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Dec
24
answered Urn problem with balls
Dec
24
answered Game to maintain distinct number of balls in glasses
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.).
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
25
awarded  Yearling