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 Mar 23 comment Why does “convex function” mean “concave *up*”? @Wangyan So is the question. Feb 24 comment Joint probability distribution of functions of random variables This is Calculus, pure and simple. One way to approach it is through differential forms, as explained at stats.stackexchange.com/questions/180715/…. Feb 7 comment Approximating the number $e$ through computer simulation - mathematical background Re (1) This seems like an ambiguous criterion. If we make sure to generate random variates sufficiently slowly, then (according to it) the calculation will be "Monte Carlo". How about if I time the effort to compute the head and then take twice as much time to generate a Monte-Carlo estimate of the tail--that would seem to qualify as a "Monte Carlo" calculation by this criterion. (2) The tail CDF has a simple closed form expression. Regardless, there are many ways to generate random variables from it. Feb 7 comment Approximating the number $e$ through computer simulation - mathematical background +1 Re the final paragraph: You almost got the point, but it's the opposite of what you state. We can generalize this method and simulate, say, $\sum_{i=n+1}^\infty 1/i!$ and add $1+1+1/2+\cdots+1/n!$ to it to estimate $e$. As $n$ increases this becomes extremely efficient, to the point where no iterations of the simulation are needed. Thus, there is a sequence of simulations that range from this "pretty terrible way" to methods requiring no iterations at all. At what point, exactly, do we stop calling this "simulation" and start calling it pure "calculation"? :-) Feb 2 comment integral of trace function That tells us you are unfamiliar with the multivariate Gaussian integral. That's fine--you can look it up in various places, such as en.wikipedia.org/wiki/…. Feb 2 comment integral of trace function Would it help to recognize that $\operatorname{tr}(\mu\mu^\prime\Sigma)=\mu^\prime\Sigma\mu$ (assuming $\mu$ is a column vector)? Jan 21 comment Relation between differential geometry and differential geodesy Your edit appears to answer the question. Dec 28 comment Calculate the Gamma function Γ(2.7) @Lauren Numerical analysts have found that a far better technique is to use the functional relationship $\Gamma(n) = \Gamma(n+1)/n$ repeatedly until $n+1$ is so large that Stirling's approximation to $\Gamma(n+1)$ is sufficiently accurate. Very few of these steps are needed: using five terms in the asymptotic series and stopping as soon as $n+1 \ge 6$ gives ten decimal digits of precision. Aug 31 comment How to solve for the matrix $X$ in the following equation $AXB + X = CD$ It's a system of linear equations--solve it as you would any system. Jun 25 comment What are the odds that the pattern, win lose lose, will happen 23 times in a row (69 rounds)? Thank you: we appreciate your respect for not double-posting on SE sites. Jun 25 comment What are the odds that the pattern, win lose lose, will happen 23 times in a row (69 rounds)? (Migrating to Mathematics by request of the OP.) Jun 11 comment What are the odds that the pattern, win lose lose, will happen 23 times in a row (69 rounds)? This problem actually is much easier than any of the related ones. Your questions are answered simply by applying the definition of independence (which is what you must assume to answer them without further assumptions): the probabilities of independent events multiply. So go ahead and do the multiplications. You will notice that although the patterns will determine the probabilities, they do not change the fact that in every case you just multiply them all, so there really is nothing here involving combinatorial issues. May 19 comment Proof for Mean of Geometric Distribution Your reference gives three distinct derivations. The other two seem to answer your question. Mar 7 comment Bivariate normal distribution: showing that linear combinations of joint Gaussians are Gaussian This is merely equation (21) with $\rho$ used in place of that ratio of covariances. Feb 18 comment Prove that $a_i\leq 0$ for $i=1,2,…,N-1$? You might have the wrong sign in the equation you obtained: check it against the sequence $(0,-2,0)$ for $N=2$, for instance. As a hint, relax the conditions and suppose only that $a_{i+1}-2a_i+a_{i-1}\ge 0$ for $i=1,2,\ldots,N-1$. Can you still draw the desired conclusion? Feb 11 comment Difference between Real Analysis and Probability Theory? Nice little glitch in the SE technology: I have been able to upvote this answer twice--once on CV and once again here :-). Feb 11 comment Difference between Real Analysis and Probability Theory? +1 This answer gets to the heart of the matter. Analysis and probability have different interests: although they use similar sets of tools, they ask different questions and pursue almost completely different avenues of investigation. The two disciplines will nevertheless remain closely intertwined because insights from one can lead to progress in the other, much as (say) investigations of general relativity and quantum mechanics in physics have inspired advances in low-dimensional topology and operator theory, respectively. Jan 30 comment What's a proof that the angles of a triangle add up to 180°? @JoeZ. Or $-2\pi$: the complex arithmetic keeps track of angle orientation, too. Dec 29 comment Interesting and unexpected applications of $\pi$ This is David H's answer in disguise: the ARE is algebraically related to (a) $e^0=1$ and (b) $\int_\mathbb{R}e^{-x^2}dx=\Gamma(1/2)=\sqrt{\pi}$. The appearance of $\pi$ is due to the duplication formula for the Gamma function, $\Gamma(z)\Gamma(1-z)=\pi\csc(\pi z)$, exhibiting $\Gamma$ as a kind of "square root" of a trig function. This is because $\Gamma(z)=\Gamma(z+1)/z$ implies $\Gamma$ has (simple) poles at $0,-1,-2,\ldots$, whence $\Gamma(z)\Gamma(1-z)$ has poles at $\mathbb Z$, strongly suggesting periodicity--which is why $\pi$ should show up! Dec 28 comment Prove without induction $2^n \mid (b+\sqrt{b^2-4c})^n + (b-\sqrt{b^2-4c})^n$ @Thank you for noticing that, Thomas. That is easily fixed by replacing one of the $c$'s by $1$; I will make the change.