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 the various examples during this assortment are new, and plenty of outdated ones, formerly buried within the literature, at the moment are available for the 1st time.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.

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**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 ).