Counterexamples in Probability and Real Analysis by Gary L. Wise, Eric B. Hall

By Gary L. Wise, Eric B. Hall

A counterexample is any instance or consequence that's the contrary of one's instinct or to generally held ideals. Counterexamples may have nice academic worth in illuminating complicated subject matters which are tricky to provide an explanation for in a rigidly logical, written presentation. for instance, rules in mathematical sciences that may appear intuitively visible could be proved unsuitable with using a counterexample. This monograph concentrates on counterexamples to be used on the intersection of likelihood and actual research, which makes it distinct between such remedies. The authors argue convincingly that likelihood idea can't be separated from genuine research, and this publication includes over three hundred examples with regards to either the speculation and alertness of arithmetic. the various examples during this assortment are new, and plenty of outdated ones, formerly buried within the literature, at the moment are available for the 1st time. not like a number of different collections, the entire examples during this ebook are thoroughly self-contained--no info are befell to vague outdoors references. scholars and theorists throughout fields as diversified as genuine research, chance, statistics, and engineering will desire a reproduction of this publication.

Show description

Read Online or Download Counterexamples in Probability and Real Analysis PDF

Best probability books

Credit Risk: Modeling, Valuation and Hedging

The most goal of credits chance: Modeling, Valuation and Hedging is to give a finished survey of the previous advancements within the quarter of credits chance examine, in addition to to place forth the newest developments during this box. a tremendous point of this article is that it makes an attempt to bridge the space among the mathematical concept of credits danger and the monetary perform, which serves because the motivation for the mathematical modeling studied within the publication.

Meta Analysis: A Guide to Calibrating and Combining Statistical Evidence

Meta research: A advisor to Calibrating and mixing Statistical proof acts as a resource of easy tools for scientists desirous to mix facts from various experiments. The authors goal to advertise a deeper realizing of the proposal of statistical facts. The publication is constituted of components - The guide, and the idea.

Measures, integrals and martingales

This can be a concise and easy advent to modern degree and integration concept because it is required in lots of elements of research and likelihood conception. Undergraduate calculus and an introductory direction on rigorous research in R are the single crucial must haves, making the textual content appropriate for either lecture classes and for self-study.

Stochastic Digital Control System Techniques

''This publication can be an invaluable connection with regulate engineers and researchers. The papers contained conceal good the new advances within the box of recent keep an eye on thought. ''- IEEE team Correspondence''This publication can assist all these researchers who valiantly attempt to continue abreast of what's new within the conception and perform of optimum regulate.

Extra resources for Counterexamples in Probability and Real Analysis

Example text

Xk ) (k < n) and find the necessary condition which turns the inequality into equality. 3. Show that Shannon’s entropy of a continuous random variable in R with finite mean µ and variance σ2 is bounded by Shannon’s entropy of a normal distribution with mean µ and variance σ2 . 4. Let X be a random variable with probability density function f(x). Show Z 1 x2 f (x) dx ≥ exp (2H (X)) . 2πe R 5. f. f. gθ (y), if there exists a nonnegative function h on the product space X × Y for which the following relations are satisfied: Z h(x, y)fθ (x)dµ(x) i) gθ (y) = X Z Z ii) h(x, y) ≥ 0, h(x, y)dµ(x) = h(x, y)dµ(y) = 1.

Determine the Rφ -Divergence, with φ (x) = x−x2 , between two multivariate normal distributions. Find the expression, as a particular case, for two univariate normal distributions. 13. Determine the Bhattacharyya divergence, Z B (θ1 , θ2 ) = − log (fθ 1 (x) fθ 2 (x))1/2 dµ(x), X between two univariate normal distributions. 14. 1/2 ¶2 q (θ1 , θ2 ) = fθ 1 (x) − fθ 2 (x) dµ(x) X is a metric. Find its expression for two multivariate normal distributions. 15. Evaluate the R´enyi’s divergence as well as the Kullback-Leibler divergence for two Poisson populations.

Xd ) = 10. , Yd ) be a d-variate random vector with multivariate normal distribution, with mean vector µ and nonsingular variance-covariance matrix Σ. , xd ) (x − µ)T Σ−1 (x − µ) dx. + 2 Rd Furthermore, since Σ−1 is a symmetric nonnegative definite matrix, there exists an orthogonal matrix L such LT Σ−1 L = Λ for some diagonal matrix Λ. , d. , ud ) uT Σ−1 u du. , vd ) λi vi dv = log (2π) det (Σ) Rd i=1 µ ¶ ³ ´ d P = log (2π)d/2 det (Σ)1/2 + 12 λi V ar (Vi ) . i=1 Since Cov(U ) = LCov(V )LT we have L−1 Σ(LT )−1 = Cov(V ).

Download PDF sample

Rated 4.54 of 5 – based on 14 votes