Archive for complex numbers

complex Cauchys

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , on February 8, 2018 by xi'an

During a visit of Don Fraser and Nancy Reid to Paris-Dauphine where Nancy gave a nice introduction to confidence distributions, Don pointed out to me a 1992 paper by Peter McCullagh on the Cauchy distribution. Following my recent foray into the estimation of the Cauchy location parameter. Among several most interesting aspects of the Cauchy, Peter re-expressed the density of a Cauchy C(θ¹,θ²) as

f(x;θ¹,θ²) = |θ²| / |x-θ|²

when θ=θ¹+ιθ² [a complex number on the half-plane]. Denoting the Cauchy C(θ¹,θ²) as Cauchy C(θ), the property that the ratio aX+b/cX+d follows a Cauchy for all real numbers a,b,c,d,

C(aθ+b/cθ+d)

[when X is C(θ)] follows rather readily. But then comes the remark that

“those properties follow immediately from the definition of the Cauchy as the ratio of two correlated normals with zero mean.”

which seems to relate to the conjecture solved by Natesh Pillai and Xiao-Li Meng a few years ago. But the fact that  a ratio of two correlated centred Normals is Cauchy is actually known at least from the1930’s, as shown by Feller (1930, Biometrika) and Geary (1930, JRSS B).

Bayes at the Bac’ [again]

Posted in Kids, Statistics with tags , , , , , , , , on June 19, 2014 by xi'an

When my son took the mathematics exam of the baccalauréat a few years ago, the probability problem was a straightforward application of Bayes’ theorem.  (Problem which was later cancelled due to a minor leak…) Surprise, surprise, Bayes is back this year for my daughter’s exam. Once again, the topic is a pharmaceutical lab with a test, test with different positive rates on two populations (healthy vs. sick), and the very basic question is to derive the probability that a person is sick given the test is positive. Then a (predictable) application of the CLT-based confidence interval on a binomial proportion. And the derivation of a normal confidence interval, once again compounded by  a CLT-based confidence interval on a binomial proportion… Fairly straightforward with no combinatoric difficulty.

The other problems were on (a) a sequence defined by the integral

\int_0^1 (x+e^{-nx})\text{d}x

(b) solving the equation

z^4+4z^2+16=0

in the complex plane and (c) Cartesian 2-D and 3-D geometry, again avoiding abstruse geometric questions… A rather conventional exam from my biased perspective.

Bayes at the Bac’ [and out!]

Posted in Kids, Statistics with tags , , , , , , on June 24, 2011 by xi'an

In the mathematics exam of the baccalauréat my son (and 160,000 other students) took on Tuesday, the probability problem was a straightforward application of Bayes’ theorem. Given a viral test with 99% positives for infected patients and 97% negatives for non-infected patients, in a population with 2% of infected patients, what is the probability that the patient is infected given that the test is positive? (It looks like another avatar of Exercise 1.7  in The Bayesian Choice!) A lucky occurrence, given that I had explained to my son Bayes’ formula earlier this year (neither the math book nor the math teacher mentioned Bayes, incidentally!) and even more given that, in a crash revision Jean-Michel Marin gave him the evening before, they went over it once again! The other problems were a straightforward multiple choice about complex numbers (with one mistake!), some calculus around the functional sequence xne-x, and some arithmetic questions around Gauss’s and Bezout’s theorems. A few hours after I wrote the above, the (official) news came that this question had been posted on the web prior to the exam by someone and thus that it would be canceled from the exam by the Ministry for Education! The grade will then be computed on the other problems, which is rather unfair for the students. (On the side, the press release from the Ministry contains a highly specious argument that regulation allows for three to five exercises in the exam, hence that there is nothing wrong with reducing the number of exercises to three!) Not so lucky an occurrence, then, and I very deadly hope this will not impact in a drastic manner my son’s result! (Most likely, the grading will be more tolerant and students will not unduly suffer from the action of a very few….)

Typos in Chapters 1, 4 & 8

Posted in Books, R, Statistics with tags , , , , on February 10, 2010 by xi'an

Thomas Clerc from Fribourg pointed out an embarassing typo in Chapter 8 of “Introducing Monte Carlo Methods with R”, namely that I defined on page 247 the complex number \iota as the squared root of 1 and not of -1! Not that this impacts much on the remainder of the book but still an embarassment!!!

An inconsistent notation was uncovered by Bastien Boussau from Berkeley this time for the book The Bayesian Choice. In Example 1.1.3, on page 3, I consider an hypergeometric \mathcal{H}(30,N,20/N) distribution, while in Appendix A, I denote hypergeometric distributions as \mathcal{H}(N;n;p), inverting the role of the population size and of the sample size. Sorry about that, inconsistencies in notations are alas occuring in my books… In case I have not mentioned it so far, Example 4.3.3 further involves a typo (detected by Cristiano Passerini from Pontecchio Marconi) again with the hypergeometric distribution  \mathcal{H}(N;n;p)! The ratio should be

\dfrac{{n_1\choose n_{11}} {n-n_1\choose n_2-n_{11}}\big/ {n\choose n_2}\pi(N=n)}{\sum_{k=36}^{50} {n_1\choose n_{11}} {k-n_1\choose n_2-n_{11}}\big/ {k\choose n_2}\pi(N=k)}