Normal Probability Distribution Function Research Articles

Electrical surface resistivity of conductive polymers – A non-Gaussian approach for determination of confidence intervals

Polyanilines doped with different acids (HCl, H 2SO 4 and dodecylbenzenesulfonic acid, DBSA) were prepared and their surface electrical conductivities were characterized in a four-probe device, connected to a real-time data acquisition board. Collected data were synchronized and conductivity calculations were performed. The conductive behavior of the polyanilines was investigated along the electrification time. This allowed for introduction of a non-Gaussian technique for determination of the confidence intervals of surface resistivity data. It is shown that the distribution of experimental surface resistivity data does not follow a normal probability distribution function (PDF). Thus, ordinary assumptions related to normal distribution of the experimental errors are wrong and must be corrected for proper determination of the confidence limits of measured resistivity values. It is shown here that confidence limits of resistivity values are asymmetrical and that distribution of experimental values can follow multimodal distributions.

Fuzzy Adaptive Particle Swarm Optimization for Bidding Strategy in Uniform Price Spot Market

In a deregulated electricity market, generators have to optimally bid to maximize their profit under incomplete information of other competing generators. This paper addresses an optimal bidding strategy of a thermal generator in a uniform price spot market considering a precise model of nonlinear operating cost function and minimum up/down constraints of unit commitment. The bidding behaviors of other competing generators are described using normal probability distribution function. Bidding strategy of a generator for each trading period in a day-ahead market is solved by fuzzy adaptive particle swarm optimization (FAPSO), where inertia weight is dynamically adjusted using fuzzy evaluation. FAPSO can dynamically follow the frequently changing market demand and supply in each trading interval. The effectiveness of the proposed approach is tested with examples and the results are compared with the solutions obtained using genetic algorithm (GA) approach and other versions of PSO.

The Preferred Structure of the Interannual Indian Monsoon Variability

The leading empirical orthogonal function (EOF) of the June-Sept. mean, rotational horizontal wind at 850 hPa and 200 hPa (over the region 12.5°S–42.5°N, 50°E–100°E) from 56 years (1948–2003) of reanalysis (from the National Centers for Environmental Prediction) shows strong anti-cyclonic circulation at upper levels, strong Indian Ocean cross-equatorial flow and on-shore flow over western India at lower levels . The associated principal component (PC) is correlated at the 0.75 level with the seasonal mean observed Indian Monsoon rainfall (IMR). Composite differences of vertically integrated divergence (surface to 800 hPa) and vorticity (surface to 500 hPa) between ``strong'' years (PC-1 exceeds one standard deviation σ) and ``weak'' years (PC-1 less than − σ) suggest increased rising motion and storminess over the Bay of Bengal and central India. Composite difference maps of station rainfall from the India Meteorological Department (IMD) between strong years and normal years (weak years and normal years) are statistically significant over central India, with strong (weak) years associated with increased (decreased) precipitation. In both cases the maps of rainfall anomalies are of one sign throughout India. The correlation of PC-1 with global seasonal mean SST is strong and negative over the eastern equatorial Pacific, but positive in a surrounding horse-shoe like region. Significant negative correlation occurs in the northwestern Indian Ocean. The lag/lead correlation between the NINO3 SST index and PC-1 is similar to but stronger than the NINO3/IMR correlation. Modest (but significant) negative correlation is seen when NINO3 leads PC-1 (or IMR) by one-two months. Strong negative correlation is seen when PC-1 (or IMR) leads NINO3. The projections of running five-day means of horizontal rotational winds at 850 and 200 hPa onto EOF-1 (after removing the seasonal mean for each year) were pooled for strong, normal and weak years. The strong and normal year probability distribution functions (pdfs) are nearly indistinguishable, but the weak year pdf has more weight for moderate negative values and in both extreme tails and shows some hint of bi-modality.

SBS/Pani · DBSA mixture plasticized with DOP and NCLS – Effect of the plasticizers on the probability density of volume resistivity measurements

Plasticized mixtures of styrene–butadiene–styrene block copolymer (SBS) and Polyaniline (Pani) were prepared in a Haake internal mixer. Two different plasticizers were used: dioctyl phthalate (DOP) and cashew nut shell liquid (CNSL). Pani and plasticizers were characterized by FTIR and the resistive behavior of plasticized mixtures was investigated along the electrification time. It is shown that obtained experimental data are subject to deterministic dynamic fluctuations that cannot be described by single normal probability distribution functions (PDF) along the time. The PDF analysis shows that obtained PDFs must be described as sums of at least three distinct Gaussian distributions with different areas. It is also shown that the Gaussian component with larger area may provide better representation of the measured volume resistivity values.

Statistical Distribution of Fiber-Reinforced Concrete Beam Test Data

Abstract Experience in post-crack performance testing of fiber-reinforced concrete (FRC) and shotcrete (FRS) using beams has demonstrated that variability both at first crack and in the post-crack range is high. When variability in performance is high it is important to carefully consider the within-set standard deviation in order to distinguish significant from insignificant differences between sets of data. It is also important to consider the characteristic distribution of results and use an appropriate probability distribution function (PDF) to estimate the standard deviation. The present investigation has sought to identify whether the Normal probability distribution function is the most appropriate model describing variability in cracking and post-crack performance parameters derived using beam tests.

PROBABILITY DISTRIBUTION OF CADMIUM PARTITIONING COEFFICIENTS OF CROPLAND SOILS

The solution to solid partitioning coefficient (Kd) is an important parameter in assessing the environmental and health risks of potentially toxic metals in soils. Ideally, the metal in solution phase should be determined at the field moisture condition under which the exposures take place. In reality, it is often represented by the concentration in extracts of a given soil-to-water ratio. Using cadmium (Cd) as an example, we demonstrated the uncertainties in determining the soil solution concentrations in cropland soils where the metal contents are only slightly higher than the baselines. Results of extraction experiments showed the Cd concentration of soil solution under simulated field moisture conditions (i.e., water-to-soil ratio of 0.5) tends to be stochastic and follows a normal distribution. The distribution of Kd of two California cropland soils was quantitatively described by a normal probability distribution function. The normal distribution pattern was further validated by data collected in the field. When Kd is characterized in the probabilistic terms, the risks of environmental and health harm of metals in the cropland soils may be more appropriately assessed. (Soil Science 2007;172:132‐140)

Uncertainty assessment for the thermal expansion of simulated fuel via the Monte Carlo method

The uncertainty of the thermal expansion of simulated fuel was evaluated based on the Monte Carlo method. The uncertainties of the thermal expansion were divided into three components: initial sample length, system calibration factor, and an iterative measurement. For each component, a normal probability distribution function was assumed and the extended uncertainty was evaluated for various samples and batches. The results were compared with an international standard guide. Furthermore, the Shapiro–Wilk test was performed to establish a normality of the estimated sample means.

Probability Analysis of Rainfall of Bankura for Design of Soil and Water Conservation Structures

Daily rainfall data of 21 years (from 1983 to 2003) for Bankura district of West Bengal were analyzed to ascertain their fit to several probability distributions. The series generated by using Weibull's plotting position formula was treated as the Observed series. The expected values were estimated by Gumbel, Log Pearson Type -III and Log Normal probability distribution functions. The goodness of fit was determined by X2 test. Linear probability model was also developed using the parameter Frequency Factor for the best distribution. The analysis indicated that Log Pearson Type -III gave the closest fit to the observed data and it is recommended to forecast the annual maximum I-day rainfall for the design return period. The theoretical annual maximum one day rainfall values using Log Pearson Type -III for 5, 10, 15,25 and 50 years return period were 116, 155, 165, 185 and 207 mm, respectively. These expected I-day maximum rainfall values for the desired return period could be used for hydrological design of different soil and water conservation structures in watershed development projects being implemented under different programme in Bankura district. In this paper, the usefulness of predicted annual maximum I-day rainfall for the desired return period has been verified in the design of a masonry check dam in Kashibedia micro- watershed, which is located at Bankura I block of Bankura district.

Plume structure in high-Rayleigh-number convection

Near-wall structures in turbulent natural convection at Rayleigh numbers of at A Schmidt number of 602 are visualized by a new method of driving the convection across a fine membrane using concentration differences of sodium chloride. The visualizations show the near-wall flow to consist of sheet plumes. A wide variety of large-scale flow cells, scaling with the cross-section dimension, are observed. Multiple large-scale flow cells are seen at aspect ratio (AR)= 0.65, while only a single circulation cell is detected at AR= 0.435. The cells (or the mean wind) are driven by plumes coming together to form columns of rising lighter fluid. The wind in turn aligns the sheet plumes along the direction of shear. the mean wind direction is seen to change with time. The near-wall dynamics show plumes initiated at points, which elongate to form sheets and then merge. Increase in rayleigh number results in a larger number of closely and regularly spaced plumes. The plume spacings show a common log–normal probability distribution function, independent of the rayleigh number and the aspect ratio. We propose that the near-wall structure is made of laminar natural-convection boundary layers, which become unstable to give rise to sheet plumes, and show that the predictions of a model constructed on this hypothesis match the experiments. Based on these findings, we conclude that in the presence of a mean wind, the local near-wall boundary layers associated with each sheet plume in high-rayleigh-number turbulent natural convection are likely to be laminar mixed convection type.

Ajustes de funções de distribuição de probabilidade à radiação solar global no Estado do Rio Grande do Sul

O objetivo deste trabalho foi verificar o ajuste das séries de dados de radiação solar global média decendial, de 22 municípios do Estado do Rio Grande do Sul, às funções de distribuições de probabilidade normal, log-normal, gama, gumbel e weibull. Aplicou-se o teste de aderência de Kolmogorov-Smirnov, nas 792 séries de dados (22 municípios x 36 decêndios) de radiação solar global média decendial, para verificar o ajuste dos dados às distribuições normal, log-normal, gama, gumbel e weibull, totalizando 3.960 testes. Os dados decendiais de radiação solar global média se ajustam às funções de distribuições de probabilidade normal, log-normal, gama, gumbel e weibull, e apresentam melhor ajuste à função de distribuição de probabilidade normal.

Investment Forecasting With Multivariate Linear Regression

The procedure for estimating probabilities of future investment returns using time-shifted indexes is based on the simple principle that a multi-dimensional conditional probability distribution can be envisioned involving investment total returns (for a single investment or a fixed portfolio of investments as the dependent variate) and the total returns of one or more market indexes shifted forward in time by one year as the independent variate(s). Furthermore, it is shown that the probability function involving the dependent investment return is decoupled from the independent index return variate(s) by linear regression. The deviations of the investment returns from the regressed line, surface, or hypersurface is characterized with a one-dimensional normal cumulative probability distribution function, (estimated by the Students t-distribution), to determine the probabilities for the next years returns. Thus, the expected investment total returns at a given probability level is the regressed return based on last year's index return(s) plus the accepted risk (deviation) increment. Adjustments are made in the statistical parameters to account for the extrapolation of the regression information beyond the period covered by the data in the return records. A hierarchy of regression variates is determined by selecting the variates, one at a time, which will provide the greatest error reduction; then, the viability of the last variate chosen is assessed by the F-significance test. It is concluded that the use of multiple time-shifted indexes can improve the estimates of future investment returns. Also, it was shown that properly diversified asset allocations could be established to take advantage of the predictive power of the time-shifted index technique. It is obvious that the approach could be employed to further refine the allocations within a broad asset class - to establish, for example, the relative allocations between large company, small company, and specialty company stock.

Autorun Persistence of Hydrologic Design

Persistence is the most important property in any hydrologic design concerning the storage capacity of reservoirs, average return periods, failure risks, and drought properties. Its consideration in analytical derivations of design criteria presents difficulties, especially in autocorrelated hydrologic processes, and for this reason, most often the analytical expressions are obtained on the basis of non-persistent (independent) cases. Although the conventional autocorrelation coefficients and function are used in many hydrological design problems, the very definition of the autocorrelation function requires that the underlying hydrologic process generating mechanism abide with normal (Gaussian) probability distribution function in addition to other restrictive assumptions. Since almost all of the analytical stochastic approaches are based on the normality assumption, it is necessary to transform non-Gaussian distributions to the normal distribution to use analytical expressions. During the transformation process, the very genuine persistence property of the basic hydrologic variables is not preserved, even when the statistical parameters such as the average, standard deviation, skewness coefficient, kurtosis, etc., are maintained in the transformed data. This shortcoming in the autocorrelation function is by-passed by the introduction and use of an autorun function, which is probability distribution free and a robust parameter. Its basis is the conditional probability statement which does not require any assumptions in practical applications. Various practical and simple hydrological design quantities are developed on the basis of the autorun coefficients without considering the conventional autocorrelation structure. The application of methodology is presented for John’s river in Florida.

Crop‐Yield Distributions Revisited

AbstractThis article revisits the issue of crop‐yield distributions using improved model specifications, estimation, and testing procedures that address the concerns raised in recent literature, which could have invalidated previous findings of yield nonnormality. It concludes that some aggregate and farm‐level yield distributions are nonnormal, kurtotic, and right or left skewed, depending on the circumstances. The advantages of utilizing nonnormal versus normal probability distribution function models, and the consequences of incorrectly assuming crop‐yield normality are explored.

Aerosol Retrievals from Individual AVHRR Channels. Part II: Quality Control, Probability Distribution Functions, Information Content, and Consistency Checks of Retrievals

This second part of a two-part study evaluates retrievals of aerosol optical depths, τ1 and τ2, in Advanced Very High Resolution Radiometer (AVHRR) channels 1 and 2 centered at λ1 = 0.63 and λ2 = 0.83 μm, and an effective Ångström exponent, α, derived therefrom as α = −ln(τ1/τ2)/ln(λ1/λ2). The retrievals are made with the Second Simulation of the Satellite Signal in the Solar Spectrum (6S) radiative transfer model from four NOAA-14 AVHRR datasets, collected between February 1998 and May 1999 in the latitudinal belt of 5°–25°S. A series of quality control (QC) checks applied to the retrievals to identify outliers are described. These remove a total of ∼1% of points, which presumably originate from channel misregistration, residual cloud in AVHRR cloud-screened pixels, and substantial deviations from the assumptions used in the retrieval model (e.g., bright coastal and high altitude inland waters). First, from examining histograms of the derived parameters it is found that τ and α are accurately fit by lognormal and normal probability distribution functions (PDFs), respectively. Second, the scattergrams τ1 versus τ2 are analyzed to see if they form a coherent pattern. They do indeed converge at the origin, as expected, but frequently are outside of the expected domain in τ1–τ2 space, defined by two straight lines corresponding to α = 0 and α = 2. This results in a low bias in α, which tends to fill in an interval of α ∈ [−1, 1] rather than α ∈ [0, 2]. Third, scattergrams of α versus τ are used to empirically confirm a previously drawn theoretical conclusion that errors in α are inversely proportional to τ. More in-depth quantitative analyses suggest that the AVHRR-derived Ångström exponent becomes progressively more meaningful when τ > 0.2. Geographical trends are studied to demonstrate that the selected ocean area is reasonably uniform to justify application of consistency checks to reveal angular trends in the retrievals. These checks show that in most cases, the artifacts in the retrieved τ and α are statistically insignificant. On average, the analyses suggest that the retrieved τ1, τ2, and α show a high degree of self- and interconsistency, with the exception of a troublesome May 1999 dataset. The most prominent problem noticed so far is the inconsistency between τ1 and τ2, persistent from one dataset to another, which calls for fine-tuning some (not aerosol-model related) elements of the retrieval algorithm. These adjustments will be discussed elsewhere.

Probability bounds given by hypercherry trees

In this article new lower and upper bounds are given for the probability of the union of events. For this purpose the new concept of hypercherry trees has been introduced. Earlier the concept of cherry tree and its application for bounding the probability of union of events was introduced by Bukszár and Prékopa. This, based on the cherry tree bound, is always an upper bound, and it can be regarded as a generalisation of the upper bound introduced by Hunter by means of maximum weight spanning trees. Later the Hunter bound was generalised by Tomescu. He used the concept of hypertrees in the framework of uniform hypergraphs and on the basis of these new hypergraph structures it became possible to define not only upper but also lower bounds on the probability of union of events. The new bounds of the paper are generalisations of Tomescu's bounds in the same sense as the upper bound by Bukszár and Prékopa was a generalisation of the Hunter bound. The efficiency of the new bounds is illustrated on some test problems according to multivariate normal probability distribution function calculations.

A statistical analysis of a growth series of the Permian nectridean Diplocaulus magnicornis showing two-stage ontogeny

ABSTRACT We use probability plots to compare metric data for a growth series of the Permian nectridean Diplocaulus magnicornis Cope to a normal (Gaussian) probability distribution function and demonstrate that the ontogeny of Diplocaulus is not a single continuous process. The plots are bimodal, revealing a population of small, ontogenetically young animals whose growth is governed by a distinct process. This population, which comprises ~20% of the growth series, is not evident in allometric studies. These young animals have a mean mid-line skull length of ~21 mm, and their orbits are relatively larger, and rounder, than the orbits of the older animals, whose mean skull length is ~94 mm. Although there is no evidence of lissamphibian-type metamorphosis in the Nectridea, the probability plots clearly demonstrate a two-stage ontogeny, best described as an ontogenetic morphological saltation. This saltation may have been an alternative strategy to the metamorphic heterochrony found in lissamphibians and their relatives.

On the concavity of multivariate probability distribution functions

We prove that the multivariate standard normal probability distribution function is concave for large argument values. The method of proof allows for the derivation of similar statements for other types of multivariate probability distribution functions too. The result has important application, e.g., in probabilistic constrained stochastic programming problems.

Drought parameters in relation to truncation levels

Drought parameters, namely the longest duration, largest severity and the intensity have been calculated in relation to the truncation levels ranging from 0 to −0\5 in the standardized domain for the normal, gamma and log-normal probability distribution functions (PDFs) of the drought variables. The drought variables taken in the investigation are the annual rainfall and runoff time-series evolving randomly and obeying the Markovian dependence. The analysis showed that the assumption of independence of the drought duration and intensity works well in deriving the expression for drought severity at various truncation levels. An estimate of drought intensity has been realized from the concept of the truncated normal distribution of the standardized form of the time-series of drought variables in the normalized domain. Furthermore the non-normality and the dependence in the time-series have a significant effect on the drought parameters at the truncation levels under consideration. Copyright © 2000 John Wiley & Sons, Ltd.

Impact of Random Variables Probability Distribution on Public Health Risk Assessment from Contaminated Soil

Probabilistic methods are now being applied increasingly to public health risk assessment instead of the deterministic, conservative, point estimates. An essential part of the probabilistic methods is the selection of probability distribution functions to represent the uncertainty of the random variables considered. We study the effect of selection of different probability distribution functions on the probabilistic outcome using the first-order reliability method (FORM). An example of cancer risk resulting from dermal contact with benzo(a)pyrene (BaP)-contaminated soil is given. Cancer potency factor, soil concentration, and fraction of skin area exposed were assigned normal, lognormal, and uniform probability distribution functions, and the effect of probability of exceeding a target risk level (termed the probability of failure) and sensitivity measures were studied. We investigated the question: what happens when one assumes different distribution shapes with the same mean and standard deviation? The results indicate that the selection of a probability distribution function for the random variables had a moderate impact on the probability of failure when the target risk is at the 50th percentile level, while the impact was much larger for a 95th target risk percentile. We conclude that the probability distribution will have a large impact because in most cases the regulatory threshold risk is at the tail end of the risk distribution. The impact of the distributions on probabilistic sensitivity, however, showed a reversed trend, where the impact was slightly more appreciable for the 50th percentile than for the 95th percentile. The selection of distribution shape did not, however, alter the order of probabilistic sensitivity of the basic random variables.

Improved Bounds and Simulation Procedures on the Value of the Multivariate Normal Probability Distribution Function

Improved bounds and simulation procedures on the value of the multivariate normal probability distribution function value are given in the paper. The author's variance reduction technique was based on the Bonferroni bounds involving the first two binomial moments only. The new variance reduction technique is adapted to the most refined new bounds developed in the last decade for the estimation the probability of union respectively intersection of events. Numerical test results prove the efficiency of the simulation procedures described in the paper.