Showing papers on "Cumulative distribution function published in 1982"

Probability and Statistics for Engineering and the Sciences

Abstract: 1. OVERVIEW AND DESCRIPTIVE STATISTICS. Populations, Samples, and Processes. Pictorial and Tabular Methods in Descriptive Statistics. Measures of Location. Measures of Variability. 2. PROBABILITY. Sample Spaces and Events. Axioms, Interpretations, and Properties of Probability. Counting Techniques. Conditional Probability. Independence. 3. DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Random Variables. Probability Distributions for Discrete Random Variables. Expected Values. The Binomial Probability Distribution. Hypergeometric and Negative Binomial Distributions. The Poisson Probability Distribution. 4. CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Probability Density Functions. Cumulative Distribution Functions and Expected Values. The Normal Distribution. The Exponential and Gamma Distributions. Other Continuous Distributions. Probability Plots. 5. JOINT PROBABILITY DISTRIBUTIONS AND RANDOM SAMPLES. Jointly Distributed Random Variables. Expected Values, Covariance, and Correlation. Statistics and Their Distributions. The Distribution of the Sample Mean. The Distribution of a Linear Combination. 6. POINT ESTIMATION. Some General Concepts of Point Estimation. Methods of Point Estimation. 7. STATISTICAL INTERVALS BASED ON A SINGLE SAMPLE. Basic Properties of Confidence Intervals. Large-Sample Confidence Intervals for a Population Mean and Proportion. Intervals Based on a Normal Population Distribution. Confidence Intervals for the Variance and Standard Deviation of a Normal Population. 8. TESTS OF HYPOTHESIS BASED ON A SINGLE SAMPLE. Hypotheses and Test Procedures. z Tests for Hypotheses About a Population Mean. The One-Sample t Test. Tests Concerning a Population Proportion. Further Aspects of Hypothesis Testing. 9. INFERENCES BASED ON TWO SAMPLES. z Tests and Confidence Intervals for a Difference between Two Population Means. The Two-Sample t Test and Confidence Interval. Analysis of Paired Data. Inferences Concerning a Difference between Population Proportions. Inferences Concerning Two Population Variances. 10. THE ANALYSIS OF VARIANCE. Single-Factor ANOVA. Multiple Comparisons in ANOVA. More on Single-Factor ANOVA. 11. MULTIFACTOR ANALYSIS OF VARIANCE. Two-Factor ANOVA with Kij = 1. Two-Factor ANOVA with Kij > 1. Three-Factor ANOVA 11. 4 2p Factorial Experiments. 12. SIMPLE LINEAR REGRESSION AND CORRELATION. The Simple Linear Regression Model. Estimating Model Parameters. Inferences About the Slope Parameter ss1. Inferences Concerning Y*x* and the Prediction of Future Y Values. Correlation. 13. NONLINEAR AND MULTIPLE REGRESSION. Assessing Model Adequacy. Regression with Transformed Variables. Polynomial Regression. Multiple Regression Analysis. Other Issues in Multiple Regression. 14. GOODNESS-OF-FIT TESTS AND CATEGORICAL DATA ANALYSIS. Goodness-of-Fit Tests When Category Probabilities Are Completely Specified. Goodness-of-Fit Tests for Composite Hypotheses. Two-Way Contingency Tables 15. DISTRIBUTION-FREE PROCEDURES. The Wilcoxon Signed-Rank Test. The Wilcoxon Rank-Sum Test. Distribution-Free Confidence Intervals. Distribution-Free ANOVA. 16. QUALITY CONTROL METHODS. General Comments on Control Charts. Control Charts for Process Location. Control Charts for Process Variation. Control Charts for Attributes. CUSUM Procedures. Acceptance Sampling.

On the distribution function and moments of power sums with log-normal components

Abstract: An approximate technique is presented for the evaluation of the mean and variance of the power sums with log-normal components. Exact expressions for the moments with two components are developed and then used in a nested fashion to obtain the moments of the desired sum. The results indicate more accurate estimates of these quantities over a wider range of individual component variances than any previously reported procedure. Coupling our estimates with the Gaussian assumption for the power sum provides a characterization of the cumulative distribution function which agrees remarkably well with a Monte Carlo simulation in the 1 to 99 percent range of the variate. Simple polynomial expressions obtained for the moments lead to an effective analytical tool for various system performance studies. They allow quick and accurate calculation of quantities such as cochannel interference caused by shadowing in mobile telephony.

Microstructure of two‐phase random media. I. The n‐point probability functions

Abstract: The microstructure of a two‐phase random medium can be characterized by a set of general n‐point probability functions, which give the probability of finding a certain subset of n‐points in the matrix phase and the remainder in the particle phase. A new expression for these n‐point functions is derived in terms of the n‐point matrix probability functions which give the probability of finding all n points in the matrix phase. Certain bounds and limiting values of the Sn follow: the geometrical interpretation of the Sn and their relationship with n‐point correlation functions associated with fluctuating bulk properties is also noted. For a bed or suspension of spheres in a uniform matrix we derive a new hierarchy of equations, giving the Sn in terms of the s‐body distribution functions ρs associated with a statistically inhomogeneous distribution PN of spheres in the matrix, generalizing expressions of Weissberg and Prager for S2 and S3. It is noted that canonical ensemble of mutually impenetrable spheres a...

Experimental and theoretical statistics of microwave amplitude scintillations on satellite down-links

Abstract: Extensive experimental results are presented on the statistics of tropospheric amplitude scintillations on an X -band satellite down-link obtained using the orbital test satellite beacon transmissions. The experimentally found distribution is shown to depart significantly from the expected log-normal distribution, and this is explained in terms of a Gaussian process with a time variable standard deviation from which a universal model is derived. It has been found that on average no less than about 100 h of data are required before the probability density and cumulative probability distribution functions approach stationarity. The statistics of the scintillation intensity are also presented, and a log-normal distribution of intensity is shown to be in good agreement with observations from other experimental sites. Link budget implications are outlined together with a simple strategy for the investigation of the scintillation process at any ground station.

Performance Analysis for the Expanding Search PN Acquisition Algorithm

Abstract: An approach is presented for approximating the cumulative probability distribution of the acquisition time of the serial PN search algorithm. The results are applicable to variable as well as fixed dwell time systems. The theory is developed for the case where some a priori information on the PN code epoch is available (reacquisition problem or acquisition of very long codes). The special case of a search over the whole code is also treated. The accuracy of the approximation is demonstrated by comparing with published exact results for the fixed dwell time algorithm.

Freund's bivariate exponential distribution and censoring

Abstract: In some problems, a bivariate random vector (T ,T ) with bivariate cumulative distribution function F is observed for each of ? independent subjects, but the coordinates may be subject to censoring In the first section, we describe several mechanisms which can generate the censorship The usual nonparametric approaches to estimation of F are then shown to be unsatisfactory Therefore, in the third section, we describe a parametric model due to Freund (1961) This model is studied not because all data can be forced to fit this specific parametric form, but because this model suggests some approaches to the nonparametric problem These ideas, together with some relationships to the work of other authors, are outlined in the fourth section

A note on linear functions of ordered correlated normal random variables

Abstract: SUMMARY The probability density function of linear functions of ordered components of a bivariate normal random variable is evaluated.

Adaptive constant false alarm rate (cfar) processor

Abstract: The incoming radar signals to a constant false alarm rate (CFAR) processor, having some known cumulative distribution function F(X) or probability density function f(X), are transformed into new signals Y according to the equation Y=σ[-1n {1-Fx (X)}]1/K where σ and K are a scale parameter and shape parameter, respectively, and can be arbitrarily set. The threshold to which the processed signal is compared in the CFAR processing is variable in accordance with the selected value of K.

Calculating Detection Probabilities for Systems Employing Noncoherent Integration

Abstract: Cumulative probability distributions that occur in radar and sonar detection problems are calculated directly from the characteristic function by using a Fourier series. The error in the result is controlled by two parameters which can be adjusted to suit the application. The technique is applied to the problem of determining the detection performance of consecutive discrete Fourier transforms (DFTs) for a narrowband Gaussian signal with a rectangular spectrum. Since the characteristic function is used directly in its product form this technique does not suffer from the numerical problems associated with the partial fraction approach. The technique can handle many different problems in a single computational structure making it a valuable tool in system performance studies.

Rapid generation of discrete random variates from general distributions

Abstract: Method and apparatus for the rapid generation of discrete random variates, of a general probability distribution, from random variates of a uniform distribution uses simple logic and moderate storage requirements. Access to a first table at an address which is a function of a uniform variate gives the desired variate value directly for a large percentage of cases and in the remaining cases the desired variate value is obtained by an indexed search of a second table of cumulative distribution function values. Results of greater precision are realized in faster execution times with less logic complexity and reduced storage bulk as compared with prior methods and apparatus.

Hardness Assurance and Overtesting

Abstract: A procedure is developed for estimating the advantage gained by testing electronic piece parts at levels of radiation higher than the specification level which they must survive. When the probability distribution of radiation stress to failure is approximately lognormal, and a test shows that with confidence, C, the survival probability is at least PT at the test level of stress RT, then with the same confidence the survival probability is at least PS at a lower specification level of stress RS where, Ps = F[ F?(PT) + ln(RT/RS/?ln(MAX)] The standard deviation, ?ln(MAX), is the estimated maximum s.d. in the logarithms of stress to failure; the function, F(X), is the standard normal cumulative probability distribution function; and the function, F(P), is the antifunction of F(X) - that is, F(P) standard deviations above the mean of a normal distribution includes fraction P of the distribution. A discussion is given of how this formula also applies to those tests which are more properly acceptance/rejection tests rather than tests which establish a confidence that parts will survive a given radiation level with a given probability. The suggested overtesting technique is compared to other standard testing techniques.

On the Use of a Cumulative Distribution as a Utility Function in Educational or Employment Selection

Abstract: Formal decision theory can make important contributions to educational or employment decision-making, provided one can quantify the utilities of different possible outcomes such as test scores and grade-point averages. Because utility is usually a monotonic increasing function of performance score, a cumulative probability function may be convenient for describing one’s utilities. Moreover, calculations of expected utility of a decision are greatly simplified when the utility and the probability function have the same functional form, e.g., both are normal. A least-squares procedure, developed by Lindley and Novick for fitting a utility function, is applied to truncated normal and extended beta distribution functions. The truncated normal and beta distributions avoid the symmetry and infinite range restrictions of the normal distribution and can provide fits in some cases in which the normal functional forms cannot provide a reasonable fit.

Skinner's Method for Computing Bounds on Distributions and the Numerical Solution of Continuous-Time Queueing Problems

Abstract: Skinner's method provides a means of computing; numerically, upper and lower bounds on a cumulative distribution function resulting from the convolution of probability density functions. The method thus provides approximate numerical results whose accuracy is known precisely. This paper provides an exposition of Skinner's method. It shows how the method Can be applied to the computation of numerical solutions of other problems, as well as the waiting time distribution of the M/G/1 queue, for which Skinner presented the method.

New expressions for the pdf and cdf of the filtered output of an analog cross correlator (Corresp.)

Abstract: A general expression is derived for the probability density function (pdf) associated with the filtered output of an analog cross correlator, the inputs of which consist of deterministic signals having the same frequency plus correlated stationary Gaussian noise processes. The correlator is composed of input bandpass filters, a multiplier, and a zonal Iowpass filter. The noise components in each channel have distinct variances and obey no spectral symmetry condition. Using the derived expression for the pdf, the cumulative distribution function (cdf) under the same general conditions is calculated. All previous known special cases are subsequently deduced from the general expressions given here. The general results derived here are more suitable for numerical computations than comparable previous results found in the literature due to the fact that expressions here do not involve multiple infinite series. Lastly, asymptotic formulas are developed for both the pdf and cdf which may be helpful in certain calculations.

Isotropic random flights: random numbers of flights

Abstract: Isotropic random flights, where the number of individual flights N is random, are studied. N is taken to be governed by a Poisson distribution and also by a negative binomial distribution, each with mean (N). The probability density function of the length of the vector sum is shown to be mixed, in that it contains impulse components (Dirac delta functions) as well as the absolutely continuous component. The limiting density functions are also obtained, and in the negative binomial case lead to the random flight version of the K-density function introduced by Jakeman and collaborators (1976, 1978). Finally, the moments about the origin are explicitly evaluated for both fixed N and random N.

A study of generalized intensity statistics: extension of the theory and practical examples

Abstract: Generalized probability density functions, cumulative distribution functions and moments of the normalized structure amplitude |E|, depending on space-group symmetry of the crystal and on the composition of the asymmetric unit, were extended to include the tenth moment of |E| and five-term expansions. The formalism was also simplified and is presented in a concise and unified form. The equations linking the formalism to practical problems, the composition and space-group terms, are discussed from a practical point of view and a convenient implementation of the above statistics in a computer program is indicated. The generalized cumulative distributions of |E| and of the normalized intensity z = |E|2 are compared with corresponding distributions based on five published structures, each containing one outstandingly heavy atom (Pt, Rh and Br) and about twenty light ones in the asymmetric unit, excluding hydrogens. These examples indicate that the above formalism is a valuable tool for resolving space-group ambiguities which cannot be treated by conventional methods because of effects of atomic heterogeneity. N(z) distributions for a structure belonging to the space group Fddd show that the theoretical expressions correctly predict the existence of different intensity distributions in reflection subsets with hkl all even and hkl all odd for this space group.

Fourier Analysis For Estimating Probability Of Sliding For The Plane Shear Failure Mode

Abstract: The probabilistic nature of appropriate geologic variables can be included in analyzing the stability of potential slope failure modes. Random variables in the two-dimensional plane shear analysis of a specified structural discontinuity include the shear strength and waviness angle of the discontinuity and, in some cases, the rock mass density. Estimated probability density functions that describe these random variables are combined by Fourier analysis to produce an estimate of the safety factor probability density function, which appears to approximate a gamma distribution. The probability that sliding will occur along the specified structure is equal to the area under this density function where the safety factor is less than one. For the covering abstract of the symposium see TRIS 452576. (Author/TRRL)

Generalizations of a contamination model for continuous type random variables

Abstract: In 1965, Stanley Warner (Warner, 1965) introduced a model for contaminating discrete type random variables. He presented this scheme as being potentially useful in survevs where sensitive in-formation is being gathered. Since that time much research has been conducted and many papers written on the development of these discrete type randomized response models. More recently, atten-tion has been focused on the application of randomized response type models for preservation of confidentiality in existing data files (Boruch 1971 and 1972, Ranney 1975, Felligi 1974, and Inge-marsson 1975). In 1974, Poole (Poole, 1974) introduced a randomized response model for a positive continuous type random variable which was basically a continuous variable analog of the discrete variable Warner model. In this paper the results of the 1974 paper are extended to a lt-dimensional continuous type random variable in k-dimensional Euclidean space.

A Data Based Random Number Generator for a Multivariate Distribution - A Users' Manual

Abstract: : Let X be a k-dimensional random variable serving as input for a system with output Y(not necessarily of dimension k). Given X, an outcome Y or a distribution of outcomes G(Y/X) may be obtained either explicitly or implicitly. We consider here the situation in which we have a real world data set X (J) to the nth power (j=1) to the n power and a means of simulating an outcome Y. A method for empirical random number generation based on the sample of observations of the random variable X without estimating the underlying density is discussed.

Turbulence effects on coherent laser radar target statistics

Abstract: This paper reports theoretical probability density and cumulative distribution functions for glint and speckle target returns in a compact coherent laser radar Calculator programs are given to facilitate use of these results

A modified form of the first Gumbel distribution: model for the occurrence of large earthquakes. Part. I. Derivation of distribution

Abstract: The followin cumulative distribution G(x)={(exp[-λ((A_2-A(x))/(A_2-A_1 ))]@1, x>M_max )┤,M_min≤x〖≤M〗_max Is proposed for the magnitudes of largest earthquake where A(x)=exp(-βx), A_1= exp(-βM_min), and A_2= exp(-βM_max). M_min is the threshold magnitude value and M_max is the maximum regional value. M_max, β and λ are the parameters to be determined. When no upper bound of magnitude is assumed (M_max→∞), the new proposed distribution take the form of the well-known first Gumbel distribution. The distribution is obtained under the following assumptions: the annual number of earthquake is a Poisson random variable with the mean λ, and the earthquake magnitude is random variable distributed according to a double truncated exponential distribution. The proposed distribution provides an estimation of the probability of occurrence of strong earthquakes significantly more realistic than the classical first Gumbel distribution. As an example, the results of calculations for earthquakes felt in Norway during the period 1899-1979 are given.

On Generation of Random Numbers with Specified Distributions or Densities.

Abstract: : Generation of random numbers with specified probability density functions or cumulative distribution functions is reviewed and employed to generate some standard random variables with common densities and distributions. Combinations of random variables then afford a quick method of generating variables with more-involved distribution functions. Application to the OM cumulative distribution function is made as an example. Timing results for all cases are listed. Numerous statistical tests on the Hewlett-Packard 9845 Desk Calculator confirm it as a reliable generator of uniformly distributed statistically independent random numbers. (Author)

Nonparametric maximum likelihood estimation of a probability density via mathematical programming

Abstract: Based on sample values of a one-dimensional random variable a nonparametric maximum likelihood estimate for the unknown probability density is introduced as the solution of an optimization problem in an appropriate Hilbert space. This solution turns out to be a polynomial spline function, and a complete characterization is given using recent results on the differentiability of the optimal value of a parametrized family of optimization problems. An important feature of this estimate is that its support interval results in a quite natural way from the formulation of the problem and is not fixed in advance. The estimator is shown to have a certain consistency property for a special class of density functions. Numerical results will be given in a subsequent paper.

On the Probability of a Negative Estimator of Heteroscedasticity in the Random Coefficient Model

Abstract: In the estimation of a linear regression model with random coefficients, sometimes negative estimates of variances of random coefficients are obtained–an undesirable feature. In this paper we have obtained the asymptotic bounds of the probability of obtaining negative estimators. A simple illustration is also provided for the purpose.