An algorithm to compute the cdf of the product of two normal random variables

Determining distributions of the functions of random variables is a very important problem with a wide range of applications in Risk Management, Finance, Economics, Science, and many other areas. This paper develops the theory on both density and distribution functions for the quotient Y = X 1 X 2 and the ratio of one variable over the sum of two variables Z = X 1 X 1 + X 2 of two dependent or independent random variables X 1 and X 2 by using copulas to capture the structures between X 1 and X 2 . Thereafter, we extend the theory by establishing the density and distribution functions for the quotients Y = X 1 X 2 and Z = X 1 X 1 + X 2 of two dependent normal random variables X 1 and X 2 in the case of Gaussian copulas. We then develop the theory on the median for the ratios of both Y and Z on two normal random variables X 1 and X 2 . Furthermore, we extend the result of median for Z to a larger family of symmetric distributions and symmetric copulas of X 1 and X 2 . Our results are the foundation of any further study that relies on the density and cumulative probability functions of ratios for two dependent or independent random variables. Since the densities and distributions of the ratios of both Y and Z are in terms of integrals and are very complicated, their exact forms cannot be obtained. To circumvent the difficulty, this paper introduces the Monte Carlo algorithm, numerical analysis, and graphical approach to efficiently compute the complicated integrals and study the behaviors of density and distribution. We illustrate our proposed approaches by using a simulation study with ratios of normal random variables on several different copulas, including Gaussian, Student-t, Clayton, Gumbel, Frank, and Joe Copulas. We find that copulas make big impacts from different Copulas on behavior of distributions, especially on median, spread, scale and skewness effects. In addition, we also discuss the behaviors via all copulas above with the same Kendall’s coefficient. The approaches developed in this paper are flexible and have a wide range of applications for both symmetric and non-symmetric distributions and also for both skewed and non-skewed copulas with absolutely continuous random variables that could contain a negative range, for instance, generalized skewed-t distribution and skewed-t Copulas. Thus, our findings are useful for academics, practitioners, and policy makers.

The multilevel Monte Carlo (MLMC) method has been used for a wide variety of stochastic applications. In this paper we consider its use in situations in which input random variables can be replaced by similar approximate random variables which can be computed much more cheaply. A nested MLMC approach is adopted in which a two-level treatment of the approximated random variables is embedded within a standard MLMC application. We analyse the resulting nested MLMC variance in the specific context of an SDE discretisation in which Normal random variables can be replaced by approximately Normal random variables, and provide numerical results to support the analysis.

ABSTRACTConvolutions of normal random variables often arise in engineering problems, and the probability densities of the sums of these random variables are known in the literature. There are practical situations where specification limits on a process are imposed externally, and the product is typically scrapped if its performance does not fall in the specification range. As such, the actual distribution after inspection is truncated. Despite the practical importance of the role of truncated distributions, there has been little work on the theoretical foundation of convolutions associated with truncated random variables. This is paramount, since convolutions are often used as an important standard in statistical tolerance analysis. In this article, the convolutions of the combinations of truncated normal and truncated skew normal random variables on double and triple truncations are developed. The successful completion of this research task on convolution could help obtain a better understanding of integ...

On the Moments of Ratio of Quadratic Forms in Normal Random Variables

The literature is replete with acknowledgments that spatial autocorrelation (SA) inflates the variance of a random variable (RV), and that it also may alter other RV distributional properties. In most studies, impacts of SA have been examined only for the three most commonly used distributions: the normal, Poisson (and its negative binomial counterpart), and binomial distributions; much less is known about its effects on two other RVs that are utilized in GIScience research: the beta and the multinomial. The beta distribution—which is considered to be very flexible because it can mimic a uniform, exponential, sinusoidal, and normal RV—can be utilized to analyze the radiance of a remotely sensed image, for example. The multinomial distribution, a generalization of the binomial distribution, has been widely used for land use classification, and to describe land use change. The literature also suggests that RV impacts of negative SA, a neglected topic in spatial analysis, may differ from those of positive SA, at least for some RVs (e.g., the Poisson RV). The purpose of this article is to extend the investigation of effects of SA to beta and multinomial RVs, with both positive SA and negative SA assessed and contrasted with each other, using simulation experiments. The simulation experiments are designed to support this assessment. One of the major discoveries is that impacts of positive SA and negative SA behave similarly when a RV conforms to a normal distribution; however, maximum negative SA is unable to materialize for asymmetric RV, whereas positive SA always converges upon its maximum.

The state of technical objects can be checked by comparing the average values ​​of samples of measurements of their parameters. Non-parametric statistics allow you to draw statistical conclusions, in particular, to evaluate distribution characteristics and test statistical hypotheses, without, as a rule, weakly justified assumptions that the distribution function of the sample elements belongs to one or another parametric family. For example, it is widely believed that statistical data often follow a normal distribution. Meanwhile, the analysis of practical results, in particular, measurement errors, always leads to the same conclusion - in the vast majority of cases, real distributions differ significantly from normal ones. In the tasks of non-destructive testing, technical diagnostics and monitoring of the state of objects, non-parametric statistics are the basis for research and decision-making about suitability or quality. Despite the large number of publications dedicated to research on specific issues of the use of non-parametric statistics methods, the internal structure of this scientific direction remains an actual problem even at the present time. Most often, in practice, the problem of belonging of samples to one general population is solved. One of the main verification criteria is the Student criterion. But, in order to use them, it is necessary to check samples for "normality". Uncritical use of the normality hypothesis often leads to significant errors, for example, when rejecting the results of observations (outliers), during statistical quality control, and in other cases. If the samples are short (n⩽20), and the measurements are normal random variables, then classical mathematical statistics suggests solving this problem with a modified Student's test [1]. The power of this criterion is no worse than the classic Student's criterion. This is especially evident with short samples (n≤20). In this article, by conducting computational experiments, the informativeness of the modified and classic Student's criterion was compared. The study was conducted for samples of random variables with three distribution laws - logistic, exponential and Rayleigh. The result showed that the modified Student's test has more power than the classical test. But, as expected, both criteria are sensitive to the symmetry of the distribution law - the greater its asymmetry, the greater the error of accepting the null hypothesis.

This paper addresses the problem of profit-oriented disassembly line design and balancing considering partial disassembly, presence of hazardous parts and uncertainty of task processing times. Few papers have studied the stochastic disassembly line balancing problem and existing approaches have focused on heuristic and metaheuristic methods. Most existing work has concentrated on complete disassembly where task times are assumed to be normal random variables and where AND/OR graphs are not considered. The objective of this paper is the design of a serial line that obtains the maximum revenue and then balances the workload under uncertainty. The processing time of a disassembly task is assumed to be a random variable with any known probability distribution. An AND/OR graph is used to model the precedence relationships among tasks. Stochastic programming models and exact-based solution approaches combining the L-shaped algorithm and Monte Carlo sampling techniques are proposed. The relevance and applicability of the proposed models and solution methods are shown by solving efficiently a set of disassembly problem instances from the literature.

This chapter provides a self-contained review of the foundations of probability theory, in order to fix notations and introduce mathematical objects employed in the remaining chapters. In particular we stress the notions of measurability, related to what it is actually possible to observe when performing an experiment, and statistical independence. We present several tools to deal with random variables; in particular we focus on the normal random variables, a cornerstone in the theory of random phenomena. We conclude presenting the law of large numbers and the central limit theorem.

A goodness-of-fit statistic D 2 is introduced for use in multinomial distributions. Pearson's X2 and D 2 are both approximately normally distributed when the sample size N is not large relative to the number of multinomial categories k. Under sequences of local alternative hypotheses the test based on D 2 exhibits moderate power when the X2 test is biased. Application is made to the analysis of large sparse contingency tables. A theorem is presented that describes the likelihood ratio Λ for testing a simple multinomial distribution against a mixture of multinomial distributions. A wide variety of mixing distributions is considered, and the D 2 statistic is a special case of log Λ when testing for Dirichlet mixtures of multinomial distributions. In the case where N is large, relative to k, X2 and D 2 + k behave approximately as chi-squared random variables and differ by a very small amount under the null hypothesis. In this situation X2 and D 2 + k will generally yield the same inference. With a large, sparse multinomial distribution where N and k are both large, X2 and D 2 will usually behave as normal random variables with means and variances that are unrelated to the chi-squared distribution. In the sparse distribution, X2 and D 2 are not equivalent and X2 will accept the null hypothesis too often under certain alternative hypotheses. When testing for independence of rows and columns in a large, sparse two-dimensional contingency table, estimate the cell means conditional on the marginal totals in the usual manner. The [Dcirc]2 statistic resulting from substituting these estimated means for the cell means is highly correlated with D 2. The normalizing mean and variance of [Dcirc]2 under the null hypothesis of independence are given at the end of the article.

Target association in sensor data fusion often assumes that both the target tracks and the measurements are described with Gaussian random variables. For some sensor reports, such as passive acoustics, this assumption creates a poor approximation. For the association of measurement to target track, significant errors can occur as a uniform distribution is warped such that the centroid is weighted significantly more than the edges when a single Gaussian is used as the approximation. Using the chi-squared metric to associate a new measurement to an existing target track under this condition can increase the likelihood of association or reduce it significantly. In this paper, a fuzzy-logic approach to data association is enhanced. The original approach showed that it could emulate the chi-squared metric comparing two Gaussian random variables. Here, the approach is enhanced to handle the cases of two uniformly distributed random variables and the case where a uniform measurement is compared to a Gaussian track

The exact distribution of the linear combination α X + β Y is derived when X and Y are normal and logistic random variables distributed independently of each other. Tabulations of the associated percentage points are given along with a computer program to generate them. This work is motivated by problems in reliability engineering.

The exact distribution of the linear combination ?X+βY is derived when X and Y are normal and Laplace random variables distributed independently of each other. A program in MAPLE is provided to compute the associated percentage points.

The normal and Laplace are the two earliest known continuous distributions in statistics and the two most popular models for analyzing symmetric data. In this note, the exact distribution of the ratio | X / Y | is derived when X and Y are respectively normal and Laplace random variables distributed independently of each other. A MAPLE program is provided for computing the associated percentage points. An application of the derived distribution is provided to a discriminant problem.

We present a pricing kernel that summarizes well the main features of the dynamics of interest rates and risk in postwar U.S. data and use it to uncover how the pricing kernel has moved with the short rate in this data. Our findings imply that standard monetary models miss an essential link between the central bank instrument and the economic activity that monetary policy is intended to affect, and thus we call for a new approach to monetary policy analysis. We sketch a new approach using an economic model based on our pricing kernel. The model incorporates the key relationships between policy and risk movements in an unconventional way: the central bank's policy changes are viewed as primarily intended to compensate for exogenous business cycle fluctuations in risk that threaten to push inflation off target. This model, while an improvement on standard models, is considered just a starting point for their revision. It leads to critical questions that researchers need to answer as they continue to revise their approach to monetary policy analysis. Copyright © 2009 by the National Bureau of Economic Research.

Calculations of level probabilities for normal random variables with unequal variances with applications to Bartholomew's test in unbalanced one-way models