An infinite series for the computation of the complementaryprobability distribution function of a sum of independent randomvariables and its application to the sum of Rayleigh random variables

The properties of the series are studied for both bounded and unbounded random variables. The technique is used to find efficient series for computation of the distributions of sums of uniform random variables and sums of Rayleigh random variables. A useful closed-form expression for the characteristic function of a Rayleigh random variable is presented, and an efficient method for computing a confluent hypergeometric function is given. An infinite series for the probability density function of a sum of independent random variables is also derived. The inversion of characteristic functions, a trapezoidal rule for numerical integration, and the sampling theorem in the frequency domain are related to, and interpreted in terms of, the results     

Estimating the distribution of a sum of independent lognormalrandom variables

Four methods that can be used to approximate the distribution function (DF) of a sum of independent lognormal random variables (RVs) are compared. The aim is to determine the best method to compute the DF considering both accuracy and computational effort. The investigation focuses on values of the dB spread, σ, valid for practical problems in wireless transmission (6 dB ⩽σ⩽12 dB). Contrary to some previous reports, our results show that the simpler Wilkinson's approach gives a more accurate estimate, in some cases of interest, than Schwartz and Yeh's (1982) approach     

Critical time of the lognormal distribution

The critical time is the time point as the failure rate starts to decrease and also as the mean residual lifetime starts to increase. The estimated critical time is useful for determining the duration of a burn-in process. The method for estimating the critical time of the failure rate for lognormal lifetime distribution is discussed. A single time censored data is used as a example for illustration.     

On the hazard rate of the lognormal distribution

It is argued that plots of the hazard rate for the lognormal random variable which have appeared in some recent literature are incorrect and/or misleading; the hazard rate always begins at zero, rises to a maximum, then decreases very slowly to zero. An equation for the location of the maximum of the hazard rate is derived. The maximum lies in a finite interval for all positive values of the standard deviation of the associated normal random variable. Approximations that can be used to compute the hazard rate for parameter values outside of the usual range in the tables associated with the normal (Gaussian) random variable are presented     

Highly accurate simple closed-form approximations to lognormal sum distributions and densities

Sums of lognormal random variables occur extensively in wireless communications, in part, because a shadowing environment is well modeled by a lognormal distribution. A closed-form expression does not exist for the sum distribution and, furthermore, it is difficult to numerically calculate the distribution. Numerous approximations exist that are based on approximating a sum of lognormal random variables as another lognormal random variable. A new paradigm to calculate an approximation to the lognormal sum distribution, based on curve fitting on lognormal probability paper, is introduced in this letter. Highly accurate, simple closed-form approximations to lognormal sum distributions are presented.     

Confidence limits on the failure rate and a rapid projection nomogram for the lognormal distribution

In order to predict the sample failure rate from a limited number of early malfunctions, we have developed the rapid lognormal projection nomogram (RLPN) which we present at 1, 5, and 10 years of service. In the core of each diagram is a family of curves showing the variation of failure rate as a function of sample median life and standard deviation. Superimposed are Arrhenius plots for a series of activation energies, all referenced with respect to a fixed time. We also display the standard deviation as a function of relative median life, using cumulative failure as a parameter. The RLPN provides graphical means to estimate the failure rate at the planned service life and operating temperature in an efficient manner from the time to failure at the aging temperature and sample size for known or assumed values of the standard deviation and activation energy. To obtain the confidence limits for the failure rate of the lognormal distribution at the 90% confidence level, we have derived an approximate formula that relates the bounds to the service life and the sample's size, median life, and standard deviation. Then, by an appropriate selection of the independent variables and parameters, the confidence intervals are displayed as a series of curve families. The applications of the diagrams are discussed by means of illustrative examples taken from the field of GaAs FETs.     

Estimation of typical sum of lognormal random variables using log shifted gamma approximation

The sum of lognormal distributions is a well-known problem that no closed-form expression exists and it is difficult to evaluate numerically. In this paper, log shifted gamma (LSG) approximation method is proposed to represent the sum of lognormal distributions and to derive a closed-form expression of the typical value of the sum. Illustrative results show that the LSG model provides much more accurate approximation than other previous methods for a wide range of lognormal variances.     

Predictive densities for the lognormal distribution and their applications

Maximum likelihood predictive densities (MLPDs) for a future lognormal observation are obtained and their applications to reliability and life testing are considered. When applied to reliability and failure rate estimations, they give estimators that can be much less biased and less variable than the usual maximum likelihood estimations (MLEs) obtained by replacing the unknown parameters in the density function by their MLEs. When applied to lifetime predictions, they give prediction intervals that are shorter than the usual frequentist intervals. Using the MLPDs, it is also rather convenient to construct the shortest prediction intervals. Extensive simulations are performed for comparisons. A numerical example is given for illustration.     

Some informational properties of the lognormal distribution (Corresp.)

The marginal, joint, and conditional entropy and the trans-information are derived for random variables with lognormal probability distributions, revealing some interesting deviations from its sister distribution, the normal. A maximal entropy property, capacity theorem for the lognormal channel, and implications for some nonlinear transformations are also presented. Potential applications of the measures to psychophysics are mentioned.     

Bounds on the delay distribution of window random-access algorithms

A method for analyzing the delay distribution of window random-access algorithms is presented. The window size is allowed to vary during the operation of the algorithm. It is shown that the quantities of interest in the computation of the delay distribution can be related to the solution of appropriate infinite systems of linear equations. Once the constants and the coefficients of the unknowns of the system are determined, bounds on the solution can be developed by applying previously developed methodologies. The method is applied to the delay distribution analysis of the Capetanakis window random-access algorithm and the part-and-try algorithm, both under binary feedback     

General expression for pdf of a sum of independent exponential random variables

In order to evaluate exactly the performance of some diversity schemes, the probability density function (pdf) of a sum of independent exponential random variables (r.v.'s) must be known. This paper proposes a simple method to find it by using characteristic function., The resultant pdf is successfully applied to formulate the closed-form BER expression of 2 Tx-J Rx transmit diversity as well as the outage probability of repetition coding.     

Application of three-parameter lognormal distribution in EM data analysis

The three-parameter lognormal distribution has been demonstrated for applications in electromigration data analysis, especially for Cu interconnect structures with insufficient redundancy. Examples are given on estimating parameter values from experimental data using the maximum likelihood method. Detailed analyses are presented on confidence bound estimations of the parameters and their propagation to lifetime projections.     

Bounds for independent regulated inputs multiplexed in a service curve network element

We consider the problem of bounding the probability of buffer overflow in a network node fed with independent arrival processes that are each constrained by arrival curves, but that are served as an aggregate. Existing results assume that the node is a constant rate server. However, in practice, one finds complex network nodes that do not provide a constant service rate, and thus, to which the existing bounds do not apply. Now many nodes can be adequately abstracted by a service curve property. We extend previous results to such cases. As a by-product, we also provide a slight improvement to the bound in Chang et al. (see Proc. Sigmettics 2001, Cambridge, MA, May 2001, p.184-193). Our bounds are valid for both discrete and continuous time models.     

Nakagami-m approximation to the sum of M non-identical independent Nakagami-m variates

A Nakagami-m approximate distribution to the sum of M non-identical independent Nakagami-m variates is derived. Comparisons between exact and approximate sum distributions show excellent agreement. This new approach finds applicability in important communications issues such as equal-gain combining, signal detection, linear equalisers, outage probability, intersymbol interference and phase jitter.     

对数正态分布下的无线光通信系统误码率分析

对开关键控(OOK)强度调制直接检测(IM/DD)方式的无线光通信接收发射系统的光电信号进行分析,并使用最大后验概率(MAP)方法确定判决阈值。将对数正态分布作为湍流信道上光强闪烁模型,建立了系统误码率与湍流强度、光源相干参数、激光发射器光功率等系统参数之间的定量关系。采用部分相干光作为信号光能降低光源相干度,有效抑制湍流效应。计算结果表明：在一定条件下,湍流强度改变0.5个量级系统误码率相差6-8个量级,并且最优源相干参数可以通过计算得到,其对应的部分相干光最有利于提高通信系统性能。     

Reliability sampling plans for lognormal distribution, based onprogressively-censored samples

This paper presents reliability sampling plans for the lognormal distribution based on progressively censored samples. In constructing these sampling plans, large-sample approximations to the best linear unbiased estimators of the location and scale parameters are used. For some selected progressive censoring schemes, reliability sampling plans are tabulated for p_α and p_β to match MIL-STD-105. While in general, variable-sampling plans require smaller sample size when compared with attribute-sampling plans, the ordinary complete and right-censored life test experiments are special cases of the progressively censored experiment. Hence, the progressively censored reliability sampling plans in this paper are widely applicable. General application of the procedure is discussed, and two examples are provided     

Information inequalities for the sum of independent random vectors (Corresp.)

Lower bounds are given for the rate-distortion and distortion-rate functions of the sums of linear transformations of independent random vectorsXandY. These lower bounds are sharp in the sense that they are exactly achieved when the random vectorsXandYare Gaussian and identically distributed.     

A comprehensive review of the lognormal failure distribution with application to LED reliability

Life-test data for many semiconductor devices obey the lognormal law of failure. In this paper we offer a survey of the key statistical properties of the lognormal distribution which are relevant in device engineering. The statistical developments are illustrated throughout by specific examples drawn from life-test data for GaP red LEDs which show lognormal failure behavior. Additional theoretical justification for accepting the lognormal distribution in the case of these devices is provided by the statistical recasting of the diffusion theory of red LED degradation.The treatment of the fundamental concepts of reliability in terms of cumulative failure function, instantaneous failure rate, mean time for failure (MTTF), etc., although concise, is self-contained. Moreover, the estimation of the two characteristic parameters (median life, standard deviation) of the lognormal distribution from sample data by means of a number of alternative procedures (graphical, least-square, maximum likelihood) is discussed in detail. The efficient determination of the MTTF and failure rate is facilitated by convenient charts developed here which relate these quantities to the parameters median life and standard deviation at 5, 10, 20 and 40 years of device service life. In order to assess the uncertainty associated with the estimates, the confidence limits on the median life and standard deviation of the population as a function of confidence level and sample size are given in novel graphical forms. Finally, error bounds for the MTTF and failure rate of the population are also presented.     

Bounds on a probability for the heavy tailed distribution and the probability of deficient decoding in sequential decoding

Although sequential decoding of convolutional codes gives a very small decoding error probability, the overall reliability is limited by the probability P_G of deficient decoding, the term introduced by Jelinek to refer to decoding failures caused mainly by buffer overflow. The number of computational efforts in sequential decoding has the Pareto distribution and it is this "heavy tailed" distribution that characterizes P_G. The heavy tailed distribution appears in many fields and buffer overflow is a typical example of the behaviors in which the heavy tailed distribution plays an important role. In this paper, we give a new bound on a probability in the tail of the heavy tailed distribution and, using the bound, prove the long-standing conjecture on P_G, that is, P_G ap constanttimes1/(sigma^rhoN^rho-1) for a large speed factor sigma of the decoder and for a large receive buffer size N whenever the coding rate R and rho satisfy E(rho)=rhoR for 0 les rho les 1     

Bounds on the Spectrum of a CHIRP Pulse

A simple method is described for calculating and plotting strong bounds on the frequency spectra of linear frequency modulated (LFM) pulses used in CHIRP radar transmitters. Bounds on the voltage density (amplitude), energy density, power density, or power spectrum can be hand calculated and plotted in 10 to 20 minutes with a computational accuracy that is adequate for most electromagnetic compatibility (EMC) analyses. The bounds obtained are within one dB of the peaks of the central lobe and major sidelobes. The method is based on the fact that the bounds on the far sidelobes are essentially the same whether or not the pulse is LFM. Several numerical examples are given for Hamming-weighted and trapezoidal pulses. The method is a convenient mathematical tool for engineers involved in compatibility analysis or concerned with radar transmitters complying with military standards pertaining to EMC.