Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [estimation]

For questions about estimation and how and when to estimate correctly

1691 questions
57
votes
12 answers

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 places of accuracy when $n\geq180$. The third…
Argon
  • 24,757
  • 10
  • 93
  • 132
29
votes
3 answers

How to verify if a curve is exponential by eyeballing?

A plane curve is printed on a piece of paper with the directions of both axes specified. How can I (roughly) verify if the curve is of the form $y=a e^{bx}+c$ without fitting or doing any quantitative calculation? For example, for linear curves, I…
arax
  • 2,759
  • 1
  • 18
  • 24
26
votes
1 answer

Prove that $e^\pi+\frac{1}{\pi} < \pi^e+1$

Prove that: $$e^\pi+\frac{1}{\pi}< \pi^{e}+1$$ Using Wolfram Alpha $\pi e^{\pi}+1 \approx 73.698\ldots$ and $\pi(\pi^{e}+1) \approx 73.699\ldots$ Can this inequality be proven without brute-force estimations (anything of the sort $e\approx…
LHF
  • 8,461
  • 1
  • 22
  • 41
21
votes
8 answers

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 estimate sines and cosines quite well, but I consider…
brianmearns
  • 817
  • 7
  • 21
16
votes
3 answers

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 the correct answer. The scoring guidelines are such…
Ahaan S. Rungta
  • 7,486
  • 5
  • 26
  • 64
15
votes
1 answer

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 to estimate an expression of the form $|x+y+z|$…
Bartek
  • 6,225
  • 4
  • 28
  • 66
14
votes
2 answers

Probability vs Confidence

My notes on confidence give this question: An investigator is interested in the amount of time internet users spend watching TV a week. He assumes $\sigma = 3.5$ hours and samples $n=50$ users and takes the sample mean to estimate the population…
hongsy
  • 495
  • 2
  • 8
  • 16
12
votes
1 answer

Proving an integral is finite

I have the following integral: $$\displaystyle \int_{\mathbb{R}^2} \left( \int_{\mathbb{R}^2} \frac{J_{1}(|\alpha|)J_{1}(|k- \alpha|)}{|\alpha||k-\alpha|} \ \mathrm{d}\alpha \right)^2 \ \mathrm{d}k,$$ where both $\alpha$ and $k$ are vectors in…
user363087
  • 1,105
  • 1
  • 9
  • 19
10
votes
2 answers

How to know when to use t-value or z-value?

I'm doing 2 statistics exercises: The 1st: An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with a standard deviation of 40 hours. If a sample of 30 bulbs has an average life of 780…
L.I.B L.I.B
  • 163
  • 1
  • 1
  • 6
10
votes
3 answers

Approximate the second largest eigenvalue (and corresponding eigenvector) given the largest

Given a real-valued matrix $A$, one can obtain its largest eigenvalue $\lambda_1$ plus the corresponding eigenvector $v_1$ by choosing a random vector $r$ and repeatedly multiplying it by $A$ (and rescaling) until convergence. Once we have the…
mitchus
  • 520
  • 1
  • 4
  • 17
9
votes
4 answers

Why does finding the $x$ that maximizes $\ln(f(x))$ is the same as finding the $x$ that maximizes $f(x)$?

I'm reading about maximum likelihood here. In the last paragraph of the first page, it says: Why does the value of $p$ that maximizes $\log L(p;3)$ is the same $p$ that maximizes $L(p;3)$. The fact that it mentions that log is an increasing…
mauna
  • 3,250
  • 2
  • 28
  • 47
9
votes
2 answers

Estimating the sum $\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$

By integral test, it is easy to see that $$\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$$ converges. [Here $\ln(x)$ denotes the natural logarithm, and $\ln^2(x)$ stands for $(\ln(x))^2$] I am interested in proving the following inequality (preferrably…
Prism
  • 10,483
  • 4
  • 39
  • 115
9
votes
0 answers

Geometric Interpretation of Product of Two Multivariate Gaussians densities

I am trying to understand the high-dimensional geometry behind Bayesian estimation. When you multiply two Normal densities with respective means $\mu_1, \mu_2$ and covariances $\Sigma_1, \Sigma_2$, the renormalized product is again a Normal density…
Maximilian Sölch
  • 91
  • 5
9
votes
1 answer

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 $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - \Phi(x)\right|$$ From the Berry-Essen theorem I can…
Stephan Kulla
  • 6,911
  • 1
  • 28
  • 66
8
votes
1 answer

Probability that a sample comes from one of two distributions

Let's say I have two normal distributions with means $\mu_1$, $\mu_2$ and standard deviations $\sigma_1$, $\sigma_2$ (which I know). I am handed a random variate from one of the distributions (I don't know which). What is the likelihood that my…
David G
  • 335
  • 1
  • 7
1
2 3
…
99 100