Formulas for the distribution of sums of independent exponential random variables

This paper derives a new type of formula for the probability that, among a collection of items with s-independent exponential times to failure, a certain subset of them fails in a given order before a certain time, and all the remaining items survive beyond that time. This formula is in the form of a power series that satisfies a certain constant coefficient linear differential equation with specified initial conditions. This provides an alternative to existing closed-form formulas of the "exponomial" variety, viz., a nonlinear combination of exponential terms, where the coefficients of the exponential terms are polynomials in the mission time. Some results are given which quantify the computation effort required to achieve a specified accuracy using partial sums of the infinite series; a simple example illustrates these results. This approach can be very efficient for system reliability analysis where the product of the mission time and the sum of the failure rates down any path leading to system failure is small. Further work is needed to expand the practical applicability of this approach to cases where some rates are large and/or the mission time is long.     

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     

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     

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.     

On the ratio of independent complex Gaussian random variables

In this paper, we derive a closed form equation for the joint probability distribution \({{f_{{R}_{z}}},{\varTheta _{z}}}({r_{z}},{\theta _{z}})\) of the amplitude \({R_{z}}\) and phase \({\varTheta _{z}}\) of the ratio \({Z=\frac{X}{Y}}\) of two independent non-zero mean Complex Gaussian random variables \(X\sim CN(\nu _{x} \mathrm {e}^{j\phi _{x}},{\sigma ^{2}_{x}})\) and \(Y\sim CN(\nu _{y} \mathrm {e}^{j\phi _{y}},{\sigma ^{2}_{y}})\). The derived joint probability distribution only contains a confluent hypergeometric function of the first kind \({_1F_{1}}\) without infinite summations resulting in computational efficiency. We further derive the probability distribution for the ratio of two non-zero mean independent real Rician random variables containing an infinite summation generated by the estimation of the Cauchy product of equivalent series of two modified Bessel functions.     

On the maximum entropy of the sum of two dependent random variables

Investigates the maximization of the differential entropy h(X+Y) of arbitrary dependent random variables X and Y under the constraints of fixed equal marginal densities for X and Y. We show that max[h(X+Y)]=h(2X), under the constraints that X and Y have the same fixed marginal density f, if and only if f is log-concave. The maximum is achieved when X=Y. If f is not log-concave, the maximum is strictly greater than h(2X). As an example, identically distributed Gaussian random variables have log-concave densities and satisfy max[h(X+Y)]=h(2X) with X=Y. More general inequalities in this direction should lead to capacity bounds for additive noise channels with feedback     

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.     

On the application of the sum of generalized Gaussian random variables: maximal ratio combining

In most wireless communication systems, the additive noise is assumed to be Gaussian. However, there are many practical applications where non-Gaussian noise impairs the received signal. Examples include co-channel and adjacent channel interference in mobile cellular systems, impulsive noise in wireless and power-line communications, ultra-wide-band interference and multi-user interference in wireless systems, and spectrum sensing. To cover this issue, we consider in this paper the application of the sum of generalized Gaussian (GG) random variables (RVs). To this end, we consider single-input multiple-output (SIMO) systems that operate over Nakagami-m fading channels in the presence of an additive white generalized Gaussian noise (AWGGN). Specifically, we derive a closed-form expression for the bit error rate (BER) of several coherent digital modulation schemes using maximal ratio combining diversity in the Nakagami-m fading channels subject to an AWGGN. The derived expression is obtained based on the fact that the sum of L GG RVs can be approximated by a single GG RV with a suitable shaping parameter. In addition, the obtained BER expression is valid for integer and non-integer value of the fading parameter m. Analytical results are supported by Monte-Carlo simulations to validate the analysis.     

Closed-form expressions for distribution of sum of exponentialrandom variables

In many systems which are composed of components with exponentially distributed lifetimes, the system failure time can be expressed as a sum of exponentially distributed random variables. A previous paper mentions that there seems to be no convenient closed-form expression for all cases of this problem. This is because in one case the expression involves high-order derivatives of products of multiple functions. The authors prove a simple intuitive multi-function generalization of the Leibnitz rule for high-order derivatives of products of two functions and use this to simplify this expression, thus giving a closed form solution for this problem. They similarly simplify the state-occupancy probabilities in general Markov models     

Failure-rate functions for doubly-truncated random variables

The failure rate function is generalized for doubly-truncated random variables. This generalized function is used to characterize the distribution function and to obtain the properties that any function must have to be a generalized failure rate function, both for continuous and discrete random variables. From the theoretical results are constructed nonparametric estimators of the Kaplan-Meier type and cumulative-hazard type     

General Nakagami approximation for sum of Ricean interferers

The statistical distribution of the sum of interfering signals at the receiving end of a wireless system is important to effect theoretical evaluation of its performances. The authors show that the sum of generic, narrow-band, Ricean interferers can be approximated by a single Nakagami interferer. A theoretical insight about this fact is given and some simulated results are shown to confirm the result     

A conditional entropy bound for a pair of discrete random variables

LetX, Ybe a pair of discrete random variables with a given joint probability distribution. For0 leq x leq H(X), the entropy ofX, define the functionF(x)as the infimum ofH(Ymid W), the conditional entropy ofYgivenW, with respect to all discrete random variablesWsuch that a)H(Xmid W) = x, and b)WandYare conditionally independent givenX. This paper concerns the functionF, its properties, its calculation, and its applications to several problems in information theory.     

Bounds on the distribution of a sum of independent lognormal randomvariables

The distribution function of a sum of lognormal random variables (RVs) appears in several communication problems. Approximations are usually used for such distribution as no closed form nor bounds exist. Bounds can be very useful in assessing the performance of any given system. In this letter, we derive upper and lower bounds on the distribution function of a sum of independent lognormal RVs. These bounds are given in a closed form and can be used in studying the performance of cellular radio and broadcasting systems     

Quantization schemes for bivariate Gaussian random variables

The problem of quantizing two-dimensional Gaussian random variables is considered. It is shown that, for all but a finite number of cases, a polar representation gives a smaller mean square quantization error than a Cartesian representation. Applications of the results to a transform coding scheme known as spectral phase coding are discussed.     

Commentary: failure-rate functions for doubly-truncated random variables

Navarro and Ruiz (see ibid., vol.45, p.685-90, 1996) express the nonparametric maximum likelihood estimator (NPMLE) of the distribution of a failure-time random variable as a function of the NPMLE of generalized failure-rate functions. These generalized failure-rate functions are equal to the probability density functions of a doubly-truncated failure-time random variable at the endpoints of the truncating interval. Readers can infer from this paper that this simple estimator can be applied to a doubly-truncated sample of failure times. This commentary explains why that estimator does not apply to the general setting in which the observed failure times are doubly-truncated with subject-specific truncating intervals. A doubly-truncated sample of times to brain tumor progression illustrates the deviation of that estimator from the NPMLE for these data.     

Minimum entropy of error estimation for discrete random variables

The principle of minimum entropy of error estimation (MEEE) is formulated for discrete random variables. In the case when the random variable to be estimated is binary, we show that the MEEE is given by a Neyman-Pearson-type strictly monotonous test. In addition, the asymptotic behavior of the error probabilities is proved to be equivalent to that of the Bayesian test     

Generation of random variables for modelling VBR multimedia sources

The design and performance evaluation of multimedia systems require the availability of adequate models to mimic the statistical properties of the traffic generated by a multimedia source. In this paper the authors propose a very simple method for the generation of discretetime and discretestate autocorrelated random variables which can be used to model a traffic source by simulation. The probability distribution and autocorrelation sequence of the variables generated exactly match the corresponding experimental histograms of the source to be modelled when the experimental autocorrelation sequence is decreasing with downward convexity. In this paper an analytical demonstration of the method proposed is given and its use is illustrated by three telecommunications examples.     

Bayesian prediction for the range based on a two-parameter exponential distribution with a random sample size

This paper is concerned with the problem of predicting the range of a future sample when the underlying distribution is a two-parameter exponential distribution. A Bayesian prediction distribution of the range will be derived under the assumption that the old sample is a censored type II and the sample size of the future sample is a random variable. A numerical illustration is provided.     

On the PDF of the sum of random vectors

There are various cases in physics and engineering sciences (especially communications) where one requires the envelope probability density function (PDF) of the sum of several random sinusoidal signals. According to the correspondence between a random sinusoidal signal and a random vector, the sum of random vectors can be considered as an abstract mathematical model for the above sum. Now it is desired to obtain the PDF of the length of the resulting vector. Considering the common and reasonable assumption of uniform distributions for the angles of the vectors, many researchers have obtained the PDF of the length of the resulting vector only for special cases. However in this paper, the PDF is obtained for the most general case in which the lengths of vectors are arbitrary dependent random variables. This PDF is in the form of a definite integral, which may be inappropriate for analytic manipulations and numerical computations. So an appropriate infinite Laguerre expansion is also derived. Finally, the results are applied to solve a typical example in computing the scattering cross section of random scatterers