Activity for Michael Hardy
| Type | On... | Excerpt | Status | Date |
|---|---|---|---|---|
| Edit | Post #289532 |
Post edited: some MathJax usage improvements |
— | 3 months ago |
| Edit | Post #291432 |
Post edited: Proper horizontal spacing between f(x) and dx. Also, without \left and \right, the absolute value signs lack the right horizontal spacing when \sin is used. |
— | 3 months ago |
| Edit | Post #292073 |
Post edited: |
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| Edit | Post #292073 |
Post edited: |
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| Edit | Post #292073 | Initial revision | — | 3 months ago |
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A: What is the probability density function for the tau distribution? It is well worth knowing how to do what I show you how to do below. The books in which you find formulas were not brought down from Heaven by an archangel. They are derived by humans, and you are one of those. First, notice that (assuming $\nu>1$) as $t^2{\nu-1}$ gets bigger, so does $\tau\nu \sim... (more) |
— | 3 months ago |
| Edit | Post #292071 | Initial revision | — | 3 months ago |
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A: The effect of measurement accuracy and rounding on hypothesis testing If the standard deviation of your measurements is in the trillions, then rounding to integers will make little difference – probably no practical difference – since the rounding error is so tiny by comparison to the value. If the standard deviation is $1/2$ then you would be throwing away huge amount... (more) |
— | 3 months ago |
| Edit | Post #292070 |
Post edited: |
— | 3 months ago |
| Edit | Post #292070 |
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— | 3 months ago |
| Edit | Post #292070 |
Post edited: |
— | 3 months ago |
| Edit | Post #292070 | Initial revision | — | 3 months ago |
| Answer | — |
A: Example of $f:[0,1]\to\mathbf{R}$ with $\lim_{a\to 0^+}\int_a^1f(x)dx=L $ for some real number $L$ but $\int_0^1|f(x)|dx=\infty $ Such an example is $\dfrac{\sin(1/u)} u$ for $0 < u < 1. $ \begin{align} \text{Let } & u = \frac1x. \\[6pt] \text{Then } & du = -\frac{dx}{x^2}. \end{align} For $1\le a < b,$ as $x$ goes from $a$ to $b,$ $u$ goes from $1/a$ to $1/b$, so $$ \int{1/b}^{1/a} \frac{\sin(1/u)} u du = -\intb^a \f... (more) |
— | 3 months ago |
| Suggested Edit | Post #289532 |
Suggested edit: some MathJax usage improvements (more) |
helpful | 3 months ago |
| Edit | Post #292069 | Initial revision | — | 3 months ago |
| Answer | — |
A: Is it known whether most numbers are normal or not The measure of the set of all real numbers that are not normal is $0.$ Notice first that we can restrict attention to numbers between $0$ and $1.$ A number in that interval is normal precisely of that number plus any integer you choose is normal. Take the probability that a number between $0$ a... (more) |
— | 3 months ago |
| Edit | Post #292068 |
Post edited: |
— | 3 months ago |
| Edit | Post #292068 | Initial revision | — | 3 months ago |
| Answer | — |
A: The derivatives of a function at a boundary point The endpoint fails to be an "interior" point only when the space $\left[0,L\right[$ is embedded in a larger space. If a function is defined on a certain domain, one may take the value of the derivative of the function $f$ at a point $a$ in the domain to be $$ f'(a) = \lim{x\toa \atop x\in\text{doma... (more) |
— | 3 months ago |
| Comment | Post #290509 |
Contrast the typography in the expression $\displaystyle \frac1x\cdot|\sin x|$ with that in $\displaystyle \frac1x\cdot\left|\sin x\right|.$ The first does not have proper horizontal spacing. The second is coded as `\frac1x\cdot\left|\sin x\right|`. This is a consequence of the fact that \sin has con... (more) |
— | 3 months ago |
| Edit | Post #292067 | Initial revision | — | 3 months ago |
| Answer | — |
A: How can you forebode that the answer shall be independent of the number of sides of the base of the polygon ? Cavalieri's principle will tell you that the shape of the base doesn't matter. The area of the base is the only information about the base that is needed. Cavalieri's principle (which I suspect was enunciated by Archimedes of Syracuse a couple of millennia before the man named Cavalieri was born) ... (more) |
— | 3 months ago |
| Suggested Edit | Post #291824 |
Suggested edit: Mathematical notation should be in math mode in MathJax and in LaTeX. (more) |
declined | 3 months ago |
| Suggested Edit | Post #291432 |
Suggested edit: Proper horizontal spacing between f(x) and dx. Also, without \left and \right, the absolute value signs lack the right horizontal spacing when \sin is used. (more) |
helpful | 3 months ago |
| Edit | Post #292059 |
Post edited: |
— | 3 months ago |
| Comment | Post #289712 |
In point (2), just as you replaced $\mu,$ the population mean, with $\overline x,$ the sample mean, I would have replaced $\sigma^2,$ the population variance, with $s^2.$ And the conclusion that this is an unbiased estimator of the variance relies on the assumption that this is an i.i.d. sample, i.e.... (more) |
— | 3 months ago |
| Edit | Post #292062 | Initial revision | — | 3 months ago |
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A: What is the formula for sample standard deviation of a small sample size? This popular meme of $\text{“}N<30\text{”}$ is only a popular meme (except that it's mentioned in statistics courses for the mathematically unschooled), and is about the central limit theorem, which says (loosely speaking) that the sample mean or the sample sum is approximately normally distributed i... (more) |
— | 3 months ago |
| Edit | Post #292059 | Initial revision | — | 3 months ago |
| Answer | — |
A: Minimal non-standard number in non-standard models of PA The nonstandard models would be elementarily equivalent to the standard models, i.e. they would satisfy the same first-order formulas. In particular, for every first-order formula with one free variable, the set of numbers satisfying it would have a least element. However, for every number $x>0,$ ... (more) |
— | 3 months ago |