scholarly journals Material intrinsic heterogeneity: statistical derivation

TOS forum ◽  
2020 ◽  
Vol 2020 (10) ◽  
pp. 31
Author(s):  
Geoff Lyman
Keyword(s):  
Poisson Distribution ◽  
Random Quantity ◽  
Random Variable ◽  
Sampling Variance ◽  
Statistical Sampling ◽  
Analyte Content ◽  
Type Size ◽  
Mean Variance ◽  
Entire Distribution ◽  
Number Of Particles

The value of a fully statistical sampling theory is that it is possible to quantify a measure of material intrinsic heterogeneity and, on this basis, provide the entire distribution of the analyte content of potential samples to be extracted from the lot. The analyte content of a sample of a given mass is a random quantity as samples of nominally equal masses taken from a lot in a given state of comminution will not have exactly the sample analyte content. The analyte content of a sample is correctly described as a random variable and to characterise a random variable completely it is necessary to know either the probability density function or distribution function for the random variable, or all of the moments of the random variable (mean, variance and all the higher moments). The following discussion derives the fundamental sampling variance from a purely mathematical statistics basis, relying on the assumption that the number of particles of any one type (size and analyte content) that fall into a sample taken in a mechanically correct manner (following the principle of equiprobable sampling) follows a Poisson distribution. In addition, the Poisson distributions of particle numbers are statistically independent. A more fully argued substantiation of this fundamental assumption, partial experimental evidence and standard statistical introduction to the definition and properties of the Poisson distribution, and reasons for its use, can be found at the end of this article. © Materials Sampling & Consulting 2020

A MEAN–VARIANCE BOUND FOR A THREE-PIECE LINEAR FUNCTION

2007 ◽  
Vol 21 (4) ◽  
pp. 611-621 ◽  
Author(s):  
Karthik Natarajan ◽  
Zhou Linyi
Keyword(s):  
Convex Function ◽  
Closed Form ◽  
Linear Function ◽  
Upper Bound ◽  
Random Variable ◽  
Expected Value ◽  
Variance Bound ◽  
The Mean ◽  
Mean Variance

In this article, we derive a tight closed-form upper bound on the expected value of a three-piece linear convex function E[max(0, X, mX − z)] given the mean μ and the variance σ2 of the random variable X. The bound is an extension of the well-known mean–variance bound for E[max(0, X)]. An application of the bound to price the strangle option in finance is provided.

scholarly journals Mean-Variance Hedging and Forward-Backward Stochastic Differential Filtering Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
Author(s):  
Guangchen Wang ◽  
Zhen Wu
Keyword(s):  
Control Systems ◽  
Partial Information ◽  
Random Variable ◽  
Contingent Claim ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Quadratic Optimal Control ◽  
Hedging Problem ◽  
Mean Variance ◽  
Markov Control

This paper is concerned with a mean-variance hedging problem with partial information, where the initial endowment of an agent may be a decision and the contingent claim is a random variable. This problem is explicitly solved by studying a linear-quadratic optimal control problem with non-Markov control systems and partial information. Then, we use the result as well as filtering to solve some examples in stochastic control and finance. Also, we establishbackwardandforward-backwardstochastic differential filtering equations which aredifferentfrom the classical filtering theory introduced by Liptser and Shiryayev (1977), Xiong (2008), and so forth.

scholarly journals Numerical Modelling of Squeeze Film Performance between Rotating Transversely Rough Curved Circular Plates under the Presence of a Magnetic Fluid Lubricant

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Nikhilkumar D. Abhangi ◽  
G. M. Deheri
Keyword(s):  
Magnetic Fluid ◽  
Random Variable ◽  
Squeeze Film ◽  
Circular Plates ◽  
Random Roughness ◽  
Load Carrying ◽  
Nonzero Mean ◽  
Mean Variance ◽  
Positive Effect ◽  
Bearing System

An endeavour has been made to study and analyze the behaviour of a magnetic fluid-based squeeze film between curved transversely rough rotating circular plates when the curved upper plate lying along a surface determined by an exponential function approaches the curved lower plate along the surface governed by a secant function. A magnetic fluid is used as the lubricant in the presence of an external magnetic field oblique to the radial axis. The random roughness of the bearing surfaces is characterised by a stochastic random variable with nonzero mean, variance, and skewness. The associated nondimensional averaged Reynolds equation is solved with suitable boundary conditions in dimensionless form to obtain the pressure distribution, leading to the expression for the load carrying capacity. The results establish that the bearing system registers an enhanced performance as compared to that of the bearing system dealing with a conventional lubricant. This investigation proves that albeit the bearing suffers due to transverse surface roughness, there exist sufficient scopes for obtaining a relatively better performance in the case of negatively skewed roughness by properly choosing curvature parameters and the rotation ratio. It is appealing to note that the negative variance further enhances this positive effect.

85.76 Another Discrete Random Variable Having the Property 'Mean = Variance'

2001 ◽  
Vol 85 (504) ◽  
pp. 516 ◽  
Author(s):  
M. N. Deshpande
Keyword(s):  
Random Variable ◽  
Discrete Random Variable ◽  
Mean Variance

scholarly journals Gaussian expansions and bounds for the Poisson distribution applied to the Erlang B formula

2008 ◽  
Vol 40 (01) ◽  
pp. 122-143 ◽  
Author(s):  
A. J. E. M. Janssen ◽  
J. S. H. van Leeuwaarden ◽  
B. Zwart
Keyword(s):  
Distribution Function ◽  
Asymptotic Expansion ◽  
Poisson Distribution ◽  
Integral Transformation ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Normal Distribution Function ◽  
Gaussian Form ◽  
The Central Limit Theorem ◽  
Leading Term

This paper presents new Gaussian approximations for the cumulative distribution function P(A λ ≤ s) of a Poisson random variable A λ with mean λ. Using an integral transformation, we first bring the Poisson distribution into quasi-Gaussian form, which permits evaluation in terms of the normal distribution function Φ. The quasi-Gaussian form contains an implicitly defined function y, which is closely related to the Lambert W-function. A detailed analysis of y leads to a powerful asymptotic expansion and sharp bounds on P(A λ ≤ s). The results for P(A λ ≤ s) differ from most classical results related to the central limit theorem in that the leading term Φ(β), with is replaced by Φ(α), where α is a simple function of s that converges to β as s tends to ∞. Changing β into α turns out to increase precision for small and moderately large values of s. The results for P(A λ ≤ s) lead to similar results related to the Erlang B formula. The asymptotic expansion for Erlang's B is shown to give rise to accurate approximations; the obtained bounds seem to be the sharpest in the literature thus far.

Some characterizations for the Poisson distribution starting with a power-series distribution

1972 ◽  
Vol 71 (3) ◽  
pp. 517-522 ◽  
Author(s):  
D. N. Shanbhag ◽  
R. M. Clark
Keyword(s):  
Power Series ◽  
Poisson Distribution ◽  
Random Variable ◽  
Discrete Random Variable ◽  
Power Series Distribution ◽  
Destructive Process ◽  
Image Position

Let X be a non-negative discrete random variable with distribution {Px} and Y be a random variable denoting the undestroyed part of the random variable X when it is subjected to a destructive process such that

scholarly journals FORECASTING OF SAFETY TRANSPORT BY EXTREME STATISTICS / PROGNOZOWANIE BEZPIECZEŃSTWA OBIEKTÓW TRANSPORTOWYCH METODAMI STATYSTYKI EKSTREMALNEJ

2013 ◽  
Vol 26 (1) ◽  
pp. 5-12
Author(s):  
Arefyev Igor ◽  
Volovik Аleksandr ◽  
Klavdiev Аleksandr
Keyword(s):  
Random Variable ◽  
Transport Problem ◽  
Sample Survey ◽  
Statistical Sampling ◽  
Extreme Statistics ◽  
Functional Elements

Abstract Recently, the transport problem is acute to minimize accidents and disasters, caused by the failure of the functional elements. Today it is still not a fully developed theory of the solution of such problems. The authors propose an approach to perform this task, based on the methodology of extreme statistics and information Janes principle. Example given in the article and the calculations prove this possibility, up to an extreme level of statistical sampling, when it reaches capacity. The method allows for a decision on a sample survey, when the nature of the distribution is not known anything other than the expectation of a random variable.

scholarly journals Theorizing Probability Distribution in Applied Statistics

2020 ◽  
Vol 1 (1) ◽  
pp. 79-95
Author(s):  
Indra Malakar
Keyword(s):  
Probability Distribution ◽  
Normal Distribution ◽  
Poisson Distribution ◽  
Binomial Distribution ◽  
Identical Condition ◽  
Random Variable ◽  
Fixed Number ◽  
Theoretical Knowledge ◽  
Applied Statistics ◽  
Discrete Random Variable

This paper investigates into theoretical knowledge on probability distribution and the application of binomial, poisson and normal distribution. Binomial distribution is widely used discrete random variable when the trails are repeated under identical condition for fixed number of times and when there are only two possible outcomes whereas poisson distribution is for discrete random variable for which the probability of occurrence of an event is small and the total number of possible cases is very large and normal distribution is limiting form of binomial distribution and used when the number of cases is infinitely large and probabilities of success and failure is almost equal.

scholarly journals Multi-type branching processes with time-dependent branching rates

2018 ◽  
Vol 55 (3) ◽  
pp. 701-727 ◽  
Author(s):  
D. Dolgopyat ◽  
P. Hebbar ◽  
L. Koralov ◽  
M. Perlman
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Random Variable ◽  
Time Dependent ◽  
Exponential Random Variable ◽  
Number Of Particles

Abstract Under mild nondegeneracy assumptions on branching rates in each generation, we provide a criterion for almost sure extinction of a multi-type branching process with time-dependent branching rates. We also provide a criterion for the total number of particles (conditioned on survival and divided by the expectation of the resulting random variable) to approach an exponential random variable as time goes to ∞.

Unequal probability sampling

2019 ◽  
pp. 140-172
Author(s):  
David G. Hankin ◽  
Michael S. Mohr ◽  
Ken B. Newman
Keyword(s):  
Variance Estimation ◽  
Random Variable ◽  
Auxiliary Variable ◽  
Second Order ◽  
Systematic Sampling ◽  
Sampling Variance ◽  
Probability Sampling ◽  
Unequal Probability ◽  
Inclusion Probabilities ◽  
Selection Probabilities

Equal probability selection is a special case of the general theory of probability sampling in which population units may be selected with unequal probabilities. Unequal selection probabilities are often based on auxiliary variable values which are measures of the sizes of population units, thus leading to the acronym (PPS)—“Probability Proportional to Size”. The Horvitz–Thompson (1953) theorem provides a unifying framework for design-based sampling theory. A sampling design specifies the sample space (set of all possible samples) and associated first and second order inclusion probabilities (probabilities that unit i, or units i and j, respectively, are included in a sample of size n selected from N according to some selection method). A valid probability sampling scheme must have all first order inclusion probabilities > 00 (i.e., every population unit must have a chance of being in the sample). Unbiased variance estimation is possible only for those schemes that guarantee that all second order inclusion probabilities exceed zero, thus providing theoretical justification for the absence of unbiased estimators of sampling variance in systematic sampling and other schemes for which some second order inclusion probabilities are zero. Numerous generalized Horvitz–Thompson (HT) estimators can be formed and all are consistent estimators because they are functions of consistent HT estimators. Unequal probability systematic sampling and Poisson sampling (the unequal probability counterpart to Bernoulli sampling for which sample size is a random variable) are also considered. Several R programs for selecting unequal probability samples and for calculating first and second order inclusion probabilities are posted at http://global.oup.com/uk/companion/hankin.

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