Mathematics

Questions tagged [estimation]

For questions about estimation and how and when to estimate correctly

1804 questions
78
votes
3 answers

An explanation of the Kalman filter

In the past 3 months, I've been trying to understand the Kalman filter. I have tried to implement it, I have watched YouTube tutorials, and I have read some papers about it and its operation (update, predicate, etc.). However, I still am unable to…
KnowledgeDude
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58
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
  • 26,059
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
  • 2,815
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
  • 8,575
23
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
  • 877
18
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
  • 6,583
17
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
  • 537
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…
Ahaan S. Rungta
  • 7,736
15
votes
3 answers

Estimating $\int_{0}^{\pi}\sin(\sin(x))\,\mathrm{d}x$

Which of the following is closest to this integral? $$\int_{0}^{\pi}\sin(\sin(x))\,\mathrm{d}x$$ $$\text{(A) }1+\frac{\sqrt{2}}{2}\quad\quad\quad\quad\text{(B) }\sqrt{3}\quad\quad\quad\quad\text{(C) }\sqrt{\pi}\quad\quad\quad\quad\text{(D)…
Hussain-Alqatari
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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
  • 1,175
11
votes
1 answer

What are some examples of asymptotic series that cannot be differentiated term by term?

When I was studying the asymptotic series section in the foundations of order estimation, I saw that the asymptotic series of a function has the property of being integrable: $$\text{If } F(x) \approx \sum_{k=2}^{\infty} \frac{C_{k}}{x^{k}} \text{,…
Grant Russell
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11
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
  • 303
10
votes
6 answers

How to estimate $10^{\frac{1}{2}} + 10^{\frac{1}{3}} + \ldots + 10^{\frac{1}{10}}$

I heard an interesting interview question recently, which was as follows: Estimate the value of $X = 10^{\frac{1}{2}} + 10^{\frac{1}{3}} + \ldots + 10^{\frac{1}{10}}$ . You have 30 seconds to Compute it . My approach was as follows: $10^\frac{1}{2}…
Christopher Miller
  • 668
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 first…
mitchus
  • 550
9
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
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