Probability density function

About: Probability density function is a research topic. Over the lifetime, 22321 publications have been published within this topic receiving 422885 citations. The topic is also known as: probability function & PDF.

Modelling of error covariances by 4D‐Var data assimilation

Abstract: The extended Kalman filter is presented as a good approximation to the optimal assimilation of observations into a numerical weather prediction (NWP) model, as long as the evolution of errors stays close to linear. The error probability distributions are approximated by Gaussians, characterized by their mean and covariance. The full nonlinear forecast model is used to propagate the mean, and a linear model (not necessarily tangent to the full model) the covariances. Since it is impossible to determine the covariances in detail, physically based assumptions about their behaviour must be made; for instance, three-dimensional balance relationships are used. The linear model can be thought of as extending the covariance relationships to the time dimension. Incremental four-dimensional variational (4D-Var) is derived as a practical implementation of the extended Kalman filter, optimally using these modelled covariances for a finite time window. It is easy to include a simplified model of forecast errors in the representation. This Kalman filter based paradigm differs from more traditional derivations of 4D-Var in attempting to estimate the mean, rather than the mode, of the posterior probability density function. The latter is difficult for a NWP system representing scales which exhibit chaotic behaviour over the period of interest. The covariance modelling assumptions often result in a null space of error modes with little variance. It is argued that this is as important as the variance and correlation structures usually examined, since the implied constraints allow optimal use of observations giving gradient and tendency information. Difficulties arise in the approach when the NWP system is capable of resolving significant structures (such as convective cells) not always determined by the observations.

On weak convergence and optimality of kernel density estimates of the mode

Abstract: A mode of a probability density $f(t)$ is a value $\theta$ that maximizes $f$. The problem of estimating the location of the mode is considered here. Estimates of the mode are considered via kernel density estimates. Previous results on this problem have several serious drawbacks. Conditions on the underlying density $f$ are imposed globally (rather than locally in a neighborhood of $\theta$). Moreover, fixed bandwidth sequences are considered, resulting in an estimate of the location of the mode that is not scale-equivariant. In addition, an optimal choice of bandwidth depends on the underlying density, and so cannot be realized by a fixed bandwidth sequence. Here, fixed and random bandwidths are considered, while relatively weak assumptions are imposed on the underlying density. Asymptotic minimax risk lower bounds are obtained for estimators of the mode and kernel density estimates of the mode are shown to possess a certain optimal local asymptotic minimax risk property. Bootstrapping the sampling distribution of the estimates is also discussed.

Serial Free-Space Optical Relaying Communications Over Gamma-Gamma Atmospheric Turbulence Channels

Abstract: In this paper, a study on the end-to-end performance of multihop free-space optical wireless systems over turbulence-induced fading channels, modeled by the gamma-gamma distribution, is presented. Our analysis is carried out for systems employing amplify-and-forward channel-state-information-assisted or fixed-gain relays. To assess the statistical properties of the end-to-end signal-to-noise ratio for both considered systems, we derive novel closed-form expressions for the moment-generating function, the probability density function, and the cumulative distribution function of the product of rational powers of statistically independent squared gamma-gamma random variables. These statistical results are then applied to studying the outage probability and the average bit error probability of binary modulation schemes. Also, for the case of channel-state-information-assisted relays, an accurate asymptotic performance analysis at high SNR values is presented. Numerical examples compare analytical and simulation results, verifying the correctness of the proposed mathematical analysis.

Effect of Molecular Vibrations on Gas Electron Diffraction. I. Probability Distribution Function and Molecular Intensity for Diatomic Molecules

Abstract: General expressions and their practical approximations are presented for the molecular intensity of gas electron diffraction and the radial distribution function for an atom pair in the molecule which exerts moderately anharmonic intramolecular motion in thermal equilibrium. The calculations are based on a polynomial expansion of the probability distribution function of the internuclear distance around a Gaussian function representing harmonic vibration. The coefficients of the expansion for a diatomic system are calculated by the second-order perturbation method in terms of the coefficients of the potential energy expanded to the fourth order of the displacement in the internuclear distance. The structural parameters under various definitions (the mean internuclear distances, the mean square amplitudes, and the phase parameters) of a number of diatomic molecules are tabulated by the use of the spectroscopic values ωe, αe, and χe taken from the literature. The temperature dependence of the structural para...

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