scholarly journals On The Quotient of a Centralized and a Non-centralized Complex Gaussian Random Variable

2020 ◽  
Vol 125 ◽  
Author(s):  
Dazhen Gu
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Complex Signal ◽  
Tolerance Intervals ◽  
Practical Applications ◽  
Second Moments ◽  
Complex Gaussian Random Variable ◽  
The Mean ◽  
Theoretical Developments ◽  
Straightforward Extension

A detailed investigation of the quotient of two independent complex random variables is presented. The numerator has a zero mean, and the denominator has a non-zero mean. A normalization step is taken prior to the theoretical developments in order to simplify the formulation. Next, an indirect approach is taken to derive the statistics of the modulus and phase angle of the quotient. That in turn enables a straightforward extension of the statistical results to real and imaginary parts. After the normalization procedure, the probability density function of the quotient is found as a function of only the mean of the random variable that corresponds to the denominator term. Asymptotic analysis shows that the quotient closely resembles a normally-distributed complex random variable as the mean becomes large. In addition, the first and second moments, as well as the approximate of the second moment of the clipped random variable, are derived, which are closely related to practical applications in complex-signal processing such as microwave metrology of scattering-parameters. Tolerance intervals associated with the ratio of complex random variables are presented.

Uniform saddlepoint approximations

1988 ◽  
Vol 20 (3) ◽  
pp. 622-634 ◽  
Author(s):  
J. L. Jensen
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Tail Probability ◽  
Poisson Random Variable ◽  
Compound Poisson ◽  
Saddlepoint Approximations ◽  
The Mean ◽  
General Conditions

The validity of the saddlepoint expansion evaluated at the point y is considered in the limit y tending to ∞. This is done for the expansions of the density and of the tail probability of the mean of n i.i.d. random variables and also for the expansion of the tail probability of a compound Poisson sum , where N is a Poisson random variable. We consider both general conditions that ensure the validity of the expansions and study the four classes of densities for X1 introduced in Daniels (1954).

The Largest Missing Value in a Sample of Geometric Random Variables

2014 ◽  
Vol 23 (5) ◽  
pp. 670-685 ◽  
Author(s):  
MARGARET ARCHIBALD ◽  
ARNOLD KNOPFMACHER
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Missing Value ◽  
Asymptotic Expressions ◽  
Mean And Variance ◽  
The Mean ◽  
Special Case

We consider samples of n geometric random variables with parameter 0 < p < 1, and study the largest missing value, that is, the highest value of such a random variable, less than the maximum, that does not appear in the sample. Asymptotic expressions for the mean and variance for this quantity are presented. We also consider samples with the property that the largest missing value and the largest value which does appear differ by exactly one, and call this the LMV property. We find the probability that a sample of n variables has the LMV property, as well as the mean for the average largest value in samples with this property. The simpler special case of p = 1/2 has previously been studied, and verifying that the results of the present paper coincide with those previously found for p = 1/2 leads to some interesting identities.

Uniform saddlepoint approximations

1988 ◽  
Vol 20 (03) ◽  
pp. 622-634 ◽  
Author(s):  
J. L. Jensen
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Tail Probability ◽  
Poisson Random Variable ◽  
Compound Poisson ◽  
Saddlepoint Approximations ◽  
The Mean ◽  
General Conditions

The validity of the saddlepoint expansion evaluated at the point y is considered in the limit y tending to ∞. This is done for the expansions of the density and of the tail probability of the mean of n i.i.d. random variables and also for the expansion of the tail probability of a compound Poisson sum , where N is a Poisson random variable. We consider both general conditions that ensure the validity of the expansions and study the four classes of densities for X 1 introduced in Daniels (1954).

Simultaneous prediction of functionally dependent random variables by maximum likelihood estimation

2021 ◽  
Vol 16 (2) ◽  
pp. 143-150
Author(s):  
Nikita A. Moiseev
Keyword(s):  
Forecast Error ◽  
Random Variables ◽  
Likelihood Estimation ◽  
Random Variable ◽  
Parametric Approach ◽  
Functional Link ◽  
Dependent Random Variables ◽  
Steady Decrease ◽  
Simultaneous Prediction ◽  
The Mean

The paper presents a fundamental parametric approach to simultaneous forecasting of a vector of functionally dependent random variables. The motivation behind the proposed method is the following: each random variable at interest is forecasted by its own model and then adjusted in accordance with the functional link. The method incorporates the assumption that models’ errors are independent or weekly dependent. Proposed adjustment is explicit and extremely easy-to-use. Not only does it allow adjusting point forecasts, but also it is possible to adjust the expected variance of errors, that is useful for computation of confidence intervals. Conducted thorough simulation and empirical testing confirms, that proposed method allows to achieve a steady decrease in the mean-squared forecast error for each of predicted variables.

scholarly journals Possibility of using CMS Maple to study laws of distribution of random variables

Radiotekhnika ◽  
2021 ◽  
pp. 128-134
Author(s):  
I. Moshchenko ◽  
O. Nikitenko ◽  
Yu.V. Kozlov
Keyword(s):  
Data Processing ◽  
Random Variables ◽  
Random Variable ◽  
Large Set ◽  
Practical Applications ◽  
Symbolic Calculations ◽  
Variable Distribution ◽  
Topic Distribution ◽  
Distribution Laws ◽  
Independent Work

The use of CMS Maple for students' practical and independent work is described. The study of random variable distribution laws is actual. Statistical calculations without computer are difficult and require many functional and quintiles tables of standard distributions. This does not contribute to feeling the element of novelty in the material being studied, to be able to arbitrarily change the conditions of tasks, etc., it takes a lot of time in solving applied production problems, which is inappropriate Thus to determine and research random variable distribution laws both in practical applications and in studying we must use special mathematical packages. The most extended of them are Mathcad, MatLab, Mathematica, Maple. Specialized statistical packages (SAS, SPSS, STATISTIKA, STATGRAPHICS) are not relevant to study. Their use for studying requires very high education level in mathematical statistics. Most of the existing math packages allow users to operate at random variables, including the Computer Mathematics System (CMS) Maple. Thus, the purpose of this article is a description of the studying possibilities of the random variables distribution laws with CMS Maple and the application of the acquired skills to the independent work of students. The Maple Statistics Library has a large set of commands for analyzing data, computing various numerical characteristics of random variables, graphing their distribution laws, and for statistical data processing. Thanks to a powerful set of statistical tools, the possibility of symbolic calculations and data processing of CMS Maple, wide possibilities of graphical interpretation of the results obtained not only in a static but also in a dynamic form, it is advisable to use it when studying the topic "Distribution Laws of Random Variables" in students' practical and independent work to use their acquired skills in solving applied problems of science and technology.

Notes on the variance of a widow's pension

1974 ◽  
Vol 101 (1) ◽  
pp. 103-107 ◽  
Author(s):  
P. P. Boyle
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Stochastic Approach ◽  
Higher Order ◽  
Expected Value ◽  
Confidence Limits ◽  
Higher Order Moments ◽  
Variable Time ◽  
Second Moments ◽  
Constant Rate

Pollard and Pollard have considered a stochastic approach to certain actuarial problems. In their notation ãx denotes the random variable whose expected value is ax. The random variable ãx depends on the random variable, time until death and the (assumed constant) rate of interest. The authors show how to calculate the second and higher-order moments of some common actuarial random variables such as ãx and Ãx. The second moments, in particular, are useful in estimating confidence limits for the total liability in respect of a group of contracts.

Design Sensitivity Method for Sampling-Based RBDO With Fixed COV

2015 ◽  
Author(s):  
Hyunkyoo Cho ◽  
K. K. Choi ◽  
Ikjin Lee ◽  
David Lamb
Keyword(s):  
Random Variables ◽  
Score Function ◽  
Random Variable ◽  
Probability Of Failure ◽  
Design Sensitivity ◽  
Mean Value ◽  
Reliability Based Design ◽  
Design Variables ◽  
Change In The Mean ◽  
The Mean

Conventional reliability-based design optimization (RBDO) uses the means of input random variables as its design variables; and the standard deviations (STDEVs) of the random variables are fixed constants. However, the fixed STDEVs may not correctly represent certain RBDO problems well, especially when a specified tolerance of the input random variable is presented as a percentage of the mean value. For this kind of design problem, the coefficients of variations (COVs) of the input random variables should be fixed, which means STDEVs are not fixed. In this paper, a method to calculate the design sensitivity of probability of failure for RBDO with fixed COV is developed. For sampling-based RBDO, which uses Monte Carlo simulation for reliability analysis, the design sensitivity of the probability of failure is derived using a first-order score function. The score function contains the effect of the change in the STDEV in addition to the change in the mean. As copulas are used for the design sensitivity, correlated input random variables also can be used for RBDO with fixed COV. Moreover, the design sensitivity can be calculated efficiently during the evaluation of the probability of failure. Using a mathematical example, the accuracy and efficiency of the developed method are verified. The RBDO result for mathematical and physical problems indicates that the developed method provides accurate design sensitivity in the optimization process.

scholarly journals Stochastic Comparisons of Some Distances between Random Variables

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 981
Author(s):  
Patricia Ortega-Jiménez ◽  
Miguel A. Sordo ◽  
Alfonso Suárez-Llorens
Keyword(s):  
Financial Risk ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Random Variable ◽  
Distribution Functions ◽  
Stochastic Comparisons ◽  
Stochastic Orderings ◽  
Absolute Value ◽  
The Difference

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given.

scholarly journals Fully degenerate Bell polynomials associated with degenerate Poisson random variables

2021 ◽  
Vol 19 (1) ◽  
pp. 284-296
Author(s):  
Hye Kyung Kim
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Combinatorial Identities ◽  
Bell Polynomials ◽  
Poisson Random Variable ◽  
Central Moments ◽  
Special Polynomials ◽  
Special Numbers And Polynomials ◽  
New Type ◽  
Two Parameters

Abstract Many mathematicians have studied degenerate versions of quite a few special polynomials and numbers since Carlitz’s work (Utilitas Math. 15 (1979), 51–88). Recently, Kim et al. studied the degenerate gamma random variables, discrete degenerate random variables and two-variable degenerate Bell polynomials associated with Poisson degenerate central moments, etc. This paper is divided into two parts. In the first part, we introduce a new type of degenerate Bell polynomials associated with degenerate Poisson random variables with parameter α > 0 \alpha \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the fully degenerate Bell polynomials. We derive some combinatorial identities for the fully degenerate Bell polynomials related to the n n th moment of the degenerate Poisson random variable, special numbers and polynomials. In the second part, we consider the fully degenerate Bell polynomials associated with degenerate Poisson random variables with two parameters α > 0 \alpha \gt 0 and β > 0 \beta \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the two-variable fully degenerate Bell polynomials. We show their connection with the degenerate Poisson central moments, special numbers and polynomials.

A Note on the Accuracy of Normal Approximation of Random Quantities

2021 ◽  
Vol 73 (1) ◽  
pp. 62-67
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Sample Size ◽  
Order Statistic ◽  
Normal Approximation ◽  
Order Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Limiting Distribution ◽  
Exact Order ◽  
Fixed Sample ◽  
Independent Identically Distributed

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

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