distributions are discrete. A multinomial process is assumed to generate the random variables, and the parameter vector of that process is assumed to be unknown. A Dirichlet prior distribution on the parameter vector is used, and the computation of the value of information thus involves a Dirichlet‐multinomial distribution on the random variables. An example involving producing to meet uncertain demands is presented.

3 - Random Variables and Their Probability Distributions

Random variables, each defined as numerical function on some sample space, will initially be regarded merely as useful tools for describing events. With every random variable, this chapter associates its probability distribution, or simply distribution. It begins with the conceptually and technically simplest case of transformations of one discrete random variable. The chapter introduces an important type of continuous distributions called an exponential distribution. In practical situations, we often deal with functions (or transformations) of random variables. The chapter considers a special class of random variables that are particularly important in medical research (survival analysis) but also in engineering (reliability analysis) and social sciences. In an engineering context, especially in problems of reliability of equipment, the hazard function is often called failure rate function.