Search results for "probability density function"
showing 10 items of 183 documents
The Influence of Spatial Correlation and Severity of Fading on the Statistical Properties of the Capacity of OSTBC Nakagami-m MIMO Channels
2009
This paper deals with the analysis of statistical prop- erties of the capacity of spatially uncorrelated orthogonal space- time block coded (OSTBC) Nakagami-m multiple-input multiple- output (MIMO) channels. We have derived exact closed-form expressions for the probability density function (PDF), cumula- tive distribution function (CDF), level-crossing rate (LCR), and average duration of fades (ADF) of the channel capacity. We have also investigated the statistical properties of the approximated capacity of spatially correlated OSTBC Nakagami-m MIMO channels. The results are studied for different values of the fading parameter m, corresponding to different fading conditions. It is observed …
On the statistical properties of the capacity of OSTBC Nakagami-lognormal MIMO channels
2010
This article presents a thorough statistical analysis of the capacity of orthogonal space-time block coded (OSTBC) Nakagami-lognormal (NLN) multiple-input multipleoutput (MIMO) channels. The NLN channel model allows to study the joint effects of fast fading and shadowing on the statistical properties of the channel capacity. We have derived exact analytical expressions for the probability density function (PDF), cumulative distribution function (CDF), level-crossing rate (LCR), and average duration of fades (ADF) of the capacity of NLN MIMO channels. It is observed that an increase in the MIMO dimension1 or a decrease in the severity of fading results in an increase in the mean channel capa…
Is there an absolutely continuous random variable with equal probability density and cumulative distribution functions in its support? Is it unique? …
2014
This paper inquires about the existence and uniqueness of a univariate continuous random variable for which both cumulative distribution and density functions are equal and asks about the conditions under which a possible extrapolation of the solution to the discrete case is possible. The issue is presented and solved as a problem and allows to obtain a new family of probability distributions. The different approaches followed to reach the solution could also serve to warn about some properties of density and cumulative functions that usually go unnoticed, helping to deepen the understanding of some of the weapons of the mathematical statistician’s arsenal.
On the use of fractional calculus for the probabilistic characterization of random variables
2009
In this paper, the classical problem of the probabilistic characterization of a random variable is re-examined. A random variable is usually described by the probability density function (PDF) or by its Fourier transform, namely the characteristic function (CF). The CF can be further expressed by a Taylor series involving the moments of the random variable. However, in some circumstances, the moments do not exist and the Taylor expansion of the CF is useless. This happens for example in the case of $\alpha$--stable random variables. Here, the problem of representing the CF or the PDF of random variables (r.vs) is examined by introducing fractional calculus. Two very remarkable results are o…
Path integral solution for non-linear system enforced by Poisson White Noise
2008
Abstract In this paper the response in terms of probability density function of non-linear systems under Poisson White Noise is considered. The problem is handled via path integral (PI) solution that may be considered as a step-by-step solution technique in terms of probability density function. First the extension of the PI to the case of Poisson White Noise is derived, then it is shown that at the limit when the time step becomes an infinitesimal quantity the Kolmogorov–Feller (K–F) equation is fully restored enforcing the validity of the approximations made in obtaining the conditional probability appearing in the Chapman Kolmogorov equation (starting point of the PI). Spectral counterpa…
A method for the probabilistic analysis of nonlinear systems
1995
Abstract The probabilistic description of the response of a nonlinear system driven by stochastic processes is usually treated by means of evaluation of statistical moments and cumulants of the response. A different kind of approach, by means of new quantities here called Taylor moments, is proposed. The latter are the coefficients of the Taylor expansion of the probability density function and the moments of the characteristic function too. Dual quantities with respect to the statistical cumulants, here called Taylor cumulants, are also introduced. Along with the basic scheme of the method some illustrative examples are analysed in detail. The examples show that the proposed method is an a…
Probabilistic response of nonlinear systems under combined normal and Poisson white noise via path integral method
2011
In this paper the response in terms of probability density function of nonlinear systems under combined normal and Poisson white noise is considered. The problem is handled via a Path Integral Solution (PIS) that may be considered as a step-by-step solution technique in terms of probability density function. A nonlinear system under normal white noise, Poissonian white noise and under the superposition of normal and Poisson white noise is performed through PIS. The spectral counterpart of the PIS, ruling the evolution of the characteristic functions is also derived. It is shown that at the limit when the time step becomes an infinitesimal quantity an equation ruling the evolution of the pro…
Levy flights in confining environments: Random paths and their statistics
2013
We analyze a specific class of random systems that are driven by a symmetric L\'{e}vy stable noise. In view of the L\'{e}vy noise sensitivity to the confining "potential landscape" where jumps take place (in other words, to environmental inhomogeneities), the pertinent random motion asymptotically sets down at the Boltzmann-type equilibrium, represented by a probability density function (pdf) $\rho_*(x) \sim \exp [-\Phi (x)]$. Since there is no Langevin representation of the dynamics in question, our main goal here is to establish the appropriate path-wise description of the underlying jump-type process and next infer the $\rho (x,t)$ dynamics directly from the random paths statistics. A pr…
Approximations in Statistics from a Decision-Theoretical Viewpoint
1987
The approximation of the probability density p(.) of a random vector x∊X by another (possibly more convenient) probability density q(.) which belongs to a certain class Q is analyzed as a decision problem where the action space is the class Qof available approximations, the relevant uncertain event is the actual value of the vector x and the utility function is a proper scoring rule. The logarithmic divergence is shown to play a rather special role within this approach. The argument lies entirely within a Bayesian framework.
Dynamic graphics in Excel for teaching statistics: understanding the probability density function
2011
In this article, we show a dynamic graphic in Excel that is used to introduce an important concept in our subject, Statistics I: the probability density function. This interactive graphic seeks to facilitate conceptual understanding of the main aspects analysed by the learners.