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39
votes
10answers
2k views

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 ...
21
votes
8answers
728 views

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 ...
5
votes
1answer
94 views

Estimating a gaussian distribution from a GMM

Suppose that we have a Gaussian mixture model (GMM) in n-dimensional space: $$P_1(x) = \sum_{i=1}^{C}\pi(c_i)\mathcal{N}(\mu_i,\Sigma_i)$$ We want to estimate a single Gaussian distribution from ...
5
votes
2answers
147 views

Naively estimating the factorial

A naive way to estimate the factorial is $n! \geq (a+1) (a+2) \dots n \geq a^{n-a}$ for any $a$. For example, it gives $n! \geq (n/2)^{n/2}$ and slightly better $n! \geq (n/3)^{2n/3}$. I am interested ...
5
votes
2answers
330 views

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 ...
5
votes
1answer
64 views

An estimate of $C^2$

Let $u$ be a function on a domain $\Omega\subset R^n$, and $D^2u=\left(\frac{\partial^2 u}{\partial x_i\partial x_j}\right)_{n\times n}$ be the Hessian of $u$. If $D^2u$ is positively definite and ...
4
votes
1answer
208 views

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 ...
4
votes
1answer
148 views

Estimate the scale of $e^{-(m+1) t} \sum _{k=0}^{\infty } \frac{t^k}{k!}\left(\sum _{r=0}^k \frac{t^r}{r!}\right)^{m}$

I would like to estimate the scale of the following series, $$S(m,t)=e^{-(m+1) t} \sum _{k=0}^{\infty } \frac{t^k}{k!}\left(\sum _{r=0}^k \frac{t^r}{r!}\right)^{m},$$ where $e$ is the base of ...
4
votes
2answers
227 views

Bound for the Legendre function of the second kind of degree $1/2$

Let $Q_{1/2}(u)$ be the Legendre function of the second kind of degree $1/2$. One can show that $Q_{1/2}(u) = O(u^{-3/2})$ as $u\to \infty$; see Equation 21 in Section 3.9.2 of Higher transcendental ...
4
votes
1answer
63 views

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 ...
3
votes
4answers
181 views

Using $(1+x)^k \approx 1+kx$ to approximate?

Use the approximation $(1+x)^k \approx 1+kx$ to estimate $(1.0003)^{26}$ and $\sqrt[4]{1.006}$. I know how to solve this step-by-step, but I don't understand what I'm doing exactly: why does ...
3
votes
2answers
1k views

Modern formula for calculating Riemann Zeta Function [duplicate]

Possible Duplicate: How to evaluate Riemann Zeta function I have an amateur interest in the Zeta Function. I have read Edward's book on the topic, which is perhaps a little dated. I would ...
3
votes
1answer
86 views

Maximum likelihood estimation - why is $\mathcal{L}$ not the joint pdf?

Here's an excerpt from my notes: Define the likelihood function: $$\mathcal{L}(\vec{x};\theta)=\prod_{i=1}^{n} f(x_i;\theta)$$ Where $f$ is the pdf of the distribution we're sampling the $x$'s ...
3
votes
1answer
32 views

Oscillations about equilibrium for coupled differentail equations

I have the following system of equations: $$\begin{align} \frac{dX}{dt} &= 2Y-2\\ \frac{dY}{dt} &= 9X-X^3 \end{align}$$ I would like to study the property of solutions to this function about ...
3
votes
1answer
756 views

Method for estimating the nth derivative?

When using numerical analysis, I often find that I am required to estimate a derivative (e.g. when using Newton Iteration for finding roots). To estimate the first derivative of a function $f(x)$ at ...
3
votes
1answer
31 views

Why is the MLE a special case of the minimum contrast estimator?

In my statistics lecture, we had two definitions, namely Let $X_1,\ldots.X_n$ be iid random variables, each with density $p_{\Theta_0}(x)$. Furthermore, let $\varrho$ be a real function such that ...
3
votes
1answer
47 views

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 ...
3
votes
1answer
226 views

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 ...
3
votes
1answer
48 views

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 ...
3
votes
1answer
74 views

Estimating the input to a system from a system state

[ Cross-posted to: http://dsp.stackexchange.com/questions/3098/estimating-the-input-to-a-system-from-a-system-state-using-ekf ] I have a system for which I have obtained a non-linear time-varying ...
3
votes
0answers
61 views

Upper bound for area of polygons

is there a formula for an upper bound for the area of a polygon, knowing the length of its edges? In the ideal situation, the answer would be a function $f_n(l_1,l_2,\dots,l_n)$ of the edges lengths ...
3
votes
1answer
59 views

Error of apprximation

Have anyone read book "Paul Glasserman Monte Carlo MIFE", it's good, but i'm stuck in chapter 6 page 341 let $$ dX_t=a(X_t)dt+b(X_t)\,dW_t $$ they said that $$ \int_{t}^{t+\Delta t}a(X_{u}) \, ...
3
votes
1answer
99 views

Is my determination of this maximum correct?

Consider $\Omega:=B_1(0)\subset\mathbb{R}^n$ (it is the open unit ball), $\mathbb{R}^n$ is provided with the euclidean norm $\lVert\cdot\rVert_2$. Now I want to determine the following ...
3
votes
2answers
2k views

Maximum likelihood estimation of $a,b$ for a uniform distribution on $[a,b]$

I'm supposed to calculate the MLE's for $a$ and $b$ from a random sample of $(X_1,...,X_n)$ drawn from a uniform distribution on $[a,b]$. But the likelihood function, ...
3
votes
1answer
140 views

Best estimate for random values

Due to work related issues I can't discuss the exact question I want to ask, but I thought of a silly little example that conveys the same idea. Lets say the number of candy that comes in a package ...
3
votes
1answer
314 views

Determine whether a statistic is sufficient, given the probability density

Let $X_1, X_2, \dots, X_n$ be a sample of i.i.d. random variables, with density $$f_\theta=\frac{2}{3\theta}\left(1-\frac{x}{3\theta}\right) $$ for $0 < x < 3\theta$. And $f_\theta=0$ if $ x ...
3
votes
0answers
81 views

Numerically estimate $a^b$ [duplicate]

Possible Duplicate: How can I calculate non-integer exponents? What is the most efficient way to estimate $a^b$ ($a > 0$) numerically? My goal is not to use built-in math functions (like ...
2
votes
4answers
204 views

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 ...
2
votes
2answers
54 views

unbiased estimator in a random sample

I Have a statistic statement here which I need to decide if it's true or false Statement: "When the sample size is random, there is no way to get an unbiased estimator for the population average." ...
2
votes
1answer
110 views

Prove $ 1 \lt \int_0^1 \frac{1+x^{30}}{1+x^{60}} \ \mathrm{d}x \lt 1 + \frac{1}{30}$ [duplicate]

Possible Duplicate: How to prove $\int_0^1 \frac{1+x^{30}}{1+x^{60}} dx = 1 + \frac{c}{31}$, where $0 \lt c \lt 1$ How can I prove the estimate $$ 1 \lt \int_0^1 \frac{1+x^{30}}{1+x^{60}} \ ...
2
votes
2answers
84 views

What's the best strategy to count the eggs in the jar?

It's Easter time, and in my workplace we have a "Count the eggs in the jar!" kind of game. What would be the best mathematical strategy to get as close as possible to the correct count? Update: ...
2
votes
1answer
52 views

Expectation of the MLE $e^{-\frac{1}{\overline{X}}}$

I am having a bit of a problem with examining the properties of a maximum likelihood estimator. I feel like I am missing something simple, but I have been unable to find someone doing an example quite ...
2
votes
1answer
84 views

Does $\operatorname{MSE}(\hat{\theta}) = \operatorname{Var}(\theta)+ \left(\operatorname{Bias}(\hat{\theta},\theta)\right)^2$?

We know that $\operatorname{MSE}(\hat{\theta})=\operatorname{E}\left[(\hat{\theta}-\theta)^2\right]$ and $\operatorname{MSE}(\hat{\theta})=\operatorname{Var}(\hat{\theta})+ ...
2
votes
3answers
53 views

Series evaluated to $m$ terms, approximating the error

Given a series $\displaystyle\sum_{n=0}^\infty a_n$, how can we bound the error (which I shall denote with $R_n$) when we evaluate it to $m$ terms? $$\sum_{n=0}^\infty a_n \approx \sum_{n=0}^m a_n$$ ...
2
votes
1answer
221 views

Bayes Estimator

Let $X_{1},...,X_{n}$ be a random sample of size n from the continuous distribution with pdf: $f_{X}(x|\alpha,\beta) = ...
2
votes
1answer
251 views

How does this calculation of $\int_0^\infty(\sin^2x)/x^2\ dx$ work?

I'm having trouble with two steps in a calculation of $$\int_0^\infty\left(\frac{\sin x}{x}\right)^2\ dx$$ in a book. They take the contours $C_R$ composed of upper half-circles ...
2
votes
2answers
79 views

Error Term in Passing from Summation to Integral

I encountered the following in a paper and do not understand how the error term is being bounded. In what follows, $n$ and $k$ are large integer constants. $$ \sum_{i=0}^{n-1} \ln\left(1 - ...
2
votes
1answer
1k views

How to efficiently estimate quantile function of Gamma distribution

I have an application that analyzes datasets comprised mostly of samples from a Gamma distribution. Mixed in with the data are an unknown number ($>= 0$) of outlier samples (which are actually taken ...
2
votes
1answer
30 views

Estimating sum of n elements by throwing away half of elements

I've got a task where i need to proove the asymptotic big-Theta equation: $$ \log n! = \Theta(n \, \log n) $$ $ \ $ Since $f(\mathit{n}) \in \Theta(g(\mathit{n}))$ means that $g(n)\cdot k{_1} \leq ...
2
votes
2answers
124 views

Finding the variance of a statistic.

$X_1,\cdots,X_n$ are independent random variables from $N(\mu,\sigma^2)$ distribution. Define $$T=\frac{1}{2(n-1)}\sum_{i=1}^{n-1}(X_{i+1}-X_i)^2$$ I have shown that it is an unbiased estimator of the ...
2
votes
1answer
54 views

Estimating the number of edges in a subset-poset in terms of the total number of elements

Let $S$ be a finite set of sets, with $N = \sum_{s \in S} |s|$ the 'total cardinality' of $S$; i.e., the sum of the cardinalities of all the sets in $S$. Now, consider the poset $(S, \subset)$ of $S$ ...
2
votes
1answer
122 views

finding maximum likelihood estimate from dependent binomial rvs

let $X_{1}$, $X_{2}$, $X_3$, $X_{4}$ be iid bernoulli rvs with $\mathbb{P}(0)=0.5$, $\mathbb{P}(1)=0.5$. $Y_{1} = X_{1}+X_{2}+X_{3}$ and $Y_{2}=X_{1}+X_{2}+X_{4}$ $Y_{1}$, $Y_{2}$ are dependent ...
2
votes
1answer
195 views

Best and most efficient way to numerically compute $e$?

There are many well-known methods for efficiently numerically computing $\pi$, such as Chudnovsky's Method or perhaps Gauss-Legendre's algorithm. I was wondering what the best method for computing $e$ ...
2
votes
1answer
54 views

Combining statistical distributions

I have a situation where a distribution is dependent on 2 variables, one of which follows the poisson distribution, and the other the normal distribution, and I want to establish the method of ...
2
votes
1answer
47 views

Estimate for integral of sine to the power of $-(1+a)$ where $a>0$

I'm trying to solve or estimate this integral $$ I=\int\limits_{\arcsin{k}}^{\pi/2}\dfrac{1}{(\sin{x})^{1+a}}\mathrm{d}x, $$ where $0<k<1/2$ and $a>0$. The estimate should depend on $k$. I ...
2
votes
4answers
53 views

Can a sampling based method estimate how many species exist?

I've got in to a bit of a debate online and I'm hoping some people here can help clear it up. The position I'm arguing against is "It's impossible even come up with a ballpark estimate for how many ...
2
votes
1answer
313 views

Error Term for Fourier Series?

Suppose I have a piecewise smooth $2 \pi$-periodic function $f$ on $\mathbf{R}$ with a Fourier series $\sum_{n \in \mathbf{Z}}a_n e^{inx}$, a number $x_0 \in \mathbf R$, and $N>0$. I would like an ...
2
votes
1answer
289 views

Method of Moments and Maximum Likelihood estimators?

The random variables $X_1,...X_n$ are independent draws from continuous unifirm distribution with support $[0,\theta]$. Derive a method of moments and maximum likelihood estimators of $\theta$. Your ...
2
votes
1answer
98 views

What is the probability of the number 1 and number 2 employees getting the bonus at a call center?

Two weeks ago, a friend working at a call center told me about their staff bonus policy. Here I paraphrase it. Suppose employee A answers the maximum number ($N_1$) of calls among the staff, and ...
2
votes
2answers
45 views

Estimating Poisson $\theta$ only from which percentage of intervals have events

Radioactive particles are emitted randomly over time from a source at an average rate of per second. In $n$ time periods of varying lengths $t_1,t_2,\dots,t_n$ (seconds), the numbers of particles ...