Mr. F
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# 264 Actions

 Mar26 comment On the integral $\int_{b'}^{b''}f(x)\ln(1+a x)\mathrm{d}x$ As an aside, I had a fun time assuming that $X$ was exponentially distributed with $\lambda=1$ and then computing the mean of $\log(1+X)$. It turns out to be Gompertz constant; I wonder if this coincidence has been noted before. Mar26 comment On the integral $\int_{b'}^{b''}f(x)\ln(1+a x)\mathrm{d}x$ Yes, I agree with that last statement but only for $x\in[0,\infty]$ Given that $X$ has finite mean over $[0,\infty]$ then the transformation of variables $X\to Y=\log(1+aX)$ should also have a finite mean on the same region. This won't be true over $(-1,\infty]$ because $|\log(1+x)|$ on that region can be an arbitrarily large. Mar26 comment On the integral $\int_{b'}^{b''}f(x)\ln(1+a x)\mathrm{d}x$ The space of functions that is integrable with respect to a given probability measure $\nu=f(x)dx$ is $L^{1}(\mathbb{R},\nu)$. The function $\log(1+x)$ doesn't decay towards $0$ as $x\to\infty$. This is a problem because then it's much harder to determine if $\log(1+x)\in L^{1}(\mathbb{R},\nu)$. If $f(x)$ does not die off to zero sufficiently quickly, then this function won't be integrable. You should look up some standard results in integrability theory to see why your stated conditions are not sufficient to determine the integrability of $\log(1+ax)$. Mar24 comment On hitting time of Brownian motion and Ito's lemma This is a good approach. More generally, you could look at the standard proofs for hitting times of the interval $(a,b)$ and then just let $a\to-\infty$ in the arguments. Mar24 comment On the integral $\int_{b'}^{b''}f(x)\ln(1+a x)\mathrm{d}x$ Even if $b'>-1$, that doesn't mean $\log(1+ab')$ is well-behaved. I think you have a backward inequality as well. I'm not sure that $\alpha$-stability buys you anything, because you're only ever working with one copy of the distribution. You're not using any central limit theorem type properties, and so all of the suggestions above about how there can be arbitrary probability density on $[b',b'']$ still applies. There's just not much you can possibly say without stronger assumptions. Mar24 answered On the integral $\int_{b'}^{b''}f(x)\ln(1+a x)\mathrm{d}x$ Mar24 answered Likelihood Ratio Test for Linear Regression Mar23 comment Error of Pearson Correlation Coefficient Just as an aside, there are many ways to evaluate fits. No single scoring method is ever "best", and in order to provide informative, truthful results to the readers of your work, you need to usually apply a wide array of model checks and fit scores. One place to start reading is Chapter 6 of the excellent resource book "Bayesian Data Analysis" by Gelman, Carlin, Stern, and Rubin. Mar23 awarded Enlightened Mar22 awarded Nice Answer Mar22 revised Gradient of a Mahalanobis distance added 2 characters in body Mar22 revised Gradient of a Mahalanobis distance deleted 117 characters in body Mar22 revised Gradient of a Mahalanobis distance edited body Mar22 answered Gradient of a Mahalanobis distance Mar22 revised Real analysis question edited body Mar22 answered Real analysis question Mar22 comment Finding displacement from origin of SVM hyperplane Scikits.learn, which is a Python layer over LIBSVM, does return the bias parameter within SVM objects. It's easy to use; I would consider it. Mar22 comment How do you filter through published papers and find the ones you should read? Yeah, like Cross-Validated, Sci-Comp, and Theoretical Computer Science for example. Mar22 comment Do non-mathematical fields use the appropriate level of analytic/probabilistic rigor? Regarding your comments on probability theory, I think Jaynes wrote the definitive rebuke of dogmatic dependence on measure theory, in Appendix B of his "Probability Theory: The Logic of Science." Reading Jaynes in grad school revived my love of math. Measure theory is important, insofar as it is an expedient tool, and if other folks like it for aesthetic reasons, that's fine. But in general, I think graduate-level education in probability has forgotten that applied math is supposed to be in the service of something... Mar22 comment P vs NP and Gödel As an interesting aside, Godel wrote a letter to Von Neumann once expressing how, if P=NP, it will be the end of human mathematics, excepting possibly the postulation of axioms. The remark is in this paper (which is well worth a read for most folks here anyway).