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 Dec24 answered Urn problem with balls Dec24 answered Game to maintain distinct number of balls in glasses Dec24 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. Dec24 answered Prove without induction $2^n \mid (b+\sqrt{b^2-4c})^n + (b-\sqrt{b^2-4c})^n$ Dec18 reviewed Approve Proof of the description of a set Dec15 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. Dec15 awarded Caucus Dec12 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. Dec11 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 $k=1$. Dec1 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. Oct16 awarded Good Answer Oct8 comment Probability/Decision- infimum over set of expectations (can be interpreted as decision problem) I edited the answer to expand on this point, Eric. Oct8 revised Probability/Decision- infimum over set of expectations (can be interpreted as decision problem) added 1087 characters in body Oct7 revised Probability/Decision- infimum over set of expectations (can be interpreted as decision problem) added 235 characters in body Oct6 answered Probability/Decision- infimum over set of expectations (can be interpreted as decision problem) Sep30 awarded Explainer Sep9 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)"). Sep9 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.). Sep8 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? Aug25 awarded Yearling