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 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 24 answered Prove without induction $2^n \mid (b+\sqrt{b^2-4c})^n + (b-\sqrt{b^2-4c})^n$ Dec 18 reviewed Approve Proof of the description of a set 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 15 awarded Caucus 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 16 awarded Good Answer 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. Oct 8 revised Probability/Decision- infimum over set of expectations (can be interpreted as decision problem) added 1087 characters in body Oct 7 revised Probability/Decision- infimum over set of expectations (can be interpreted as decision problem) added 235 characters in body Oct 6 answered Probability/Decision- infimum over set of expectations (can be interpreted as decision problem) Sep 30 awarded Explainer 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.). Aug 25 awarded Yearling Aug 21 revised Is there a name (and use) for an average based on the unique values of a set of data? Improved theTeX markup 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 30 revised How to simplify a sum with binomial coefficients multiplied by $k^3/2^k$? added 55 characters in body