For questions about estimation and how and when to estimate correctly
Questions tagged [estimation]
1806 questions
77
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…
xsari3x
- 207
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
- 25,971
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
- 1
22
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
- 867
17
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
- 6,433
- 4
- 31
- 68
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 such…
Ahaan S. Rungta
- 7,726
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
- 151
- 8
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
- 193
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}…
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
- 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
- 365
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
- 3,650