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bio website quantdec.com
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Consultant (environmental and spatial stats a specialty), expert witness, and teacher. I can be reached through (outdated but still valid) links posted on my web site.

Twitter: @WilliamAHuber // ASA-P website: http://amstatphilly.org/


Why waste time learning, when ignorance is instantaneous?

--T(iger) Hobbes.

For any complex problem there is a simple solution. And it's always wrong.

--[Mis?]attributed to H.L. Mencken by Dava Sobel, Longitude.


Dec
29
comment What are some interesting cases of $\pi$ appearing in situations that are not / do not seem geometric?
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 that $\lim_{x \to 0}\frac{x^2+x^3}{\sin^3x}$ doesn't exist.
Another way to appreciate @Chinny84's idea is to consider the limit of $1/x - (x^2+x^3)/\sin^3(x)$. This does exist and you can show it using any of the three techniques in the question. Now suppose, to assume the contrary of what you want to prove, that the limit of $(x^2+x^3)/\sin^3(x)$ exists. Then--adding the two results--you could conclude that the limit of $1/x$ exists, which it obviously does not (using any of the three methods).
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 Using triangle inequality for upper limit of integral.
Why not just compute the integral? It's dead easy--the exponential part, being analytic throughout the triangle, integrates to zero; the remaining contribution from $\bar z$ will give $-2i$ times the area of the triangle.
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.
Dec
24
comment Urn problem with balls
It might be even simpler just to count the ways of choosing one ball in one urn and the other in the other: $(n+1)^2$. Divided by $\binom{2n+2}{2}$, this gives the answer directly.
Dec
24
comment Prove without induction $2^n \mid (b+\sqrt{b^2-4c})^n + (b-\sqrt{b^2-4c})^n $
@Brian You're right; there are many inductions lurking here. The one you point to comes down to asserting that the integers are closed under addition and multiplication. I'll accept that as immediately obvious. The crux of the matter, though, is that scalars commute with matrices, whence $(2\mathbb P)^n = 2^n \mathbb P^n$: that shows where the powers of $2$ come from.
Dec
15
comment Show that $\lim_{n\rightarrow\infty}{\displaystyle\sum_{i=1}^{n}{\frac{F_n}{2^n}}}=2$
The limit is $0$, not $2$. Use $(1 +\sqrt{5})/2$ instead of $2$ in the denominator to obtain a finite limit.
Dec
12
comment How can I calculate $\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right) \left(\frac{w-a}{b}\right) f(w; \mu, \sigma²)\,\mathrm dw$
You ignore all terms except for the $k=1$ coefficient, which (upon multiplication by $1!=1$) yields the result you are looking for.
Dec
11
comment How can I calculate $\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right) \left(\frac{w-a}{b}\right) f(w; \mu, \sigma²)\,\mathrm dw$
The result you reference makes it easily possible (via linear substitutions and completing the square) to compute the integral of $\Phi((w-a)/b)f(w;\mu,\sigma^2) \exp(\lambda (w-a)/b)dw$. The MacLaurin series of that function of $\lambda$ will yield integrals proportional to $\Phi((w-a)/b)((w-a)/b)^k f(w;\mu,\sigma^2)dw$ for $k=0, 1, 2, \ldots$; you want the case $k=1$.
Dec
1
comment Show that the geometric mean is the limit of the $t$th power mean as $t \to 0$
Hint: Take logs and apply L'Hopital.
Oct
8
comment Probability/Decision- infimum over set of expectations (can be interpreted as decision problem)
I edited the answer to expand on this point, Eric.
Sep
9
comment Mean of squared “sum of squared errors”
See Advanced Theory of Statistics, Volume I, chapters 3 ("Moments and Cumulants") and 12 ("Cumulants of Sampling Distributions--(2)").
Sep
9
comment Mean of squared “sum of squared errors”
Jonas, the very first comment to your question indicates how such derivations can be done. Your example, when fully expanded, is a quartic form in the data and therefore the expectation (because it's a linear operator) becomes a homogeneous polynomial (in a suitable sense) of the first four moments of $X_1$. Mathematica merely is doing that routine algebra under the hood. An algebraic theory has been developed; it is explained in great detail in Kendall & Stuart (5th Ed.).
Sep
8
comment How to find a mapping function from n dimensional space to m dimensional space
Could you perhaps add some information to this question to show readers why it might be of interest on this site?
Aug
4
comment Does affine equivariance implies shape unbiasedness?
(Due to lack of answers on CV, this question has been migrated to Math at the OP's request.)
Jul
5
comment Combinatorics: Mean and Variance of an indicator function of items arranged in a circle.
You have hit on the crux of the matter: the question, to be unambiguous, must somehow include enough information to enable calculation of all probabilities. In some settings those probabilities are given explicitly, but usually some particular process whereby the randomization is achieved is described. When the process consists of placing each blue ball within one of the spaces between red balls, all with equal probability (of $1/15$), and doing so in a way that the placement of one blue ball affects no other blue balls, you get the multinomial probabilities.
Jul
5
comment Combinatorics: Mean and Variance of an indicator function of items arranged in a circle.
Perhaps an illustration would help. In the case $n=k=2$ there are two distinguishable configurations modulo symmetries of the circle: RBRB (red-blue-red-blue in counterclockwise order) and RBBR. In the multinomial scenario both are assumed to have probability $1/2$, while in the permutation scenario they are assumed to have probabilities $1/3$ and $2/3$, respectively. Some people have argued that absent any additional information, one should assume the maximum-entropy distribution. That's the multinomial one in this case--not the permutation one!