Michael Hardy

Minneapolis

After doing nearly all the coursework for a Ph.D. in math, I then did all the coursework for a Ph.D. in statistics and completed that degree.
56m
revised Maximal ideal in a polynomial ring
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57m
comment If $\lim_{n\to \infty}a_n = a\in \mathbb{R}$ . Prove that $\limsup_{n\to \infty}a_n x_n=a\limsup_{n\to \infty}x_n$ .
Might you have meant $\limsup\limits_{n\to\infty}$ instead of $\lim_{n\to\infty}\sup$? ${}\qquad{}$
59m
comment If $\lim_{n\to \infty}a_n = a\in \mathbb{R}$ . Prove that $\limsup_{n\to \infty}a_n x_n=a\limsup_{n\to \infty}x_n$ .
I changed $lim$ and $sup$ to $\lim$ and $\sup$, both standard usage. ${}\qquad{}$
1h
revised If $\lim_{n\to \infty}a_n = a\in \mathbb{R}$ . Prove that $\limsup_{n\to \infty}a_n x_n=a\limsup_{n\to \infty}x_n$ .
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1h
comment Distribution of random variables (normal and standard normal)
. . . or maybe this is simpler: The distribution of $(X_1-\mu,\ldots,X_n-\mu)$ is the same as that of $\sigma(Z_1,\ldots,Z_n)$. ${}\qquad{}$
1h
comment Distribution of random variables (normal and standard normal)
@NickR : The distribution of $X_i-\mu$ is the same as the distribution of $\sigma Z_i$. And $X_i-\bar X = (X_i-\mu)-(\bar X-\mu)$, so that has the same distribution as $Z_i-\bar Z$. And the joint distribution of the correlated random variables $X_i-\bar X$ and $X_j-\bar X$ is the same as that of $Z_i-\bar Z$ and $Z_j-\bar Z$. ${}\qquad{}$
1h
revised $n^\text{th}$ derivative of $\tan^{-1} x$
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1h
comment $n^\text{th}$ derivative of $\tan^{-1} x$
I changed $tan^{-1} x$ to $\tan^{-1} x$ and did a few other bits of copy-editing. ${}\qquad{}$
1h
revised $n^\text{th}$ derivative of $\tan^{-1} x$
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10h
comment Distribution of random variables (normal and standard normal)
@BruceTrumbo : In order to get that conclusion, you need to mention also that $Z$ and $Q$ are independent. ${}\qquad{}$
14h
answered Liouville's theorem application
14h
comment Distribution of random variables (normal and standard normal)
It is not true that $\sigma s_Z=s_X$, but it is true that the distribution of $\sigma s_Z$ equals the distribution of $s_X$. ${}\qquad{}$
14h
comment Puzzled at this alternating series problem.
@Elsa : The limit of the absolute value is $1$, but as you yourself said, it's alternating, so that'w where Simon got his conclusion. If the limit is not $0$ then the series diverges. ${}\qquad{}$
14h
comment Minimum of random exponential variable and time
Yes. (Except for possible boundaries whose measure is $0$, and those won't affect the value of the integral.) ${}\qquad{}$
15h
answered Distribution of random variables (normal and standard normal)
15h
comment Minimum of random exponential variable and time
Actually $[U+V\le s]$ would imply that $V<t$, so that appears to be correct. ${}\qquad{}$
15h
answered Minimum of random exponential variable and time
16h
revised Minimum of random exponential variable and time
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16h
revised Distribution of random variables (normal and standard normal)
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16h
revised Convergence of $\int_0^1 \frac{\ln(1-x)\sqrt{x-x^2}}{\sin(\pi x)} \, dx$
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