robjohn

West Hills, CA

none

Age: 56

the mean square The Mean Square
(with one standard deviation and several unusual ones)

aka Rob Johnson

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13h
revised About the integral $\int_{-1}^1 \frac{1}{\pi^2+(2 \operatorname{arctanh}(x))^2} \, dx=\frac{1}{6} $
add a link
14h
comment About the integral $\int_{-1}^1 \frac{1}{\pi^2+(2 \operatorname{arctanh}(x))^2} \, dx=\frac{1}{6} $
Would it be possible to provide some information about the $A$ that appears in the value for the supplementary integral?
16h
answered About the integral $\int_{-1}^1 \frac{1}{\pi^2+(2 \operatorname{arctanh}(x))^2} \, dx=\frac{1}{6} $
1d
comment How to find a function in $ L_{2}(0,\infty)$ but not $L_{q}(0,\infty)$, $q\neq 2 $
(+1) that works, too.
1d
comment How can I obtain this division's limit without using derivatives?
@user246608: $\lim\limits_{x\to0}\frac{\sin(x)}x=1$ is derived geometrically in this answer.
1d
revised Show that this difference goes to zero,
add the error estimate for the Riemann Sum
1d
comment Show that this difference goes to zero,
Ah, I forgot to include the approximation error. I will add that.
1d
revised How to find a function in $ L_{2}(0,\infty)$ but not $L_{q}(0,\infty)$, $q\neq 2 $
improve exposition
1d
answered How to find a function in $ L_{2}(0,\infty)$ but not $L_{q}(0,\infty)$, $q\neq 2 $
1d
answered Show that this difference goes to zero,
1d
revised How to take into account uncertainty on number of events
approximate the expected value of $\frac1n$ when $n$ has a Poisson distribution.
1d
comment How to take into account uncertainty on number of events
I think I misunderstood the question previous to the last edit. I now believe that your question is asking for the variance of $\frac1n\sum\limits_{k=1}^nX_k$. Is that correct? I have amended my answer to cover this case, too.
1d
revised How to take into account uncertainty on number of events
answer the question that I think was asked
1d
answered How to take into account uncertainty on number of events
2d
revised the best constant in an inequality?
drop the variational method for a basic Cauchy-Schwarz approach
2d
revised the best constant in an inequality?
Use the equality in Cauchy-Schwarz
2d
revised the best constant in an inequality?
give an alternate approach
2d
revised the best constant in an inequality?
the sum is equal to this
2d
answered the best constant in an inequality?
2d
comment Evaluate $\displaystyle\lim_{n \to \infty} \int_{0}^1 [x^n + (1-x)^n ]^\frac{1}{n} \ \mathrm{d}x$
(+1) However, on $\left[\frac12,1\right]$, note that $\left[1+\left(\frac{1-x}x\right)^n\right]^{1/n}\le2$ and easily $\lim\limits_{n\to\infty}\left[1+\left(\frac{1-x}x\right)^n\right]^{1/n}=1$. From these observations alone, we can apply Dominated Convergence.
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