Showing papers on "Cumulative distribution function published in 1981"
TL;DR: The contrast-response function of a class of first order intemeurons in the fly's compound eye approximates to the cumulative probability distribution of contrast levels in natural scenes, showing that this matching enables the neurons to encode contrast fluctuations most efficiently.
Abstract: The contrast-response function of a class of first order interneurons in the fly's compound eye approximates to the cumulative probability distribution of contrast levels in natural scenes. Elementary information theory shows that this matching enables the neurons to encode contrast fluctuations most efficiently.
962 citations
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23 Sep 1981TL;DR: In this article, the authors introduce Probability One-Dimension Random Variables Functions of One Random Variable and Expectation Joint Probability Distributions Some Important Discrete Distributions some Important Continuous Distributions The Normal Distribution Random Samples and Sampling Distributions Parameter Estimation Tests of Hypotheses Design and Analysis of Single Factor Experiments The Analysis of Variance Design of Experiments with Several Factors Simple Linear Regression and Correlation Multiple Regression Nonparametric Statistics Statistical Quality Control and Reliability Engineering Stochastic Processes and Queuing Statistical Decision Theory
Abstract: Introduction and Data Description An Introduction to Probability One-Dimension Random Variables Functions of One Random Variable and Expectation Joint Probability Distributions Some Important Discrete Distributions Some Important Continuous Distributions The Normal Distribution Random Samples and Sampling Distributions Parameter Estimation Tests of Hypotheses Design and Analysis of Single-Factor Experiments The Analysis of Variance Design of Experiments with Several Factors Simple Linear Regression and Correlation Multiple Regression Nonparametric Statistics Statistical Quality Control and Reliability Engineering Stochastic Processes and Queuing Statistical Decision Theory.
407 citations
TL;DR: In this paper, a response spectrum method for stationary random vibration analysis of linear, multi-degree-of-freedom systems is developed, which is based on the assumption that the input excitation is a wideband, stationary Gaussian process and the response is stationary.
Abstract: A response spectrum method for stationary random vibration analysis of linear, multi-degree-of-freedom systems is developed. The method is based on the assumption that the input excitation is a wide-band, stationary Gaussian process and the response is stationary. However, it can also be used as a good approximation for the response to a transient stationary Gaussian input with a duration several times longer than the fundamental period of the system. Various response quantities, including the mean-squares of the response and its time derivative, the response mean frequency, and the cumulative distribution and the mean and variance of the peak response are obtained in terms of the ordinates of the mean response spectrum of the input excitation and the modal properties of the system. The formulation includes the cross-correlation between modal responses, which is shown to be significant for modes with closely spaced natural frequencies.
The proposed procedure is demonstrated for an example structure that is subjected to an ensemble of earthquake-induced base excitations. Computed results based on the response spectrum method are in close agreement with simulation results obtained from time-history dynamic analysis. The significance of closely spaced modes and the error associated with a conventional method that neglects the modal correlations are also demonstrated.
308 citations
TL;DR: In this article, it was shown that in Monte Carlo simulations due to the combined effects of randomness and finite size effects one can only measure the most probable value of the correlation functions and not their average.
Abstract: In random magnets the probability distribution of the correlation function at large distance is not concentrated around its average. To illustrate this idea two examples are studied: a random Ising chain and a random cubic chain. The extension of the findings to higher dimension and the connection to Harris' criterion (1974) are discussed heuristically. It is concluded that in Monte Carlo simulations due to the combined effects of randomness and finite size effects one can only measure the most probable value of the correlation functions and not their average.
44 citations
TL;DR: A rejection/squeeze algorithm which requires the evaluation of one integral at a crucial stage of the computer generation of a random variable X with a given characteristic function under mild conditions on the characteristic function is proposed.
Abstract: We consider the problem of the computer generation of a random variable X with a given characteristic function when the corresponding density and distribution function are not explicitly known or have complicated explicit formulas. Under mild conditions on the characteristic function, we propose and analyze a rejection/squeeze algorithm which requires the evaluation of one integral at a crucial stage.
39 citations
TL;DR: In this article, a method of estimating exponential survival distributions where the expected survival time is linear in several independent variables is presented, and a modification, applying the exponential cumulative distribution function, is made to handle data where not all individuals have an observed time of death.
39 citations
TL;DR: In this article, two alternatives to Hastings' approximation to the inverse of the normal cumulative distribution function are given, one is adequate for the majority of practical problems, while the other is of use in the extreme tails.
Abstract: Two alternatives are given to Hastings' approximation to the inverse of the normal cumulative distribution function: one is adequate for the majority of practical problems, while the other is of use in the extreme tails. Both require fewer constants than Hastings' and have greater accuracy, particularly when measured in terms of relative error.
23 citations
01 Jan 1981
TL;DR: In this article, a technique was developed for the approximation of bivariate cumulative distribution function values through the use of the bivariate Pearson family when moments of the distribution to be approximated are known.
Abstract: A technique is developed for the approximation of bivariate cumulative distribution function values through the use of the bivariate Pearson family when moments of the distribution to be approximated are known. The method utilizes a factorization of the joint density function into the product of a marginal density function and an associated conditional density, permitting the expression of the double integral in a form amenable to the use of specialized Gaussian-type quadrature techniques for numerical evaluation of cumulative probabilities. Such an approach requires moments of truncated Pearson distributions, for which a recurrence relation is presented, and moments of the conditional distributions for which quartic expressions are used. It is shown that results are of high precision when this technique is employed to evaluate cumulative distribution functions that are of the Pearson class.
18 citations
TL;DR: In this paper, the probability functions, moments and cumulative distributions of the normalized intensity z = |E|2, depending on the space-group symmetry of the crystal and its chemical composition, were reduced to a simple, unified representation, by reconsidering some mathematical properties of the corresponding asymptotic expansions.
Abstract: Probability functions, moments and cumulative distributions of the normalized intensity z = |E|2, depending on the space-group symmetry of the crystal and its chemical composition, have been investigated. The probability functions were reduced to a simple, unified representation, by reconsidering some mathematical properties of the corresponding asymptotic expansions. A subsequent unified derivation of the first four even moments of |E|, in terms of symmetry and composition, leads to (i) simple and readily computable expressions for (|E|4), (|E|6) and (|E|8) and (ii) a significant simplification of the expansion coefficients which appear in the above asymptotic expansions. The convergence of these expansions is discussed and illustrated by a numerical example. It is shown that the Edgeworth arrangement of these asymptotic expansions is superior to the frequently given Gram-Charlier one. The dependence of the fourth moment of |E| on atomic heterogeneity and the generalized cumulative distribution functions N(|E|) are illustrated for all the symmorphic space groups. The results of this study are directly applicable to practical intensity statistics for structures containing all the atoms in general positions.
14 citations
TL;DR: A simple and efficient algorithm is developed for computing the conditional cumulative distribution function for estimates of the magnitude-squared coherence between two wide-sense stationary real zero-mean Gaussian stochastic processes.
Abstract: A simple and efficient algorithm is developed for computing the conditional cumulative distribution function for estimates of the magnitude-squared coherence between two wide-sense stationary real zero-mean Gaussian stochastic processes. This algorithm is applied to computing the confidence bounds for the estimates.
01 Jan 1981
TL;DR: There are many random variables whose ranget consists of the positive real line (0, ∞) or a portion of the real line as discussed by the authors. But none of these random variables are known to us.
Abstract: There are many random variables whose ranget† consists of the positive real line (0, ∞) or a portion of the real line.
TL;DR: In this article, a nonlinear distortion that converts the magnitude-squared coherence estimate to a near-Gaussian random variable is investigated, and simple approximations for the mean and variance of the nonlinearly distorted magnitude-square coherence estimates are presented.
Abstract: We investigate a nonlinear distortion that converts the magnitude-squared coherence estimate to a near-Gaussian random variable. In particular, we present simple approximations for the mean and variance of this nonlinearly distorted magnitude-squared coherence estimate and fit a Gaussian cumulative distribution function over a wide range of parameters.
TL;DR: In this article, two variations of the problem of choosing the largest of N independent and identically distributed (iid) random variables with sampling cost are studied and both the optimal strategy and the distribution of the stopping variable are discussed.
Abstract: Two variations of the problem of choosing the largest of N independent and identically distributed (iid) random variables with sampling cost are studied. In the first case it is assumed that the underlying distribution is continuous and known, but the information obtained by sampling is whether the sampled variable is larger or smaller than some given level. In the second case it is assumed that the distribution of the random variables is continuous but unknown, and the information obtained is the rank of the sampled variable relative to the other variables already in the sample. In each case both the optimal strategy and the distribution of the stopping variable are discussed.
TL;DR: In this article, the authors extended the estimate of the probability density function, based on a fixed number of observations, studied by Yamato (1971) and Davies (1973), to the case of random observations.
Abstract: The estimate of the probability density function, based on a fixed number of observations, studied by Yamato (1971) and Davies (1973), has been extended to the case when the number of observations is random. Asymptotic properties of the estimates of She d:osity function and its derivatives, as also of the estimate of the mode, have been studied under appropriate conditions.
TL;DR: The H-function is a generalization of most special functions and of many classical statistical distributions and has unique properties that make it a powerful tool for statistical analysis as mentioned in this paper, and a new simplification of the numerical evaluation of the inversion integral is presented.
Abstract: SYNOPTIC ABSTRACTThe H-function is a generalization of most special functions and of many classical statistical distributions and has unique properties that make it a powerful tool for statistical analysis. This paper provides a background of history, references, definition, and properties for the H-function and presents a new simplification of the numerical evaluation of the H-function inversion integral. Examples using this formulation are given, and new H-function identities (for generalized gamma, power, trigonometric, and logarithmic functions) are derived. A new practical technique for determining the probability density function and cumulative distribution function of a linear combination of products, quotients, and rational powers of independent random variables is demonstrated.
TL;DR: In this paper, a comparison of results obtained from the reliability program of a reservoir management problem based on the use of unconditioned and conditioned cumulative distribution functions (CDF's) is presented.
TL;DR: In this article, the authors define the standardized spacings Dr,n=(n-r) (Xr+1,n-Xr,n), 1≤r≤n, with DO,n=nX 1,n and Dn,n = 0.
Abstract: Let X be a non-negative random variable with cumulative probability distribution function F. Suppse X1, X2, ..., Xn be a random sample of size n from F and Xi,n is the i-th smallest order statistics. We define the standardized spacings Dr,n=(n-r) (Xr+1,n-Xr,n), 1≤r≤n, with DO,n=nX1,n and Dn,n=0. Characterizations of the exponential distribution are given by considering the expectation and hazard rates of Dr,n.
TL;DR: In this article, two well known load forecasting methods are generalized to predict the entire probability density function of the load, which is not to calculate the probability density of the forecasted load, but, rather, the probability distribution function of a load itself.
Abstract: ConventionaL load forecasting involves the prediction of the mean value of the demand of an electric power system. The mean value of a quantity which is subject to uncertainty does not fully characterize that quantity. In this paper, two well known load forecasting methods are generalized to predict the entire probability density function of the load. Note that the proposed technique is not to calculate the probability density of the forecasted load, but, rather, the probability density function of the load itself. From this density function, a wide variety of quantities may be calculated: the mean value; the probability that the load will exceed some threshold; a figure of confidence of the forecast mean; conditional probabilities (under speciaL conditions such as negative generation margin), and conditional expectations. Both methods presented rely on the forecasting of the statistical moments of the demand, and using those moments to calculate the probability density function using the Gram-Charlier series type A.
TL;DR: In this article, a general method based on "delta sequences" was extended to sequences of strictly stationary mixing random variables having the same marginal distribution admitting a Lebesgue probability density function, and it was proved that the rate of mean square convergence obtained in the i.i.d. case by Walter and Blum, continues to hold.
Abstract: A general method based on “delta sequences” due to Walter and Blum [12] is extended to sequences of strictly stationary mixing random variables having the same marginal distribution admitting a Lebesgue probability density function. It is proved that, under certain conditions, the rate of mean square convergence obtained in the i.i.d. case by Walter and Blum, continues to hold.
TL;DR: In this paper, a procedure for designing rectifying sampling plans by attributes with a predetermined upper bound on the outgoing quality of the lots is developed, where the plans are derived so that the upper bound is satisfied with specified minimum probability regardless of the value of the process average.
Abstract: A procedure is developed for designing rectifying sampling plans by attributes with a predetermined upper bound on the outgoing quality of the lots. These plans are derived so that the upper bound is satisfied with specified minimum probability regardless of the value of the process average. In addition, the average outgoing quality limit (AOQL) criterion is evaluated with respect to the probability bound concept. Computational procedures are presented that allow the design of sampling plans with user specified upper bounds with designated minimum probability 1 – α. A table is presented for determining sampling plans for the use of 1 – α = .95.
TL;DR: In this paper, it is argued that stochastic dominance (SD) criteria are superior or more efficient, in many cases, than those derived from traditional models, and the case in favour of SD rests on
Abstract: which cash is to be invested. The number of competing decisions in each of these decision classes being large, the total number of sets of decisions tends to be very large. Each set would generate a separate insurance and investment portfolio ; a corporate portfolio. The financial outcomes for each corporate portfolio can be represented by a probability distribution or by a cumulative probability distribution of policyowners' surplus at a defined future date. Were this very large number of distributions specified (a formidable managerial problem in itself), established methods for identifying the probability of ruin for each corporate portfolio could be employed and those distributions that indicate intolerably high probabilities of ruin eliminated from further consideration.1 The manager would then have a smaller but still difficult problem ; that of deciding which remaining distribution, and thus, set of decisions, is optimal for the company. Optimum choice demands specification of optimization criteria. This paper addresses the problem of specifying appropriate criteria. It will be argued that Stochastic Dominance (SD) criteria are superior or more efficient, in many cases, than those derived from traditional models. The case in favour of SD rests on
01 Jan 1981
TL;DR: In this article, approximate approximations for the cumulative distribution function for the normal distribution were obtained for ln δ, where δ is the cumulative distributions function for δ.
Abstract: Approximation formulas are obtained for ln Ф, where Ф is the cumulative distribution function for the normal distribution.
TL;DR: In this paper, the authors treated the probability of failure as a statistic from the viewpoint that the probability can be determined only through experimental data, and proposed a simple policy for fatigue-proof design.
TL;DR: In this paper, the relationship between the unconditional and conditional distributions of the same random variable, say Y, when the distribution of the conditioning random variable X is of a known form is investigated.
Abstract: This paper studies the relationship between the unconditional and conditional distribution of the same random variable, say Y, when the distribution of the conditioning random variable X is of a known form. The problem is tackled in the general case where the distribution of Y and Y given X are mixed. Attention is focused to two particular cases. In the first X is assumed to follow a Poisson distribution; in the second X is allowed to have a mixed Poisson form. Potential applications are also discussed.
01 Jan 1981
TL;DR: In this paper, an asymptotic expansion is given for a class of integrals for large values of a parameter, which corresponds with the degrees of freedom in a certain type of cumulative distribution functions.
Abstract: An asymptotic expansion is given for a class of integrals for large values of a parameter, which corresponds with the degrees of freedom in a certain type of cumulative distribution functions. The expansion is uniform with respect to a variable related to the random variable of the distribution functions. Special cases include the chi-square distribution and the F-distribution.
TL;DR: In this paper, the exact null distribution of Bartlett's criterion for testing the homogeneity of variances in normal samples with unequal sizes is derived and the most general form of the density function is obtained by using contour integration.
Abstract: Summary
In this paper the exact null distribution of Bartlett's criterion for testing the homogeneity of variances in normal samples with unequal sizes is derived. The most general form of the density function is obtained by using contour integration. The expression for the cumulative distribution, being a series in simple algebraic functions, seems quite tractable for computation of the exact critical values. In the special case of equal sample sizes, some indication of the relation of the work of others to our series expansions has also been given.
01 Jan 1981
TL;DR: Three methods of estimating the inverse of a continuous cumulative distribution function for the purpose of random deviate generation are discussed and results are obtained which permit comparisons of the accuracies of each of these methods under alternative assumptions about the underlying distribution.
Abstract: Three methods of estimating the inverse of a continuous cumulative distribution function for the purpose of random deviate generation are discussed. These methods are 1) the empirical approach, 2) the maximum likelihood approach, 3) a newly developed regression based estimation procedure.Analytic results are obtained which permit comparisons of the accuracies of each of these methods under alternative assumptions about the underlying distribution. Expressions for the variance of each estimate at any given quantile of the random variable are provided.A demonstration of the procedures is given using data from the outer continental shelf oil and gas lease program.