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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)
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Oct
7
revised Probability/Decision- infimum over set of expectations (can be interpreted as decision problem)
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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 $k^3/2^k$?
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Jul
30
revised How to simplify a sum with binomial coefficients multiplied by $k^3/2^k$?
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Jul
30
answered How to simplify a sum with binomial coefficients multiplied by $k^3/2^k$?