Independent Normal Random Variables Research Articles

A remark on the distribution of products of independent normal random variables

We present a proof of the explicit formula of the probability density function of the product of normally distributed independent random variables using the multiplicative convolution formula for Meijer G functions.

Fatigue reliability analysis for structures with hybrid uncertainties combining quadratic response surface and polynomial chaos expansion

This study attempted to propose a novel efficient and accurate uncertainty analysis method for structural fatigue life prediction with hybrid uncertainties. First, PCE was adopted to deal with independent normal random variables, where polynomial coefficients were expressed as quadratic functions of interval variables. In order to determine unknown coefficients, a SCC collocation technique was applied combining with LHS. Interval bounds of the corresponding mean value and standard variance were then deduced, after which a fatigue reliability analysis method was proposed. Finally, three numerical examples were implemented to validate the feasibility and efficiency of the proposed method.

Probabilistic analysis of fatigue crack growth using efficient surrogate model

In this paper probabilistic analysis of a fatigue crack growth process governed by the Walker law is performed. Sensitivity of its parameters to uncertainties related to the material properties is considered. Based on statistical analysis of experimental data, they are modelled as independent normal random variables. The effect of these uncertainties on the fatigue crack growth behaviour of Centre Cracked Plate specimen is assessed using Monte-Carlo simulations and surrogate model based on polynomial chaos expansion. An efficient truncation scheme allows to discard the high order interactions having a weak effect on the mechanical response and consequently reduce the number of finite elements calls when identifying the unknown coefficients of the polynomial chaos expansion. The statistical moments namely the mean and the standard deviation in addition to the probability density function of the mechanical response are derived with a good accuracy. The obtained results show that the fatigue crack growth life is significantly affected by the uncertainties on the material properties since its coefficient of variation reaches 21% and that it follows a log-normal distribution law.

Tests Based on Kurtosis for Multivariate Normality

In this paper, we first transform a multivariate normal random vector into a random vector with elements that are approximately independent standard normal random variables. Then we propose the multivariate version generalized from the univariate normality test based on kurtosis from the literature. Power is investigated through the Monte Carlo Simulation with different significance level, dimension, and sample size. To assess the validity and accuracy of the new tests, we carry a comparative study with several other existing tests by selecting certain types of symmetric and asymmetric alternative distributions.

Moment-based ellipticity measurement as a statistical parameter estimation problem

We show that galaxy ellipticity estimation for weak gravitational lensing with unweighted image moments reduces to the problem of measuring a combination of the means of three independent normal random variables. Under very general assumptions, the intrinsic image moments of sources can be recovered from observations including effects such as the point-spread function and pixellation. Gaussian pixel noise turns these into three jointly normal random variables, the means of which are algebraically related to the ellipticity. We show that the random variables are approximately independent with known variances, and provide an algorithm for making them exactly independent. Once the framework is developed, we derive general properties of the ellipticity estimation problem, such as the signal-to-noise ratio, a generic form of an ellipticity estimator, and Cramér-Rao lower bounds for an unbiased estimator. We then derive the unbiased ellipticity estimator using unweighted image moments. We find that this unbiased estimator has a poorly behaved distribution and does not converge in practical applications, but demonstrates how to derive and understand the behaviour of new moment-based ellipticity estimators.

Products of normal, beta and gamma random variables: Stein operators and distributional theory

In this paper, we extend Stein’s method to products of independent beta, gamma, generalised gamma and mean zero normal random variables. In particular, we obtain Stein operators for mixed products of these distributions, which include the classical beta, gamma and normal Stein operators as special cases. These operators lead us to closed-form expressions involving the Meijer $G$-function for the probability density function and characteristic function of the mixed product of independent beta, gamma and central normal random variables.

Expected Number of Real Zeros of Gaussian Self-Reciprocal Random Algebraic Polynomials

Let $$Q_{n} (x) = \sum\nolimits_{i = 0}^{n} A_{i} x^{i}$$ be a random algebraic polynomial where the coefficients $$A_{0} ,A_{1} , \cdots$$ form a sequence of centered Gaussian random variables. Moreover, we assume that the increments $$\Delta_{j} = A_{j} - A_{j - 1}$$ , $$j = 0,1,2, \ldots ,n$$ are independent normal random variables with mean zero, finite variances $$\sigma_{j}^{2}$$ and the conventional notation of $$A_{ - 1} = 0$$ . The coefficients can be considered as $$n$$ consecutive observations of a Brownian motion. Assuming the symmetric property of $$A_{j}$$ ’s, we investigate some new and considerable results about the distribution of zeros. We prove that the expected number of real zeros of $$Q_{n} (x)$$ is asymptotically of order $$logn.$$

On Stein’s method for products of normal random variables and zero bias couplings

In this paper, we extend Stein’s method to the distribution of the product of $n$ independent mean zero normal random variables. A Stein equation is obtained for this class of distributions, which reduces to the classical normal Stein equation in the case $n=1$. This Stein equation motivates a generalisation of the zero bias transformation. We establish properties of this new transformation, and illustrate how they may be used together with the Stein equation to assess distributional distances for statistics that are asymptotically distributed as the product of independent central normal random variables. We end by proving some product normal approximation theorems.

The effects of normal mixtures and autocorrelation on the fraction non-conforming

ABSTRACTIn this article, the effects of mixtures of two normal distributions on the fraction non-conforming are studied in the context of capability analysis. When the output from several processes is mixed, the quality characteristic variables of the resulting mix may result in a normal mixture distribution. This can happen in cases such as monitoring an output from several suppliers, several machines, or several workers. This study considered the independence case and autocorrelated processes for a mixture of two normal distributions, using an autoregressive model of order one, AR(1). It is shown that the true attained process fraction non-conforming (corresponding to specific values for some capability index) can be very different from what is expected when the data are independent normal random variables.

A Very Short Proof that the Sum of Independent Normal Random Variables Is Normal

A Very Short Proof that the Sum of Independent Normal Random Variables Is Normal

Statistical Change Point Analysis in Air Temperature and Rainfall Time Series for Cocoa Research Institute of Nigeria, Ibadan, Oyo State, Nigeria

In this work, a cumsum approach is used to detect change-point in mean of an independent normal random variables. A multiple shift in the mean level was considered and show how such a problem can be straightforwardly addressed through the cumsum approach. Data gotten from Cocoa Research Institute of Nigeria were used and from the result of the analysis, a single change point was detected in the amount of rainfall and a multiple change point was detected in the amount of minimum temperature and no change point was detected in the amount of maximum temperature.

A Tutorial on Pharmacodynamic Scripting Facility inSimcyp.

The Simcyp Simulator provides a framework for mechanistic Physiologically‐Based Pharmacokinetic/Pharmacodynamic modeling of potentially interacting drugs. It also provides a scripting facility, using the Lua language, for developing customized pharmacodynamic and toxicity models driven by drug concentrations at the site of action. We present an overview of the scripting facility including the scripting language, the editor, and how scripts are embedded within the Simulator. Examples incorporating differential equations and including inter‐individual variability on parameters are presented.

Multi-choice probabilistic linear programming problem

In this paper, we present a linear programming model where the parameter space contains some multi-choice parameters. Alternative choices of multi-choice parameter are considered as random variables. Using interpolating polynomial for each multi-choice parameter, the model has been transformed into a non-linear mixed integer probabilistic programming problem. Then chance constrained programming technique is used to obtain an equivalent deterministic model of the transformed problem. To find the deterministic form of the objective function four different models namely, E-model, V-model, probability maximization model and fractile criterion model are used. Assuming the values of the multi-choice parameters as independent normal random variables, the methodology is presented. A numerical example is also presented to illustrate the methodology.

Confidence Regions and Intervals for Meta-Analysis Model Parameters

This article obtains confidence regions for the heteroscedastic, one-way random effects model’s parameters the heteroscedastic, one-way random effects model. The confidence regions are based on canonical representations of the restricted and profile likelihood functions in terms of independent normal random variables and χ2 random variables. These regions provide conservative confidence intervals for the common mean and heterogeneity variance. Mathematical details and the R code are available online as supplementary material.

Advanced Idiosyncratic Risk and Multi-Factor Models

A common scenario risk analysis employs a multiple factor model with assumed changes in the factors to obtain changes in non-factor variables. This analysis is sometimes designated as a “predictive stress scenario”. We choose to designate the factor model as a multifactor “CAPM” model, also related to APT models. We are not concerned with the details of the factor model. The risks of the scenarios are evaluated for a portfolio of instruments depending on changes of the variables. A systemic problem with this factor model approach is that the empirical correlations between pairs of non-factor variables are not maintained. We designate “advanced idiosyncratic risk” or “AI-Risk” as the correction to the risk of a predictive stress CAPM model scenario that includes a better approximation to the physical correlations. We have developed a new, effective, and economical AI-Risk formalism. There are two parts to AI-Risk. The first part, which is actually standard, has independent normal random variables with no correlations; we designate that as part of the CAPM model to get the variances correctly reproduced; we believe this is standard practice. The second part contains the corrections to the correlations; this is new. We evaluate AI-Risk for some stock portfolios. We find that the AI-Risk can be important, typically between 10%-50% of the predictive stress risk. The amount depends on the ratio of long/short (“L/S”) position values, with larger AI-Risk for higher L/S ratios.A technical remark is that we use correlations as the metric for model improvement. Therefore we work with z-scores or “unit” returns with the empirical volatility divided out.Finally, we indicate the generalization to cross-sectional regression factor models.In summary this work achieves two results: 1. A parsimonious formalism incorporating correlated idiosyncratic risk that gives better agreement with given empirical correlations (so risk is calculated closer to the physical world). 2. Indications of substantial addition to risk due to correlated idiosyncratic contributions, with respect to the smaller risk obtained in the standard model with uncorrelated idiosyncratic risk.

On the norming constants for normal maxima

Given n independent standard normal random variables, it is well known that their maxima Mn can be normalized such that their distribution converges to the Gumbel law. In a remarkable study, Hall proved that the Kolmogorov distance dn between the normalized Mn and its associated limit distribution is less than 3/log⁡n. In the present study, we propose a different set of norming constants that allow this upper bound to be decreased with dn≤C(m)/log⁡n for n≥m≥5. Furthermore, the function C(m) is computed explicitly, which satisfies C(m)≤1 and limm→∞⁡C(m)=1/3. As a consequence, some new and effective norming constants are provided using the asymptotic expansion of a Lambert W type function.

Engine Compartment UWB Channel Model for Intravehicular Wireless Sensor Networks

Intravehicular wireless sensor network (IVWSN) is a cutting edge research topic that delivers cost reduction, assembly, and maintenance efficiency by removing the wiring harnesses within the vehicle and enables the integration of new sensors into the locations inside a vehicle where cable connection is not possible. Providing energy efficiency through the low-duty-cycle operation and high reliability by exploiting the large bandwidth, ultrawideband (UWB) has been determined to be the most appropriate technology for IVWSNs. We investigate the UWB channel model for IVWSNs within the engine compartment of a vehicle by collecting an extensive amount of data for 19 × 19 links for different types and conditions of the vehicle. These include a Fiat Linea with engine off, Fiat Linea with engine on, and Peugeot Bipper with engine off. The path-loss exponent is estimated to be around 3.5 without exhibiting much variation when the engine is turned on and for different types of vehicles. The power variation around the expected path loss has lognormal distribution with zero mean and standard deviation in the range of [5.5, 6.3] dB for different types of vehicles with almost no variation when the engine of the same vehicle is turned on. The clustering phenomenon in the power delay profile (PDP) is well represented by a modified Saleh-Valenzuela (SV) model. The interarrival times of the clusters are modeled using a Weibull distribution. The cluster-amplitude and ray-amplitude decay functions are represented with a dual-slope linear model with breakpoint around 26.6 and 5.5 ns, respectively. The parameters of the Weibull distribution and these dual-slope linear models do not vary significantly for different types and conditions of the vehicle. The variations of the observed PDPs around the SV model is well modeled by independent normal random variables with zero mean and with a variance independent of the delay bin, and the type and condition of the vehicle. We propose a simulation model for the UWB channel within the engine compartment based on these findings and validate it by comparing the received energy and root mean square (RMS) delay spread of the generated and observed PDPs.

Stochastic Perturbation Analysis of Deteriorated Structures

The paper addresses a generalized perturbation method for the uncertainty analysis of concrete structures subject to reinforcement corrosion. The initiation phase is modelled by the well known Ficks 2ndlaw of diffusion. Deterioration models derived from experimental studies and a non-linear finite element (FE) model are used to simulate the structural behaviour of beams from the serviceability to the ultimate conditions. The time of corrosion initiation and the failure load are expressed by a Taylor series expansion about the origin of the space of independent standard normal random variables.

Distribution of the Ratio of Normal and Rice Random Variables

The ratio of independent random variables arises in many applied problems. The distribution of the ratio |X/Y| is studied when X and Y are independent Normal and Rice random variables, respectively. Ratios of such random variables have extensive applications in the analysis of noises in communication systems. The exact forms of probability density function (PDF), cumulative distribution function (CDF) and the existing moments are derived in terms of several special functions. As a special case, the PDF and CDF of the ratio of independent standard Normal and Rayleigh random variables have been obtained. Tabulations of associated percentage points and a computer program for generating tabulations are also given.

Interval Estimation of the Stress-Strength Reliability with Independent Normal Random Variables

This article develops a procedure to obtain highly accurate confidence interval estimates for the stress-strength reliability R = P(X > Y) where X and Y are data from independent normal distributions of unknown means and variances. Our method is based on third-order likelihood analysis and is compared to the conventional first-order likelihood ratio procedure as well as the approximate methods of Reiser and Guttman (1986) and Guo and Krishnamoorthy (2004). The use of our proposed method is illustrated by an empirical example and its superior accuracy in terms of coverage probability and error rate are examined through Monte Carlo simulation studies.