Showing papers on "Probability density function published in 1971"

Nonparametric roughness penalties for probability densities

Abstract: ONE of the most fundamental problems in statistics is the estimation of a probability density function from a sample, the smoothing of a histogram being the usual non-parametric method. This method requires a large sample and even so it is difficult to decide whether “bumps” are genuinely in the population. A method is presented here that should help to overcome this difficulty.

Probabilistic methods of signal and system analysis

Abstract: Preface 1. Introduction To Probability 1-1 Engineering Applications Of Probability 1-2 Random Experiments And Events 1-3 Definitions Of Probability 1-4 The Relative-Frequency Approach 1-5 Elementary Set Theory 1-6 The Axiomatic Approach 1-7 Conditional Probability 1-8 Independence 1-9 Combined Experiments 1-10 Bemoulli Trials 1-11 Applications Of Bemoulli Trials 2. Random Variables 2-1 Concept Of A Random Variable 2-2 Distribution Functions 2-3 Density Functions 2-4 Mean Values And Moments 2-5 The Gaussian Random Variable 2-6 Density Functions Related To Gaussian 2-7 Other Probability Density Functions 2-8 Conditional Probability Distribution And Density Functions 2-9 Examples And Applications 3. Several Random Variables 3-1 Two Random Variables 3-2 Conditional Probability-Revisited 3-3 Statistical Independence 3-4 Correlation Between Random Variables 3-5 Density Function Of The Sum Of Two Random Variables 3-6 Probability Density Function Of A Function Of Two Random Variables 3-7 The Characteristic Function 4. Elements oOf Statistics 4-1 Introduction 4-2 Sampling Theory- The Sample Mean 4-3 Sampling Theory- The Sample Variance 4-4 Sampling Distributions And Confidence Intervals 4-5 Hypothesis Testing 4-6 Curve Fitting And Linear Regression 4-7 Correlation Between Two Sets of Data 5. Random Processes 5-1 Introduction 5-2 Continuous And Discrete Random Processes 5-3 Deterministic And Nondeterministic Random Processes 5-4 Stationary and Nonstationary Random Processes 5-5 Ergodic And Nonergodic Random Processes 5-6 Measurement Of Process Parameters 5-7 Smoothing Data With A Moving Window Average 6. Correlation Functions 6-1 Introduction 6-2 Example:Autocorrelation Function Of A Binary Profess 6-3 Properties Of Autocorrelation Functions 6-4 Measurement Of Autocorrelation Functions 6-5 Examples Of Autocorrelation Functions 6-6 Crosscorrelation Functions 6-7 Properties Of Crosscorrelation Functions 6-8 Examples And Applications Of Crosscorrelation Functions 6-9 Correlation Matrices For Sampled Functions 7. Spectral Density 7-1 Introduction 7-3 Properties Of Spectral Density 7-4 Spectral Density And The Complex Frequency Plane 7-5 Mean-Square Values From Spectral Density 7-6 Relation Of Spectral Density To The Autocorrelation Function 7-7 White Noise 7-8 Cross-Spectral Density 7-9 Measurement Of Spectral Density 7-10 Periodogram Estimate Of Spectral Density 7-11 Examples And Applications Of Spectral Density 8. Repines Of Linear Systems To Random Inputs 8-1 Introduction 8-2 Analysis In The Time Domain 8-3 Mean And Mean-Swquare Value Of System Output 8-4 Autocorrelation Function Of System Output 8-5 Crosscorrelation Between Input And Output 8-6 Example Of Time-Domain Analysis 8-7 Analysis In The Frequency Domain 8-8 Spectral Density At The System Output 8-9 Cross-Spectral Densities Between Input And Output 8-10 Examples Of Frequency-Domain Analysis 8-11 Numerical Computation Of System Output 9. Optimum Linear Systems 9-1 Introduction 9-2 Criteria Of Optimaility 9-3 Restrictions On The Optimum System 9-4 Optimization By Parameter Adjustment 9-6 Systems That Minimize Mean-Square Error Appendices Appendix A: Mathematical Tables Appendix B: Frequently Encountered Probability Distributions Appendix C: Binomial Coefficients Appendix D: Normal Probability Distribution Function Appendix E: The Q-Function Appendix F: Student's T-Distribution Function Appendix G: Computer Computations Appendix H: Table Of Correlation Function-Spectral Density Pairs Appendix I: Contour Integration

On the Relation between Master Equations and Random Walks and Their Solutions

Abstract: It is shown that there is a simple relation between master equation and random walk solutions. We assume that the random walker takes steps at random times, with the time between steps governed by a probability density ψ(Δt). Then, if the random walk transition probability matrix M and the master equation transition rate matrix A are related by A = (M − 1)/τ1, where τ1 is the first moment of Ψ(t) and thus the average time between steps, the solutions of the random walk and the master equation approach each other at long times and are essentially equal for times much larger than the maximum of (τn/n!)1/n, where τn is the nth moment of ψ(t). For a Poisson probability density ψ(t), the solutions are shown to be identical at all times. For the case where A ≠ (M − 1)/τ1, the solutions of the master equation and the random walk approach each other at long times and are approximately equal for times much larger than the maximum of (τn/n!)1/n if the eigenvalues and eigenfunctions of A and (M − 1)/τ1 are approxima...

Sequential estimation of a continuous probability density function and mode

Abstract: The purpose of this paper is to discuss statistical properties of an estimate of a probability density function based on the first n observations under the assumption of continuity or uniform continuity of the probability density function in case where we observe a sequence of random vectors which come from a population with the probability density function. Let X1, X2,X3,••• be a sequence of independent identically distributed m-dimensional random vectors having a probability density function f(x). Suppose the first n observations be denoted by X1, X2, ••• Xn. Then a natural estimate of the probability density function f(x) may be denoted as follows for a suitable positive constant h

New results on avalanche multiplication statistics with applications to optical detection

Abstract: In this paper, we derive statistics of the random gain of two types of avalanche diode optical detectors. A simple optical binary receiver which could employ these devices is analyzed. In particular, we determine the moment generating function of the random gain probability density for a diode with equal hole and electron collision probabilities and for a diode with unilateral gain. For the unilateral gain case, we invert the moment generating function to obtain the probability density which turns out to be a shifted Bose Einstein density. Using the Chernoff bound, we analyze the performance of a simple binary receiver using the above devices. In addition, we exactly analyze a receiver with a deterministic gain device. We upper bound the degradation incurred from the use of a random gain rather than a deterministic gain. For the devices above, the degradation can be as small as a dB or less in certain ranges of parameter values discussed in the text.

Optimal non-linear estimation†

Abstract: For the non-linear estimation problem with non-linear plant and observation models, white gaussian excitations and continuous data, the state-vector a posteriori probabilities for prediction and smoothing are obtained via the 'partition theorem'. Moreover, for the special class of non-linear estimation problems with linear models excited by white gaussian noise, and with non-gaussian initial state, explicit results are obtained for the a posteriori probabilities, the optimal estimates and the corresponding error-covariance matrices for filtering, prediction and smoothing. In addition, for the latter problem, approximate but simpler expressions are obtained by using a gaussian sum approximation of the initial state-vector probability density. As a special case of the above results, optimal linear smoothing algorithms are obtained in a new form.

Optimal nonlinear estimation

Abstract: For the nonlinear estimation problem with nonlinear plant and observation models, white gaussian excitations and continuous data, the state-vector a-posteriori probabilities for prediction, and smoothing are obtained via the "partition theorem". Moreover, for the special class of nonlinear estimation problems with linear models excited by white gaussian noise, and with nongaussian initial state, explicit results are obtained for the a-posteriori probabilities, the optimal estimates, and the corresponding error-covariance matrices for filtering, prediction, and smoothing. In addition, for the latter problem, approximate but simpler expressions are obtained by using a gaussian sum approximation of the initial state-vector probability density. As a special case of the above results, optimal linear smoothing algorithms are obtained in a new form.

Random variables with independent binary digits

Abstract: : Let X = .b1b2b3... be a random variable with independent binary digits bn taking values 0 or 1 with probabilities pn and qn. When does X have a density function. A continuous density function. A singular distribution. This note proves that the distribution X is singular is and only if the tail of the series Summation (log(pn/qn)) squared diverges, and that X has a density that is positive on some interval if and only if log(pn/qn) is a geometric sequence with ratio 1/2 for n greater than some k, and in that case the fractional part of (2 to the power k)X has an exponential density (increasing or decreasing with the uniform density a special case). It gives a sufficient condition for X to have a density, (Summation log (2 max (pn,qn))converges), but unless the tail of the sequence log(pn/qn) is geometric, ratio 1/2, the density is a weird one that vanishes at least once in every interval.

Performance-measure densities for a class of LQG control systems

Abstract: When linear-quadratic-Gaussian (LQG) control problems are solved, the performance measure takes on a probability density function Of a type which has received very little attention in the literature because of the nonzero-mean multidimensional character of the state process. This paper combines techniques of the fast Fourier transform, recent results of Baggeroer and Schwartz, and the classic methods of Karhunen-Loeve to present what seem to be the first published curves of such densities. As a vehicle for discussion, a one-parameter family of LQG control systems of the minimal variance type serves both to provide examples of the new density curves and to display certain controlled behaviors of the moments of the densities.

Multiple Error Performance of PSK Systems with Cochannel Interference and Noise

Abstract: The multiple error performance of a phase-shift-keyed (PSK) communications system, when both cochannel interference (due possibly to other cochannel angle-modulated systems) and Gaussian noise additively perturb the transmitted signals, is considered. The results are fairly general: the main requirement is that the interference be circularly symmetric. All of our findings are also applicable to the case when only noise is present. The results indicate that one cannot approximate well the effect of interference on the performance of a PSK system by treating it as additional Gaussian noise. First, we derive the probability density function f A of the phase angle of a cosinusoid plus interference and Gaussian noise. We then obtain readily computable expressions (in terms of f A ) for the probability of any number of consecutive errors in an m -phase system when either coherent or differential detection is utilized. For numerical results, the interference is assumed to be due to other cochannel angle-modulated communications systems, and the double error probability and conditional probability of error are given for 2- and 4-phase systems.

Asymptotic inference about a density function at an end of its range

Abstract: For each n, X1(n),…Xn(n) are independent and identically distributed random variables, with common probability density function Where c, θ, α, and r(y) are all unknown. It is shown that we can make asymptotic inferences about c, θ, and α, when r(y) satisfies mild conditions.

On a sequential rule for estimating the location parameter of an exponential distribution

Abstract: Let us assume that observations are obtained at random and sequentially from a population with density function In this paper we consider a sequential rule for estimating μ when σ is unknown corresponding to the following class of cost functions In this paper we consider a sequential rule for estimating μ when σ is unknown corresponding to the following class of cost functions Where δ(XI,…,XN) is a suitable estimator of μ based on the random sample (X1,…, XN), N is a stopping variable, and A and p are given constants. To study the performance of the rule it is compared with corresponding “optimum fixed sample procedures” with known σ by comparing expected sample sizes and expected costs. It is shown that the rule is “asymptotically efficient” when absolute loss (p=-1) is used whereas the one based on squared error (p = 2) is not. A table is provided to show that in small samples similar conclusions are also true.

Bayesian full information structural analysis

Abstract: I. Bayesian Full Information Analysis of the Simultaneous Equations Model.- 1. A review of the problem of identification in a Bayesian approach and the specifications of the prior density functions.- 1.1. The statistical model and notation.- 1.2. The problem of identification in a Bayesian context and the choice of prior distributions.- 2. The extended natural conjugate density and its properties.- 2.1. The extended natural conjugate density of all the parameters of the model.- 2.2. The extended natural conjugate density bearing on the parameters of a model with prior exclusion restrictions.- 2.3. Interpretation of the extended natural conjugate density.- 3. Posterior distributions of the structural parameters (?, ?-1).- 3.1. The joint a posteriori density of (?, ?-1).- 3.2. The marginal density function of ?.- Appendix to Part I. Some properties of the Wishart density function and the matric variate-t-density function.- II. Empirical illustration of a Bayesian Full Information Analysis. The analysis of the Belgian beef market.- 1. The model and the a priori information.- 1.1. The model of Calicis.- 1.2. Two equations models for the Belgian beef market.- 1.3. The likelihood function and the a priori density function.- 1.4. A description of the sources of prior-information.- 1.5. The complete specification of the prior density function.- 2. The Posterior Analysis.- 2.1. The posterior distributions.- 2.2. Comments on the results of the posterior analysis.- Conclusions.- References.

Some Results on the Sampling Distribution of the Multiple Correlation Coefficient

Abstract: Recurrence formulae for the density function and the probability integral of the multiple correlation coefficient from a normal sample are obtained. When the number of independent variates is odd these formulae are related to the distribution of the simple correlation. Asymptotic expansions in terms of the non-central beta and of the non-central x2 distributions are derived. Methods of approximation by the central and the non-central F distributions, as well as by the normal distribution, are proposed. Numerical investigations of the accuracies of these approximations are carried out.

The computation of error probability for digital transmission

Abstract: A method is presented for calculating the probability of error for a digital signal contaminated by intersymbol interference and additive Gaussian noise. The method constructs a close approximation to the probability density function of the intersymbol interference, circumventing the heretofore formidable computational problems by decomposing the calculation into a sequence of simple calculations. This method is applicable to binary transmissions, as well as 4-, 8-, 16-, … level transmission. The method is rapid enough for use on time-sharing facilities. As part of the method, a new and rather simple scheme is presented for including the effects of partial response source coding. Several interesting examples which use and give insight into the method are included.

Properties of Waveforms Obtained by Nonrecursive Digital Filtering of Pseudorandom Binary Sequences

Abstract: This paper derives and discusses statistical properties (probability density function, moments, power spectrum, and autocorrelation function) of waveforms obtained by nonrecursive digital filtering of pseudorandom linear binary sequences. This filtering may be achieved in practice by forming a weighted summation of binary digits passing along a shift register.

Random FM in Mobile Radio with Diversity

Abstract: The probability density function and autocorrelation function of random FM in a mobile radio system are derived for the case where postdetection selection diversity is used. Closed-form solutions for the autocorrelation function have been found for no diversity and two-branch diversity and infinite series expansions for higher orders. The use of two-branch selection diversity is shown to reduce the random FM by 13.5 dB in a typical case, with a further 2.5-dB reduction when three branches are used. The effect of other forms of diversity on random FM is briefly assessed, but not pursued in detail.

On the Distribution of Linear Combinations of Non-central Chi-Squares

Abstract: Press [1] expressed the distribution of an arbitrary linear combination of noncentral chi-square variates as a mixture of distributions of weighted differences between pairs of central chi-squares. The distributions appearing in the mixture depend on the coefficients in the linear combination. Here, by modifying Press's results, we obtain a mixture representation not exhibiting that property. Following Press, let X4,d denote a non-central chi-square variate, having m degrees of freedom and non-centrality parameter d, whose probability density function is given by

Generation of Random Gamma Variates from the Two-Parameter Gamma

Abstract: An algorithm has been developed capable of generating random gamma variates from any two-parameter gamma form with arbitrary parameters. The algorithm, shown to be both accurate and computationally efficient, is given as a FORTRAN IV function routine. Extensions are also made to the generation of random deviates from other useful probability density functions.

Life Testing: Structural Inference on The Exponential Model

Abstract: Structural inference theory develops the unique probability density function for the parameter of a scale model. This density constitutes a strong basis for statistical inference because no information is required beyond that provided by the classical statistical model. This paper applies structural theory to the one-parameter exponential measurement model. On the basis of the structural density function for the exponential scale parameter, the probability density of the reliability function is derived. Prediction densities are then developed as an alternative class of life test inferences; these densities concern (i) an arbitrary vector of future measurements, (ii) the average of a number of future measurements, and (iii) the smallest and largest of such future measurements. The data transformations are derived which allow the results of this paper to be applied to progressively censored data of types I and II. An example is given to illustrate the structural densities of this paper.

Bayesian Confidence Limits for the Reliability of Mixed Exponential and Distribution-Free Cascade Subsystems

Abstract: The problem treated here is the theoretical one of deriving exact Bayesian confidence intervals for the reliability of a system consisting of some independent cascade subsystems with exponential failure probability density functions (pdf) mixed with other independent cascade subsystems whose failure pdf's are unknown. The Mellin integral transform is used to derive the posterior pdf of the system reliability. The posterior cumulative distribution function (cdf) is then obtained in the usual manner by integrating the pdf, which serves the dual purpose of yielding system reliability confidence limits while at the same time providing a check on the derived pdf. A computer program written in Fortran IV is operational. It utilizes multiprecision to obtain the posterior pdf to any desired degree of accuracy in both functional and tabular form. The posterior cdf is tabulated at any desired increments to any required degree of accuracy.

Random Pulse Train Generator with Linear Voltage Control of Average Rate

Abstract: An electronic instrument which generates a train of constant amplitude pulses occurring at random times is described. The average pulse rate of this process is a linear function of a controlling voltage. The intervals between successive pulses form a renewal process and the probability density of these intervals may be varied from an exponential distribution to a gamma distribution of variable form. Statistical measurements on the pulse train are described which support the claims on its statistical properties.

Superimposed non-stationary renewal processes

Abstract: For superposition of independent, stationary renewal processes, it is well known that the distribution of waiting time between events for the superimposed process is approximately exponential if the number of processes involved is sufficiently large, (see Khintchine (1960), Ososkov (1956)). We assume that all component processes have the same age t, and we generalize the classical result to show that even for t finite (non-stationary case), the limiting waiting time distribution (as the number of processes increases) is exponential with a scale parameter which depends on t through the average of the individual process renewal densities. For moderate values of t, this scale parameter can differ greatly from its limit due to the generally slow convergence of the renewal density function to its limit. While it has been generally assumed that this waiting time distribution is approximately exponential regardless of t, the explicit formulation of the necessary normalizations and conditions seems new. While a variety of papers on superimposed point processes have appeared recently, all assume stationarity (see 4inlar (1968) for references). Franken (1963) considered the time dependent behavior of superimposed renewal processes under the highly restrictive and generally unrealistic assumption that for each n, the superimposed process is itself a renewal process. Among other things, this requires the individual processes to have distributions which depend on n and which generally do not resemble any standard form. We believe that letting the individual processes have some standard form and imposing no particular form on the superimposed process is more realistic.

A stochastic model of human fertility.

Abstract: This paper describes human fertility from a stochastic viewpoint. The model consists of two transient states S1 fecundable state and S2 pregnant and infecundable state and an absorbing state R the state of death. The frequency of a womans pregnancy is represented by the number of her transitions from S1 to S2. The transition from S1 to S2 or the corresponding probability is a function of her age and the transition from S2 to S1 is a function of the length of time she has been pregnant as well as her age; thus the process is non-homogeneous with respect to time. Explicit general formulas are derived for the multiple transition probabilities between the three states describing a females frequency of pregnancy. The length of time required for a female to have a given number of pregnancies is also studied. Here the multiple transition times are random variables and the corresponding probability density functions are also presented. (authors)