Showing papers on "Cumulative distribution function published in 1978"
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01 Jan 1978
TL;DR: Populations and variates measures of the center of a set of observations samples and populations the measurement of variability looking at data probability probabilities of compound events discrete random variable expectation of a random variable joint distributions estimation collecting data significance testing continuous random variables the normal distribution sampling distributions of means and related quantities significance tests using the normal distributions estimation of intervals and parameters hypothesis tests using chi-squared distribution poisson distribution correlation the analysis of variance simple linear regression multiple regression.
Abstract: Populations and variates measures of the centre of a set of observations samples and populations the measurement of variability looking at data probability probabilities of compound events discrete random variable expectation of a random variable joint distributions estimation collecting data significance testing continuous random variables the normal distribution sampling distributions of means and related quantities significance tests using the normal distribution estimation of intervals and parameters hypothesis tests using the chi-squared distribution poisson distribution correlation the analysis of variance simple linear regression multiple regression.
205 citations
TL;DR: In this paper, the Laguerre distribution is approximated using the saddlepoint method of evaluating the inverse Laplace transform of the moment-generating function, which is applied to evaluate the error function integral and the Mth-order Q function, and to approximating the inverse of the chi-squared distribution.
Abstract: Cumulative probability distributions such as occur in radar detection problems are approximated by a new version of the saddlepoint method of evaluating the inverse Laplace transform of the moment-generating function. When the number of radar pulses integrated is large, the approximation of lowest order yields good accuracy in the tails of the distributions, yet requires much less computation than standard recursive methods. Greater accuracy can be achieved upon summing the residual series by converting it to a continued fraction. The method is applied to evaluating the error-function integral and the Mth-order Q function, and to approximating the inverse of the chi-squared distribution. Cumulative distributions of discrete random variables, needed for determining error probabilities in optical communication receivers that involve counting photoelectrons, can be approximated by a simple modification of the method, which is here applied to the Laguerre distribution.
118 citations
TL;DR: In this paper, an approximation to the cumulative normal distribution and its inverse is proposed, which, according to circumstances, is simpler, more accurate, or more convenient than existing formulae.
Abstract: This note proposes an approximation to the cumulative normal distribution and its inverse which, according to circumstances, is simpler, more accurate, or more convenient than existing formulae.
74 citations
TL;DR: In this paper, a counterexample is given to a proposition of Marshall and Olkin (1974, Ann. Statist) describing a probability bound, and a theorem which gives the conditions under which mixtures of associated random variables remain associated is stated.
Abstract: A counterexample is given to a proposition of Marshall and Olkin (1974, Ann. Statist.) describing a probability bound. A theorem which gives the conditions under which mixtures of associated random variables remain associated, is stated. This provides a method to obtain the required bound.
43 citations
TL;DR: In this article, the number of bosons that are created by an excitation process, in a system which initially has none, is defined as a random variable having a discrete probability distribution, defined on the nonnegative integers, which satisfies P(0)>0.
Abstract: The number of bosons that are created by an excitation process, in a system which initially has none, is a random variable having a discrete probability distribution, defined on the non-negative integers, which satisfies P(0)>0. The logarithm of the resulting probability generating function is therefore analytic at the origin, and the series expansion coefficients thereby generated can each be expressed as a finite combination of ratios of the P(n), giving them an interestingly close kinship to experimental data. These 'combinants' are additive for sums of boson multiplicity random variables which are independent, and they all vanish except for the first-order one in the important Poisson case. The combinants are readily calculated in a number of theoretical models for created boson multiplicities, including the thermal model some of the related chaotic radiation models, and in some models described by master rate equations.
35 citations
TL;DR: In this article, the smallest order statistics in a random sample of sizen from a non-negative random variable with probability distribution function F is defined and a necessary and sufficient condition for F to be exponential is given which involves the identical distribution of the random variables.
Abstract: LetX be a non-negative random variable with probability distribution functionF. SupposeX
i,n (i=1,…,n) is theith smallest order statistics in a random sample of sizen fromF. A necessary and sufficient condition forF to be exponential is given which involves the identical distribution of the random variables (n−i)(X
i+1,n−Xi,n) and (n−j)(X
j+1,n−Xj,n) for somei, j andn, (1≦i
28 citations
TL;DR: In this paper, the sine power probability density function (SP-PDF) has been extended to include cases of minimum value having a non-zero probability and means, standard deviations, skewness coefficients, kurtosis coefficients and criterion indices have been computed and tabulated for use in application to practical problems.
15 citations
TL;DR: In this paper, joint- or complex-gaussian statistics are assumed to apply to the in-phase and quadrature components of an electromagnetic wave which has propagated through a turbulent medium, and the consequences of this assumption are compared to tropospheric, ionospheric and laboratory propagation experiments.
Abstract: In this paper joint- or complex-gaussian statistics are assumed to apply to the in-phase and quadrature components of an electromagnetic wave which has propagated through a turbulent medium, and the consequences of this assumption are compared to tropospheric, ionospheric, and laboratory propagation experiments. New curves for the cumulative probability distribution of intensity and for the variance of intensity versus the variance of the logarithm of intensity are presented. Joint-gaussian statistics are shown to fit the total set of turbulence propagation results better than the log-normal, Rice-Nakagami, or Nakagami-m distributions. However, a major feature of optical observations cannot be fit by joint-gaussian statistics: the straight-line cumulative intensity distributions in the saturation regime that reflect a large probability of occurrence of high intensity spikes.
14 citations
TL;DR: In this article, a finite series approximation involving Laguerre polynomials is derived for central and noncentral multivariate gamma distributions, where the density of any k nonnegative continuous random variables up to the (n1, …, nk)th degree of the distribution is approximated by a finite-series of Lagueria.
Abstract: In this paper a finite series approximation involving Laguerre polynomials is derived for central and noncentral multivariate gamma distributions. It is shown that if one approximates the density of any k nonnegative continuous random variables by a finite series of Laguerre polynomials up to the (n1, …, nk)th degree, then all the mixed moments up to the order (n1, …, nk) of the approximated distribution equal to the mixed moments up to the same order of the random variables. Some numerical results are given for the bivariate central and noncentral multivariate gamma distributions to indicate the usefulness of the approximations.
12 citations
TL;DR: In this article, integral representations of several statistical distributions are exploited in a common structure to guarantee significant digit results over a wide range of parameters, where central distributions are treated implicitly by setting noncentrality parameters to zero.
Abstract: Integral representations of several statistical distributions are exploited in a common structure to guarantee significant digit results over wide ranges of parameters. Truncation errors are estimated from computed results and computational problems arising from extreme parameters are identified. Central distributions are treated implicitly by setting noncentrality parameters to zero. Bounds analogous to those for Mill’s ratio are also given for some nonnegative special functions.
TL;DR: In this article, a survey of methods of modeling probabilistic components which are not adequately represented by the standard continuous distributions (such as normal, gamma, and Weibull) is presented.
Abstract: Methods of modelling probabilistic components which are not adequately represented by the standard continuous distributions (such as normal, gamma, and Weibull) are surveyed. The methods are categorized as systems of distributions, approximations to the cumulative distribution function, and four-parameter distributions. Emphasis is on generality, determination of appropriate parameter values, and random variate generation.
TL;DR: In this article, the authors introduce the concept of Randomness and the use of a Random Quantity and the Use of a Distribution, which is a generalization of the theory of probability.
Abstract: 2. General Problems of Probability 2.1. Probability, Mathematics, and Logic 2.2. Humanism of the Theory of Probability 2.3. Concept of Randomness 2.4. Subjective and Objective Probabilities 2.4.1. A Problem on Subjective Probabilities 2.5. Concept of a Random Quantity and the Use of a Distribution 2.6. Cumulative Distribution Function 3. Limit Theorems 3.1. De Moivre-Laplace Limit Theorems 3.1.1. The Standard Case
TL;DR: In this article, a statistical Markovian insolation model for predicting the time-sequence of half-hour solar radiation values on a horizontal surface which uses the hourly insolation values is developed.
TL;DR: In this paper, conditional probability functions for selected days during any growing season were developed using a dynamic crop growth simulation model using one hundred and twenty sets of seasonal yield and daily crop growth data were simulated.
Abstract: CONDITIONED probability functions were devel-oped using a dynamic crop growth simulation model. One hundred and twenty sets of seasonal yield and daily crop growth data were simulated. These data sets were used to develop conditional probability functions for selected days during any growing season. The probability functions were conditioned on two state variables: leaf area and available soil water. Cumulative distribution functions computed from the probability functions were used to forecast yields at each of the selected dates as the season progressed. Forecast accuracy improved as the season progressed. Combining a deterministic crop growth model with stochastic weather data provides a realistic method of yield forecasting.
01 Jan 1978
TL;DR: In this article, the authors summarized and analyzed six locations on American railroads using cumulative probability techniques, and developed relations for defect occurrence as a function of traffic and stress, and showed that defect occurrence is correlated with stress and traffic.
Abstract: Rail defect data from six locations on American railroads are summarized and analyzed using cumulative probability techniques. Relations for defect occurrence as function of traffic and stress are developed.
TL;DR: In this paper, a computational formula for computing the cumulative distribution function of noncentral chi-squared distributions with odd degrees of freedom is given, which is the same as the one presented in this paper.
Abstract: A computational formula for computing the cumulative distribution function of noncentral chi-squared distributions with odd degrees of freedom is given.
13 Apr 1978
TL;DR: A (smooth) increasing quadratic spline is constructed which interpolates the data points and preserves the convexity of the data.
Abstract: Let F be an arbitrary continuous cumulative distribution function of a single variable specified by a finite set of points. A (smooth) increasing quadratic spline is constructed which interpolates the data points and preserves the convexity of the data [2]. The spline is compared with Akima's piecewise cubic approximation [1] for several common distributions F.
TL;DR: A nonsingular linear transformation of binary-valued random vectors y = xA which minimizes a mutual information criterion I(y) is considered and is useful in improving the approximation of probability distributions.
Abstract: A nonsingular linear transformation of binary-valued random vectors y = xA which minimizes a mutual information criterion I(y) is considered. It is shown that a nonsingular A exists such that I(y) = 0 if and only if x has a generalized binomial distribution. Computational algorithms for seeking an optimal A are developed, and dimensionality reduction is discussed briefly. This linear transformation is useful in improving the approximation of probability distributions. Numerical examples are presented.
TL;DR: In this paper, the spatial distribution of the sampling stations conforms to the expected nonuniformity of the trace metal concentrations of interest, providing for a more efficient allocation of sampling effort.
Abstract: The paper outlines a model to describe the spatial distribution of selected trace elements (chromium, copper, nickel, lead, and zinc) in New York Bight sediments. Empirical tests of the fit of existing data by means of cumulative probability plots on log-normal probability paper and goodness-of-fit tests demonstrate conformity to the log-normal distribution predicted by the model. An adaptive sampling strategy for describing the spatial distribution of the trace elements is developed, based on a knowledge of the sampling distributions of the elements. The spatial distribution of the sampling stations conforms to the expected nonuniformity of the trace metal concentrations of interest, providing for a more efficient allocation of sampling effort.
01 Aug 1978
TL;DR: MOCARS is a computer program designed for use on the Idaho National Engineering Laboratory (INEL) CDC CYBER 76-173 computer system that uses Monte Carlo techniques to determine the distribution and simulation limits for a function.
Abstract: MOCARS is a computer program designed for use on the Idaho National Engineering Laboratory (INEL) CDC CYBER 76-173 computer system that uses Monte Carlo techniques to determine the distribution and simulation limits for a function. In use, the MOCARS program randomly samples data from any of the 12 different user-specified probability distributions and either evaluates a user-specified function or cut set system unavailability using the sample data. After data ordering, the values at various quantities and associated confidence bounds are calculated for output. If the cut set unavailability function is evaluated, MOCARS can determine the importance ranking for components in the unavailability calculation. Frequency and cumulative distribution histograms from the sample data are also available for output on microfilm. 39 figures, 4 tables.
01 Sep 1978
TL;DR: The computer code PDFPLOT uses the Pearson family of distributions, augmented by the normal and exponential distributions, to obtain plots of a density function and its associated cumulative distribution function.
Abstract: The computer code PDFPLOT uses the Pearson family of distributions, augmented by the normal and exponential distributions, to obtain plots of a density function and its associated cumulative distribution function. The first four moments of the distribution are input. The code selects the distribution with these moments from its family of distributions, and calculates both the density function and the cumulative distribution function. Optional output includes a listing of the cumulative probabilities and plots of the density and cumulative distribution functions. A complete input description is included, and an example is presented. 12 figures, 1 table.
12 Dec 1978
TL;DR: A method for calculating the probability of detection of generalized adaptive threshold systems is applied to sensors employing pulsewidth-adaptive-threshold signal processing concepts and the variation of probability of Detection with signal-to-noise ratio and with the form of the input signal plus clutter is presented.
Abstract: A method for calculating the probability of detection of generalized adaptive threshold systems is appliedto sensors employing pulsewidth- adaptive -threshold signal processing concepts. These sensors typicallyuse the signal as input to a guard channel that controls the threshold against which the signal is itself com- pared. The variation of probability of detection with signal -to -noise ratio and with the form of the inputsignal plus clutter is presented. For the specific sensor analyzed, in which Gaussian statistics hold, theprobability of detection curves branch off the cumulative normal distribution curves as a function of guard channel gain. There is also a range of guard channel gains for which the detection performance of pulse -width adaptive sensors will suffer essentially zero insertion loss due to the adaptive threshold itself. Introduction Mr. R. H. Genoudl gives the detection performance equations for an infrared search set (IRSS) as follows: The probability of detection PD (which is the probability that a signal Vp plus noise Vn will exceed a
TL;DR: In this paper, a new theoretical expression for the multivariate joint probability density function of an output response is derived exactly, without any simplification of the problem or analytical approximation, in the case when a general random signal having an arbitrary probability distribution and correlation properties is passed through an arbitrary linear vibratory system of finite order.
TL;DR: This suite of six programs uses several of the combinatorial algorithms from Nijenhuis and Wilf (1975) to carry out two types of test: exhaustiverandomization tests and Monte Carlo randomization tests.
Abstract: Bradley (1968) has shown that, for data that deviate substantially from the assumption of normally distributed error variance required by parametric tests, randomization tests can be more powerful than their parametric counterparts. In addition, randomization tests offer a more finely stepped distribution of the test statistic than do conventional nonparametric tests. This is of particular advantage in cases where group sizes are small, producing very "coarse-grained" distributions of the more commonly used nonparametric test statistics. Description. This suite of six programs uses several of the combinatorial algorithms from Nijenhuis and Wilf (1975) to carry out two types of test: exhaustive randomization tests and Monte Carlo randomization tests. The exhaustive randomization test calculates values of the test statistic that are obtained by rearranging the given data in all the possible ways consistent with the null hypothesis concerned. From this information, a cumulative probability distribution of the test statistic is derived, against which the observed value can be compared, using either a one-tailed or a two-tailed test. The Monte Carlo randomization test, on the other hand, takes a random sample (with replacement) of the total number of permissible data arrangements in order to derive the distribution of values for the test statistic concerned. The latter method is particularly useful in cases where the total number of permissible arrangements is prohibitively large in terms of the computing time required to perform an exhaustive test. A fuller account of the Monte Carlo procedure is given by Edgington (1969). The first pair of programs, EMATCH and RMATCH, are the exhaustive and Monte Carlo versions, respectively, of a randomization procedure to test the significance of the difference between the means of two matched samples, XI' X2, · .. , Xn and Y I , Y2, ... , YnFor this test, the null hypothesis is that each matched pair of scores (Xi, Vi) is drawn from the same
TL;DR: The importance of S-shaped curves in business decision making is discussed and simple methods of fitting equations to such curves are presented in this article, where the authors show that expensive computational equipment is not necessary in order to make numerate assessments of such business factors as profitability, working capital required, project life, risk assessment, etc.
TL;DR: In this article, the battery state of charge of an arbitrary photovoltaic system is analyzed as a Markov process driven by random white Gaussian perturbations of periodic insolation and load-demand profiles.
TL;DR: In this article, the authors give an elementary example based on the probability integral transformation U = F(X) where X is a random variable with continuous cumulative distribution function F and define a transformation T by the formula:
Abstract: Saying that random variables X and Y are jointly normal is, of course, a much stronger statement than claiming X and Y are individually normal. The gap between the two concepts is a crucial one but often puzzling to the student. By way of illustration, Feller [1, pp. 99-100], for instance, provides examples which depend on density function constructions. We give here an elementary example based on the probability integral transformation U = F(X) where X is a random variable with continuous cumulative distribution function F. We begin with a standard normal variable X with distribution function N, and define a transformation T (see FIGURE 1) by the formula: