Physical Sciences and Mathematics Commons^™

Two Pens In A Pocket Must Be Different: A Nerd-Oriented Lesson From Statistics, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Some people always carry a pen with them, so that if an idea comes to mind, they will always be able to write it down. Pens sometimes run out of ink. So, just in case, people carry two pens. The problem is that often, when one carries two identical pens, they seem to run out of ink at about the same time -- which defeats the whole purpose of carrying two pens. In this paper, we provide a simple statistics-based explanation of this phenomenon, and show that a seemingly natural idea of carrying three pens will not help. The only …

Propagation Of Probabilistic Uncertainty: The Simplest Case (A Brief Pedagogical Introduction), Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

The main objective of this text is to provide a brief introduction to formulas describing the simplest case of propagation of probabilistic uncertainty -- for students who have not yet taken a probability course.

Is It Legitimate Statistics Or Is It Sexism: Why Discrimination Is Not Rational, Martha Osegueda Escobar, Vladik Kreinovich, Thach N. Nguyen

Departmental Technical Reports (CS)

While in the ideal world, everyone should have the same chance to succeed in a given profession, in reality, often the probability of success is different for people of different gender and/or ethnicity. For example, in the US, the probability of a female undergraduate student in computer science to get a PhD is lower than a similar probability for a male student. At first glance, it may seem that in such a situation, if we try to maximize our gain and we have a limited amount of resources, it is reasonable to concentrate on students with the higher probability of …

How To Estimate Statistical Characteristics Based On A Sample: Nonparametric Maximum Likelihood Approach Leads To Sample Mean, Sample Variance, Etc., Vladik Kreinovich, Thongchai Dumrongpokaphan

Departmental Technical Reports (CS)

In many practical situations, we need to estimate different statistical characteristics based on a sample. In some cases, we know that the corresponding probability distribution belongs to a known finite-parametric family of distributions. In such cases, a reasonable idea is to use the Maximum Likelihood method to estimate the corresponding parameters, and then to compute the value of the desired statistical characteristic for the distribution with these parameters.

In some practical situations, we do not know any family containing the unknown distribution. We show that in such nonparametric cases, the Maximum Likelihood approach leads to the use of sample mean, …

Why Copulas?, Vladik Kreinovich, Hung T. Nguyen, Songsak Sriboonchitta, Olga Kosheleva

Departmental Technical Reports (CS)

A natural way to represent a 1-D probability distribution is to store its cumulative distribution function (cdf) F(x) = Prob(X ≤ x). When several random variables X₁, ..., X_n are independent, the corresponding cdfs F₁(x₁), ..., F_n(x_n) provide a complete description of their joint distribution. In practice, there is usually some dependence between the variables, so, in addition to the marginals F_i(x_i), we also need to provide an additional information about the joint distribution of the given variables. It is possible to represent this joint …

From Unbiased Numerical Estimates To Unbiased Interval Estimates, Baokun Li, Gang Xiang, Vladik Kreinovich, Panagios Moscopoulos

Departmental Technical Reports (CS)

One of the main objectives of statistics is to estimate the parameters of a probability distribution based on a sample taken from this distribution. Of course, since the sample is finite, the estimate X is, in general, different from the actual value x of the corresponding parameter. What we can require is that the corresponding estimate is unbiased, i.e., that the mean value of the difference X - x is equal to 0: E[X] = x. In some problems, unbiased estimates are not possible. We show that in some such problems, it is possible to have interval unbiased estimates, i.e., …