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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.
Apr
28
comment Empirical likelihood method to Find the order of a parameter given a set of constraints
What is "$\lambda$"?? You provide no definition. BTW, a better way to get quick, insightful, creative help for questions like this is to provide motivation and background: explain what the important symbols $L$, $J_k$, and $S_k^r$ mean and represent. As it stands, these are just a bunch of abstract equations that would require exceptional effort from most readers (who are otherwise qualified to help you) to understand, which is going to limit (rather severely) your chances of receiving a good response, even after you tell us about $\lambda$.
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.
Dec
24
comment Probability that there is sub-sequence of exact length
There is a lot of literature on this subject. The asymptotic distribution is known. I don't believe exact formulas for the case of finite $N$ are available.
Dec
24
comment Urn problem with balls
@Brian Thank you for that observation. It's very much in the spirit of this answer, which is an attempt to find as simple a solution as possible.
Dec
24
comment The point of contact of between two circles and common tangent at this point.
+1 for the simplified elementary approach. But note that the solution is incomplete: it has not (yet) addressed whether the circles are internally or externally tangent.