# Tagged Questions

For questions about estimation and how and when to estimate correctly

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### What is the fastest/most efficient algorithm for estimating Euler's Constant $\gamma$?

What is the fastest algorithm for estimating Euler's Constant $\gamma \approx0.57721$? Using the definition: $$\lim_{n\to\infty} \sum_{x=1}^{n}\frac{1}{x}-\log n=\gamma$$ I finally get $2$ decimal ...
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### Mental estimate for tangent of an angle (from $0$ to $90$ degrees)

Does anyone know of a way to estimate the tangent of an angle in their head? Accuracy is not critically important, but within $5%$ percent would probably be good, 10% may be acceptable. I can ...
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### Approximating $100!$

I participated in an Estimathon (a speed contest of Fermi problems) not long ago. It works as follows. Contestants are given questions and they must give a closed range $[a,b]$ which should contain ...
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### Estimates for the normal approximation of the binomial distribution

I'm interested in estimates of the normal approximation for binomial distributions, i.e. in estimates of $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - \Phi(x)\right|$$...
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### Is there a lower-bound version of the triangle inequality for more than two terms?

The triangle inequality $|x+y|\leq|x|+|y|$ can be generalized by induction to $$|x_1+\ldots+ x_n|\leq|x_1|+\ldots+|x_n|.$$ Can we generalize the version $|x+y|\geq||x|-|y||$ to $n$ terms too? I need ...
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### A calculation that goes awfully wrong if we let $\pi=22/7$

Me and one of my friends had an argument and he said that using $22/7$ as value of $\pi$ is sufficient for any calculation. Can we always take it $22/7$, or is there some example of some calculation ...
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### Berry-Esseen bound for binomial distribution

From the Berry-Essen theorem I can deduce $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - \Phi(x)\right| \le \frac{C(p^2+q^2)}{\sqrt{npq}}$$ with $C \le 0.4748$. My ...
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### Finding a better upper bound for an integral of a product of $n$ terms

So I'm trying to find and upper bound for the integral $$\int\limits_{a}^b \! (x-x_1)^2 \cdots (x-x_n)^2\, \mathrm{d}x,$$ where $x_i \in [a,b], \enspace \forall i=1,\dots ,n.$ I've tried ...
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### My data is not normally distributed: what can I do to estimate a tail probability?

Continuing on from my earlier question, I'm attempting to analyse the data qualitatively. In the following plot, I make $10000$ samples where I count "the number of clashes". I plot $n$ vs. the ...
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### Unbiased Estimator Question and Understanding

I'm having some difficulty with unbiased estimators, and wondered if anyone could help me. I believe I understand the general concepts OK, however when I come to look at some sample questions to test ...
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### Why does maximum likelihood estimation for uniform distribution give maximum of data?

I am looking at parameters estimation for the uniform distribution in the context of MLEs. Now, I know the likelihood function of the Uniform distribution $U(0,\theta)$ which is $1/\theta^n$ cannot ...
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### Question for the estimation of $\sum_{i=1}^x \frac{1}{w+i}$ as $x \to \infty$.

I have a question of the estimation of this summation: $$\frac{1}{w+1}+\frac{1}{w+2}+\cdots+\frac{1}{w+x}$$ Which is: $$\sum_{i=1}^x \frac{1}{w+i}$$ What I have tried: applying limit to the ...
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### What are the bounds (upper and lower) for $|A+A|$?

Let $A$ be a finite set of real (or complex) numbers. If I consider sets with small sizes, we have that: If $A$ is the empty set, then $A+A$ is also empty. If $A$ is a singleton, then $A+A$ is ...
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### Is this estimate true or not true?

Let $\varepsilon>0$. Let $\varphi(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ the standard normal density function. Then \lim_{\varepsilon\to 0}\int_0^1 \frac{1}{\sqrt{x}}\left[ \varphi\left(\frac{\sqrt{...
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### Difficult to understand difference between the estimates on E(X) and V(X) and the estimates on variance and std.dev. on lambda-hat

I'm having a very hard time to separate estimates on population values versus estimates on sample values. I'm struggling with this exercise (not homework, self-study for my exam in introductionary ...
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### Estimate the given sum.

This is the question from "Data Structures and Algorithm Analysis in C" By Mark Weiss. It is the question 1.7. It goes as follows:- Estimate the sum $\sum\limits_{i=\lfloor{N/2}\rfloor}^N\frac{1}{i}$....
### Sufficient statistic for product of $\Gamma(x + a_i)$
Background Suppose that the national mint mints coins with bias $p_i \sim \rm{Beta}(A,B)$ for some unknown constants $A$, $B$. Given $n$ coins, you flip each coin a certain number of times. Coin $i$...
Show that if $f(x)=x^2+O(x)$, and $f$ is differentiable with non-decreasing derivative $f'(x)$, then $f'(x)=2x+O(\sqrt{x})$. I know that if $f'$ is not non-decreasing, then the statement is not true....