Showing papers on "Cumulative distribution function published in 1984"

Statistics for engineering and the sciences

Abstract: CHAPTER 1: INTRODUCTION 1.1 Statistics: The Science of Data 1.2 Fundamental Elements of Statistics 1.3 Types of Data 1.4 The Role of Statistics in Critical Thinking 1.5 A Guide to Statistical Methods Presented in this Text Statistics in Action: Contamination of Fish in the Tennessee River Collecting theData CHAPTER 2: DESCRIPTIVE STATISTICS 2.1 Graphical and Numerical Methods for Describing Qualitative Data 2.2 Graphical Methods for Describing Quantitative Data 2.3 Numerical Methods for Describing Quantitative Data 2.4 Measures of Central Tendency 2.5 Measures of Variation 2.6 Measures of Relative Standing 2.7 Methods for Detecting Outliers 2.8 Distorting the Truth with Descriptive Statistics Statistics in Action: Characteristics of Contaminated Fish in the Tennessee River CHAPTER 3: PROBABILITY 3.1 The Role of Probability in Statistics 3.2 Events, Sample Spaces, and Probability 3.3 Compound Events 3.4 Complementary Events 3.5 Conditional Probability 3.6 Probability Rules for Unions and Intersections 3.7 Bayes' Rule (Optional) 3.8 Some Counting Rules 3.9 Probability and Statistics: An Example 3.10 Random Sampling Statistics in Action: Assessing Predictors of Software Defects CHAPTER 4: DISCRETE RANDOM VARIABLES 4.1 Discrete Random Variables 4.2 The Probability Distribution for a Discrete Random Variable 4.3 Expected Values for Random Variables 4.4 Some Useful Expectation Theorems 4.5 Bernoulli Trials 4.6 The Binomial Probability Distribution 4.7 The Multinomial Probability Distribution 4.8 The Negative Binomial and the Geometric Probability Distributions 4.9 The Hypergeometric Probability Distribution 4.10 The Poisson Probability Distribution 4.11 Moments and Moment Generating Functions (Optional) Statistics in Action: The Reliability of a "One-Shot" Device CHAPTER 5: CONTINUOUS RANDOM VARIABLES 5.1 Continuous Random Variables 5.2 The Density Function for a Continuous Random Variable 5.3 Expected Values for Continuous Random Variables 5.4 The Uniform Probability Distribution 5.5 The Normal Probability Distribution 5.6 Descriptive Methods for Assessing Normality 5.7 Gamma-Type Probability Distributions 5.8 The Weibull Probability Distriibution 5.9 Beta-Type Probability Distributions 5.10 Moments and Moment Generating Functions (Optional) Statistics in Action: Super Weapons Development: Optimizing the Hit Ratio CHAPTER 6: JOINT PROBABILITY DISTRIBUTIONS AND SAMPLING DISTRIBUTIONS 6.1 Bivariate Probability Distributions for Discrete Random Variables 6.2 Bivariate Probability Distributions for Continuous Random Variables 6.3 The Expected Value of Functions of Two Random Variables 6.4 Independence 6.5 The Covariance and Correlation of Two Random Variables 6.6 Probability Distributions and Expected Values of Functions of Random Variables (Optional) 6.7 Sampling Distributions 6.8 Approximating a Sampling Distribution by Monte Carlo Simulation 6.9 The Sampling Distributions of Means and Sums 6.10 Normal Approximation to the Binomial Distribution 6.11 Sampling Distributions Related to the Normal Distribution Statistics in Action: Availability of an Up/Down System CHAPTER 7: ESTIMATION USING CONFIDENCE INTERVALS 7.1 Point Estimators and their Properties 7.2 Finding Point Estimators: Classical Methods of Estimation 7.3 Finding Interval Estimators: The Pivotal Method 7.4 Estimation of Population Mean 7.5 Estimation of the Difference Between Two Population Means: Independent Samples 7.6 Estimation of the Difference Between Two Population Means: Matched Pairs 7.7 Estimation of a Poulation Proportion 7.8 Estimation of the Difference Between Two Population Proportions 7.9 Estimation of a Population Variance 7.10 Estimation of the Ratio of Two Population Variances 7.11 Choosing the Sample Size 7.12 Alternative Estimation Methods: Bootstrapping and Bayesian Methods (Optional) Statistics in Action: Bursting Strength of PET Beverage Bottles CHAPTER 8: TESTS OF HYPOTHESES 8.1 The Relationship Between Statistical Tests of Hypotheses and Confidence Intervals 8.2 Elements and Properties of a Statistical Test 8.3 Finding Statistical Tests: Classical Methods 8.4 Choosing the Null and Alternative Hypotheses 8.5 Testing a Population Mean 8.6 The Observed Significance Level for a Test 8.7 Testing the Difference Between Two Population Means: Independent Samples 8.8 Testing the Difference Between Two Population Means: Independent Samples 8.9 Testing a Population Proportion 8.10 Testing the Difference Between Two Population Proportions 8.11 Testing a Population Variance 8.12 Testing the Ration of Two Population Variances 8.13 Alternative Testing Procedures: Bootstrapping and Bayesian Methods (Optional) Statistics in Action: Comparing Methods for Dissolving Drug Tablets - Dissolution Method Equivalence Testing CHAPTER 9: CATEGORICAL DATA ANALYSIS 9.1 Categorical Data and Multinomial Probabilities 9.2 Estimating Category Probabilities in a One-Way Table 9.3 Testing Category Probabilities in a One-Way Table 9.4 Inferences About Category Probabilities in a Two-Way (Contingency) Table 9.5 Contingency Tables with Fixed Marginal Totals 9.6 Exact Tests for Independence in a Contingency Table Analysis (Optional) Statistics in Action: The Public's Perception of Engineers and Engineering CHAPTER 10: SIMPLE LINEAR REGRESSION 10.1 Regression Models 10.2 Model Assumptions 10.3 Estimating ss0 and ss1: The Method of Least Squares 10.4 Properties of the Least Squares Estimators 10.5 An Estimator of d2 10.6 Assessing the Utility of the Model: Making Inferences About the Slope ss1 10.7 The Coefficient of Correlation 10.8 The Coefficient of Determination 10.9 Using the Model for Estimation and Pediction 10.10 A Complete Example 10.11 A Summary of the Steps to Follow in Simple Linear Regression Statistics in Action: Can Dowser's Really Detect Water? CHAPTER 11: MULTIPLE REGRESSION ANALYSIS 11.1 General Form of a Multiple Regression Model 11.2 Model Assumptions 11.3 Fitting the Model: The Method of Least Squares 11.4 Computations using Matrix Algebra Estimating and Making Inferences about the ss Parameters 11.5 Assessing Overall Model Adequacy 11.6 A Confidence Interval for E(y) and a prediction interval for a Future Value of y 11.7 A First-Order Model with Quantitative Predictors 11.8 An Interaction Model with Quantitative Predictors 11.9 A Quadratic (Second-Order) Model with a Quantitative Predictor 11.10 Checking Assumptions: Residual Analysis 11.11 Some Pitfalls: Estimability, Multicollinearity, and Extrapolation 11.12 A Summary of the Steps to Follow in a Multiple Regression Analysis Statistics in Action: Bid-Rigging in the Highway Construction Industry CHAPTER 12: MODEL BUILDING 12.1 Introduction: Why Model Building is Important 12.2 The Two Types of Independent Variables: Quantitative and Qualitative 12.3 Models with a Single Quantitative Independent Variable 12.4 Models with Two Quantitative Independent Variables 12.5 Coding Quantitative Independent Variables (Optional) 12.6 Models with One Qualitative Independent Variable 12.7 Models with Both Quantitative and Qualitative Independent Variables 12.8 Tests for Comparing Nested Models 12.9 External Model Validation (Optional) 12.10 Stepwise Regression Statistics in Action: Deregulation of the Intrastate Trucking Industry CHAPTER 13: PRINCIPLES OF EXPERIMENTAL DESIGN 13.1 Introduction 13.2 Experimental Design Terminology 13.3 Controlling the Information in an Experiment 13.4 Noise-Reducing Designs 13.5 Volume-Increasing Designs 13.6 Selecting the Sample Size 13.7 The Importance of Randomization Statistics in Action: Anti-Corrosive Behavior of Epoxy Coatings Augmented with Zinc CHAPTER 14: ANALYSIS OF VARIANCE FOR DESIGNED EXPERIMENTS 14.1 Introduction 14.2 The Logic Behind an Analysis of Variance 14.3 One-Factor Completely Randomized Designs 14.4 Randomized Block Designs 14.5 Two-Factor Factorial Experiments 14.6 More Complex Factorial Designs (Optional) 14.7 Nested Sampling Designs (Optional) 14.8 Multiple Comparisons of Teatment Means 14.9 Checking ANOVA Assumptions Statistics in Action: On the Trail of the Cockroach CHAPTER 15: NONPARAMETRIC STATISTICS 15.1 Introduction: Distribution-Free Tests 15.2 Testing for Location of a Single Population 15.3 Comparing Two Populations: Independent Random Samples 15.4 Comparing Two Populations: Matched-Pair Design 15.5 Comparing Three or More Populations: Completely Randomized Design 15.6 Comparing Three or More Populations: Randomized Block Design 15.7 Nonparametric Regression Statistics in Action: Agent Orange and Vietnam Vets CHAPTER 16: STATISTICAL PROCESS AND QUALITY CONTROL 16.1 Total Quality Management 16.2 Variable Control Charts 16.3 Control Chart for Means: x-Chart 16.4 Control Chart for Process Variation: R-Chart 16.5 Detecting Trends in a Control Chart: Runs Analysis 16.6 Control Chart for Percent Defective: p-Chart 16.7 Control Chart for number of Defectives per item: c-Chart 16.8 Tolerance Limits 16.9 Capability Analysis (Optional) 16.10 Acceptance Sampling for Defectives 16.11 Other Sampling Plans (Optional) 16.12 Evolutionary Operations (Optional) Statistics in Action: Testing Jet Fuel Additive for Safety CHAPTER 17: PRODUCT AND SYSTEM RELIABILITY 17.1 Introduction 17.2 Failure Time Distributions 17.3 Hazard Rates 17.4 Life Testing: Censored Sampling 17.5 Estimating the Parameters of an Exponential Failure Time Distribution 17.6 Estimating the Parameters of a Weibull Failure Time Distribution 17.7 System Reliability Statistics in Action: Modeling the Hazard Rate of Reinforced Concrete Bridge Deck Deterioration APPENDIX A: MATRIX ALGEBRA APPENDIX B: USEFUL STATISTICAL TABLES APPENDIX C: SAS FOR WINDOWS TUTORIAL APPENDIX D: MINITAB FOR WINDOWS TUTORIAL APPENDIX E: SPSS FOR WINDOWS TUTORIAL ANSWERS TO SELECTED EXERCISES INDEX

Modeling Distributions of Insect Development Time: a Literature Review and Application of the Weibull Function

Abstract: We describe a stochastic approach for modeling insect development based on a single, temperature-independent distribution of normalized development times. We review other stochastic approaches, as well as problems encountered in modeling distributions of development time. A computer program, assembled from the Statistical Analysis System library, constructs cumulative probability distributions from frequency data on insect development times. These data are obtained from constant temperature experiments. The computer program normalizes the times of these distributions on their median time, identifies a single empirical distribution representative of all normalized distributions, and fits a cumulative Weibull function to this standard curve. The program determines the starting values of the three Weibull parameters and computes least-square estimates of these parameters using Marquardt techniques. This normalized probability function was tested against 23 data sets with good results, and can be used in population models to distribute cohort development through time under variable temperature conditions.

Outage Probability in Mobile Telephony Due to Multiple Log-Normal Interferers

Abstract: The mobile radio channel is characterized by three important factors: path losses larger than free space, fading typically taken as Rayleigh, and shadowing generally characterized as lognormal. For cellular systems, in order to determine acceptable reuse distances between base stations and to compare modulation methods, the probability of unacceptable cochannel interference (outage probability) has to be determined in the realistic situation where both fading and shadowing occur. In this paper, the average outage probability is computed for centrally located base stations when multiple log-normal interferers are present. This is done for both the mobile-to-base and base-to-mobile communication links. An unexpected result of this study is that the outage probabilities for the two cases do not differ in a significant way. Cumulative probability curves of the short-term average-signal-toaverage-interference ratio (SIR) are presented for a variety of system parameters: channel set number, propagation law exponent (γ), and dB spread (σ) of the log-normal distribution for the signal and interferers. An important observation is the large sensitivity of the performance curves to the propagation parameters: for a system with seven channel sets with a 10 dB SIR threshold, the average outage probability varies from 10 percent for \gamma = 3.7, \sigma = 6 dB, to 70 percent for \gamma = 3, \sigma = 14 dB. Alternatively, for a fixed outage objective of 10 percent, the required SIR threshold value ranges from -17 dB to 11 dB, depending on the propagation parameters. These variations make it imperative that accurate measurements of these parameters be obtained for the different service areas. Outage probabilities are also easily related to specific modulation methods and diversity approaches; detailed results are given for several representative cases.

Residual-Based Procedures for Prediction and Estimation in a Nonlinear Simultaneous System

Abstract: This paper proposes the residual-based stochastic predictor as an alternative procedure for obtaining forecasts with a static nonlinear econometric model. This procedure modifies the usual Monte Carlo approach to stochastic simulations of the model in that calculated residuals over the sample period are used as proxies for disturbances instead of random draws from some assumed parametric distribution. In compar-ison with the Monte Carlo predictor, the residual-based should be less sensitive to distributional assumptions concerning disturbances in the system. It is also less demanding computationally. The large-sample asymptotic moments of the residual-based predictor are derived in this paper and compared with those of the Monte Carlo predictor. Both procedures are asymptotically unbiased. In terms of asymptotic mean squared prediction error (AMSPE), the Monte Carlo is efficient relative to the residual-based when the number of replications in the Monte Carlo simulations is large relative to sample size. This order of relative efficiency is reversed, however, when replication and sample sizes are similar. In any event, the amount by which the AMSPE of either predictor exceeds the lower bound for AMSPE is small as a percentage of the lower bound AMSPE when sample and replication sizes are at least of moderate magnitude. The paper also discusses the extension of the residual-based anld Monte Carlo procedures to the estimation of higher order moments and cumulative distribution functions of endogenous variables in the system.

A flexible method for empirically estimating probability functions

Abstract: This paper presents a hyperbolic trigonometric (HT) transformation procedure for empirically estimating a cumulative probability distribution function (cdf), from which the probability density function (pdf) can be obtained by differentiation. Maximum likelihood (ML) is the appropriate estimation technique, but a particularly appealing feature of the HT transformation as opposed to other zero-one transformations is that the transformed cdf can be fitted with ordinary least squares (OLS) regression. Although OLS estimates are biased and inconsistent, they are usually very close to ML estimates; thus use of OLS estimates as starting values greatly facilitates use of numerical search procedures to obtain ML estimates. ML estimates have desirable asymptotic properties. The procedure is no more difficult to use than unconstrained nonlinear regression. Advantages of the procedure as compared to alternative procedures for fitting probability functions are discussed in the manuscript. Use of the conditional method is illustrated by application to two sets of yield response data.

Computer Programs: Normal Family Distribution Functions: FORTRAN and BASIC Programs

Abstract: This paper provides a set of FORTRAN and BASIC subroutines that calculate the cumulative probability distribution functions for the standard normal, chi-square, F, and Student's t distributions...

Normal Family Distribution Functions: FORTRAN and BASIC Programs

Abstract: This paper provides a set of FORTRAN and BASIC subroutines that calculate the cumulative probability distribution functions for the standard normal, chi-square, F, and Student's t distributions.

A characterization of the exponential distribution by higher order gap

Abstract: SupposeX is a non-negative random variable with an absolutely continuous (with respect to Lebesgue measure) distribution functionF (x) and the corresponding probability density functionf(x). LetX 1,X 2,...,X n be a random sample of sizen fromF andX i,n is thei-th smallest order statistics. We define thej-th order gapg i,j(n) asg i,j(n)=X i+j,n−Xi,n′ 1≤i

Economics of Energy Storage Devices in Interconnected Systems - A New Approach

Abstract: The simulation of energy storage devices in power systems for economic analysis has generally been attempted as a linear program formulation. This report suggests a new method that includes the effect of machine outage and non-linear nature of cost of production of energy. The probability density function (PDF) of hourly loads is the starting point of the method. The method has proposed the concepts of shifting the PDF to a "target" to honor the energy constraints and has shown that the modifications to the PDF can be easily obtained from the cumulative distribution function (CDF)

A characterization of the logarithmic series distribution and its application

Abstract: This paper gives a characterization of the logarithmic series distribution with probability function . Our characterization follows that of Poisson and positive Poisson given by Bol'shev (1965) and Singh (1978), respectively, A statistic is suggested as a consequence of the characterization to test whether a random sample X1,X2..Xnfollows a logarithmic series probability law. The desirability of the test statistic over the usual goodness-of-fit test is discussed, A numerical example is considered for illustration.

Computation of output electron distributions in avalanche photodiodes

Abstract: The cumulative probability distribution of the number of electrons counted during an interval (0, T) at the output of an avalanche photodiode is calculated from Personick's and McIntyre's model by numerical contour integration in the complex plane, a method that yields accurate results with a number of operations roughly independent of the numbers of electrons involved. The incident light is assumed for the sake of illustration to produce primary electrons with a Poisson distribution.

Conditioned Gaussian Probability Density

Abstract: A workable expression is developed in the case of bivariate Gaussian variables for the probability density function of one variable conditioned on a range of values for the other variable. Such an expression is useful in evaluating sturctural reliability with correlated modes of collapse. A lengthy expression for trivariate Gaussian variables is also presented.

Separation metrics for real-valued random variables

Abstract: If W is a fixed, real-valued random variable, then there are simple and easily satisfied conditions under which the function d W , where d W ( X , Y ) = the probability that W “separates” the real-valued random variables X and Y , turns out to be a metric. The observation was suggested by work done in [1].

An evaluation method for nonstationary random noise in view of temporal change of statistical moments (General theory and its application to probabilistic estimation problem of road traffic noise)

Abstract: A unified statistical treatment of the probability distribution function is theoretically proposed in the case when a general random noise of arbitrary distribution type exhibits various type nonstationary properties due to the temporal change of statistical moments. For the purpose of finding the effect of nonstationarity caused by the temporal change of moments on the resultant probability distribution form, the explicit expression of the probability density function is derived in a general form of statistical expansion series taking the stationary term into the first term of series expansion. Next, a new trial of estimating a noise level distribution over a long time interval on the basis of level distribution within a short time interval is discussed, by using the above theoretical probability expression. The validity of the theoretical result is confirmed experimentally by applying it to the actual level data of the nonstationary road traffic noise observed in a large city.

Two-Dimensional Modeling for Lineal and Areal Probabilities of Weather Conditions.

Abstract: : Single-point probabilities of weather conditions, which are easily estimated from climatic records, have been extended to lines and areas by means of Monte Carlo simulation. Simulation was accomplished using the Boehm Sawtooth Wave (BSW) model. This model was chosen because of its speed and simplicity, and because it has a spatial correlation function similar to that of many weather elements. The BSW model generates fields (or maps) of normally distributed values called Equivalent Normal Deviates (ENDs). The procedure was to obtain the cumulative probability distribution for threshold END values. To do this, a large number of maps had to be generated, 25,000 in all, to approximate the true probability distributions. This was done for 12 different sized square areas and lines. The results were put in graphical form by plotting the probabilities as a function of areal and lineal size, and fitting them to curves through hand analysis. The curves were then fitted by equations, making it possible to obtain solutions quickly by computer. Thus, a model has been produced that can be used to estimate the probability that a certain weather condition will cover a given area or length, or fraction of an area or length.

19 Empirical distribution function

Abstract: Publisher Summary This chapter describes the empirical distribution function. A statistical estimation of F(x) based on a random sample (X 1 . . . X n ,) is the so-called empirical or sample distribution function. F(x) is considered also a (random) function of x . To apply statistical methods based on empirical distribution, such as goodness of fit or two-sample tests, confidence intervals, one needs the exact or limiting distributions of statistics concerned.

On a new distribution arising in decaying inventory systems

Abstract: This paper considers the number of de~ands for a good item during a cycle of an inventory system with initial stock Q and with the items in stock deteriorating stochastically over time. The demands occur as a Poisson process, and the lot is replenished with zero lead time, making the cycle time itself a random variable. The probability distribution of the number of demands is developed; also tabulated is the minimal reorder quantity Q(k,δ) to assure meeting a minimum of k demands in a cycle with a confidence of (1-δ) .

Tables of cumulative distribution functions and percentiles of the standardized stable random variables

Abstract: It is shown that Zolotarev's (1964) integral representation of the cumulative distribution function (c.d.f.) of stable random variables and the IMSL subroutine DCADRE (for numerical integration ) provide a natural and practically simple method for finding the values of c.d.f., the percentiles and the density function of such random variables. For symmetric stable random variables (r.v.'s ) Z∝, values of P∝(z) … P(0

Under-Ice Roughness: Shot Noise Model

Abstract: : The one-dimensional roughness of an under-ice profile of elliptical bosses is modeled in the time domain by a shot noise process of elliptical pulses of random amplitude, duration, and time of occurrence. A sample realization of 8000 data points is generated and plotted for visual comparison with experimental under-ice data. Also, theoretical and simulation results for the power density spectrum, the autocorrelation function, the characteristic function, the cumulative distribution function, and the probability density function of the shot noise process are plotted and compared.

Probabilistic seismic resistance of a Mark III steel containment

Abstract: Concern for the safety of nuclear power plants has motivated efforts to determine the statistical characteristics of the seismic resistance of steel containments. This paper represents a portion of a project whose objective is to assess the uncertainty of containment strength subjected to earthquakes (and other types of loading). The seismic response is predicted by a random vibration technique outlined in a paper previously published by the authors and calibrated to a design response spectrum. BOSOR4 is first used to predict linear vibration modes. Modal analysis methods are utilized, with the random vibration technique, to obtain a statistical description of the random process stress resultants for a unit g peak ground acceleration. Peak stress resultants are described by Extreme Value I. The mean peak values are input into BOSOR5 and the peak g-level is increased until failure (strain ductility of two) is reached. An approximate product failure function is obtained by perturbing the other random variables, namely, geometric imperfections, damping, and yield strength. The Advanced First Order Second Moment technique is then used to construct the cumulative distribution of the containment seismic resistance as a function of peak ground acceleration. The resulting distribution is given for a typical Mark IIImore » steel containment. Fragility curves for the containment at 95% and 5% certainty are also given.« less

On the Stochastic Steady-State Behavior of Optimal Asset Accumulation in the Presence of Random Wage Fluctuations and Incomplete Markets

Abstract: We establish rigorously the existence and properties of the stationary probability distribution which characterizes the accumulation of non-contingent financial claims by a risk averse individual who confronts random wage fluctuations and incomplete insurance markets We show that there exists a unique, almost-everywhere continuous stationary cumulative distribution function which characterizes the accumulation of non-contingent financial claims in a stochastic steady-state This distribution is shown to possess a single mass point coinciding with the non-negative, finite borrowing limit faced by the individual We establish that the stationary distribution which characterizes the asset accumulation of low time preference individuals is at least as large, in the sense of first-degree stochastic dominance, as that of individuals with higher rates of time preference We prove that, so long as individuals are allowed to borrow in amounts which can be repayed with probability one, additive differences in the probability distribution governing random wage earnings imply inversely proportional additive differences in the stationary probability distributions which govern the accumulation of non-contingent financial claims

Reply to 'Comment on temperature dependence of electrical conductivity and the probability density function'

Abstract: For original paper see Love, ibid., vol.16, p.5985 (1983). For comment, see Davies, ibid., vol.17, p.565 (1984). Consideration of the Fermi function as a temperature and the temperature dependence of electrical conductivity of metallic substances supports the premise that the Fermi function is a cumulative density function. The criticisms presented in the comment do not invalidate this concept.

Computing approximate random Delta v magnitude probability densities

Abstract: This paper describes the development and use of an algorithm to compute approximate statistics of the magnitude of a single random trajectory correction maneuver (TCM) Delta v vector. The TCM Delta v vector is modeled as a three component Cartesian vector each of whose components is a random variable having a normal (Gaussian) distribution with zero mean and possibly unequal standard deviations. The algorithm uses these standard deviations as input to produce approximations to (1) the mean and standard deviation of the magnitude of Delta v, (2) points of the probability density function of the magnitude of Delta v, and (3) points of the cumulative and inverse cumulative distribution functions of Delta v. The approximates are based on Monte Carlo techniques developed in a previous paper by the author and extended here. The algorithm described is expected to be useful in both pre-flight planning and in-flight analysis of maneuver propellant requirements for space missions.