Heart rates must be Lognormal in distribution. Simply, it is not possible to have a negative heart rate and at low variance (and a mean > 6 standard deviations away from 0), the lognormal behaves like a normal. Natural distribution of heart rates vs that from modern life. Incidentally I failed to understand from San-Millan’s paper(s) the 2 mmol lactate threshold. I don’t see a threshold. Even for athletes (top graph below) there is a mix outside asymptote (Lactate >5 mmol becomes 0 fat oxidation).( I put the question to him on Twitter with no response.)   From San-Millan’s pape ..read more

Prove \(I= \displaystyle\int_{-\infty }^{\infty}\sum_{n=0}^{\infty } \frac{\left(-x^2\right)^n }{n!^{2 s}}\; \mathrm{d}x= \pi^{1-s}\). We can start as follows, by transforming it into a generalized hypergeometric function: \(I=\displaystyle\int_{-\infty }^{\infty }\, _0F_{2 s-1} (\overbrace{1,1,1,…,1}^{2 s-1 \text{times}}; -x^2)\mathrm{d}x\), since, from the series expansion of the generalized hypergeometric function, \(\, _pF_q\left(a_1,a_p;b_1,b_q;z\right)=\sum_{k=0}^{\infty } \frac{\prod_{j=1}^p \left(a_j\right)_k z^k}{\prod_{j=1}^q k! \left(b_j\right)_k}\), where \((.)_k\) is the Pochhamme ..read more

  Figuring out the sampling error of rolling correlation.   Financial theory requires correlation to be constant (or, at least, known and nonrandom). Nonrandom means predictable with waning sampling error over the period concerned. Ellipticality is a condition more necessary than thin tails, recall my Twitter fight with that non-probabilist Clifford Asness where I questioned not just his empirical claims and his real-life record, but his own theoretical rigor and the use by that idiot Antti Ilmanen of cartoon models to prove a point about tail hedging. Their entire business reposes ..read more

Summary: HRV appears to be ill defined in papers and regressions; using logged variables fixes the problem. HeartRateVariability.nb   https://www.fooledbyrandomness.com/HeartRateVariability.nb.pdf ..read more

Ariely.nbThe “Ariely” fake result reveal a probability distribution totally foreign to psychologists   https://www.fooledbyrandomness.com/Ariely.nb.pdf   ..read more

Introduction and Result A maximum entropy alternative to Bayesian methods for the estimation of independent Bernouilli sums. Let \(X=\{x_1,x_2,\ldots, x_n\}\), where \(x_i \in \{0,1\}\) be a vector representing an n sample of independent Bernouilli distributed random variables \(\sim \mathcal{B}(p)\). We are interested in the estimation of the probability p. We propose that the probablity that provides the best statistical overview, \(p_m\) (by reflecting the maximum ignorance point) is \(p_m= 1-I_{\frac{1}{2}}^{-1}(n-m, m+1)\), (1) where \(m=\sum_i^n x_i \) and \(I_.(.,.)\) is the beta regula ..read more

You have zero probability of making money. But it is a great trade. One-tailed distributions entangle scale and skewness. When you increase the scale, their asymmetry pushes the mass to the right rather than bulge it in the middle. They also illustrate the difference between probability and expectation as well as the difference between various modes of convergence. Consider a lognormal \(\mathcal{LN}\) with the following parametrization, \(\mathcal{LN}\left[\mu t-\frac{\sigma ^2 t}{2},\sigma \sqrt{t}\right]\) corresponding to the CDF \(F(K)=\frac{1}{2} \text{erfc}\left(\frac{-\log (K)+\mu t-\f ..read more

The main paper Bitcoin, Currencies and Fragility is updated here . BTC-QF The supplementary material is updated here www.fooledbyrandomness.com/BTC-QF-appendix.pdf   BTC-QF-appendix https://www.fooledbyrandomness.com/BTC-QF.pdf https://www.fooledbyrandomness.com/BTC-QF-appendix.pdf ..read more

Background: We’ve discussed blood pressure recently with the error of mistaking the average ratio of systolic over diastolic for the ratio of the average of systolic over diastolic. I thought that a natural distribution would be the gamma and cousins, but, using the Framingham data, it turns out that the lognormal works better. For one-tailed distribution, we do not have a lot of choise in handling higher dimensional vectors. There is some literature on the multivariate gamma but it is neither computationally convenient nor a particular good fit. Well, it turns out that the Lognormal has some ..read more

The Lancet article: Maron, Barry J., and Paul D. Thompson. “Longevity in elite athletes: the first 4-min milers.” The Lancet 392, no. 10151 (2018): 913 contains an eggregious probabilistic mistake in handling “expectancy” a severely misunderstood –albeit basic– mathematical operator. It is the same mistake you read in the journalistic “evidence based” literature about ancient people having short lives (discussed in Fooled by Randomness), that they had a life expectancy (LE) of 40 years in the past and that we moderns are so much better thanks to cholesterol lowering pills. Something ..read more