Reputation
5,476
Top tag
Next privilege 10,000 Rep.
Access moderator tools
Badges
15 25
Newest
 Good Answer
Impact
~248k people reached

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
revised Urn problem with balls
added 2 characters in body
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
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