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1answer
70 views

Estimate divergence by gradient in H1

I am currently trying to fully understand the stationary Stokes equations of incompressible fluid. In the mixed form (homogeneous boundary data), for $\Omega\subset\mathbb{R}^d$, $d\in\{2,3\}$, a ...
0
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1answer
48 views

Order of magnitude of a variable.

What will be the order of magnitude of a variable whose value varies between 0 and 1? And why?
1
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0answers
42 views

Approximating arccos(a/(a+x)) for the sake of simplfying an integral

I recently tried to evaluate $$\int e^{\beta\arccos(a/(a+x))}dx$$ (everything constant except $x$) and got a complicated answer involving a hypergeometric series with complex arguments. Can anyone ...
2
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1answer
70 views

Inverse of the German Tank Problem?

I have a problem that maps to estimating the discrete distance to a goal. The sample space is n discrete positions on a circle labeled sequentially; n is known. A target position is randomly ...
1
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1answer
92 views

Find $\sin(1/10)$ to within error of $10^{-7}$

The maclaurin series of $\sin(x)$ is $x- x^3/3! + x^5/5! - \cdots + (-1)^n x^{2n+1}/(2n+1)!$. My teacher wants me to use Taylor's inequality theorem on page 607 to solve this problem. I know that ...
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2answers
147 views

How to calculate this ratio for use in pro-rata forecasting

This is really a very simple question, it's more the understanding I need rather than a simple answer. If I have two arrays, A with 10 elements (A1, A2, ...), and another, B, with 5 elements (B1, B2, ...
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2answers
176 views

Estimate a value knowing the values of: the function, the derivative and the second derivative in 0

Please suppose you have an unknown function r(x). This function r(x) is defined in the range: [-5; 5] You know that: r(0) = 1; r'(0) = -1; r"(0) = 1. Please estimate the value of r(x) in the ...
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2answers
133 views

Maximum likelihood function (MLE) for Levy distribution

I am a student who is writing a little thesis belonged in the applied mathematics category. I choose a "Levy distribution" defined as, \begin{equation} \lambda(t;u,c) = \begin{cases} ...
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1answer
88 views

Finding parameters for curve fitting

I have 500 observed data of variable $ x $ and corresponding $ y $. The functional model is where Is it possible to find suitable constants $ A , B $ ,$ \alpha , \beta $ so that the observed ...
1
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2answers
478 views

Biased/Unbiased estimator

I'm trying to solve a statistic exam and i got lost with this exercise. 1) Consider a sample from a continuos probability distribution with density: $$ f(x) = \begin{cases} (1+\theta x)/8 & ...
0
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1answer
126 views

square root estimator

Let's say we want to do an estimation using iid samples $X_i, i=1,2,3,..., N$ the following formula, $$\hat{X}_1 = \frac{1}{N}(\sum_i\sqrt{X_i})^2$$ square sum of square roots. This form also seems ...
1
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1answer
188 views

Using the estimation lemma

I have the question: Prove using the estimation lemma, for a function $f$ which is continuous in some region $D$ that: $\lim_{r \mapsto 0}\displaystyle\int_{\Sigma}\dfrac{f(z)}{(z-z_0)}\ dz = 2\pi ...
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0answers
49 views

How to find conditional distribution of outputs given input

Suppose we have a training dataset $x_1, y_1, \dots, x_T, y_T$. Our goal is to empirically estimate $P(y|x)$ using histogram of data. What histogram should I exactly use? One option is to use the ...
1
vote
1answer
137 views

Cauchy Derivative Estimates for entire functions with a bound.

The problem statement: Assume $f$ is an entire function and that there is an $n \in \mathbb{N}$ and a $C < \infty$ such that for $z \in C$ $$|f(z)| \le C ( 1+|z|^n)$$ Also assume that $f$ is never ...
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8answers
721 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 ...
1
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1answer
234 views

Weibull Scale Parameter Meaning and Estimation

Wikipedia: http://en.wikipedia.org/wiki/Weibull_distribution gives a nice description on what the shape parameter (they call it k) means in the Weibull distribution, but I can't find anywhere what the ...
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1answer
231 views

MLE of Poisson Variable

Consider a random sample of size $n$ from a Poisson distribution with mean $\mu$. Let $\theta=P(X=0)$. Find the MLE of $\theta$ and show that it is a consistent estimator. --We have ...
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2answers
68 views

Poisson Estimators

Consider a simple random sample of size $n$ from a Poisson distribution with mean $\mu$. Let $\theta=P(X=0)$. Let $T=\sum X_{i}$. Show that $\tilde{\theta}=[(n-1)/n]^{T}$ is an unbiased estimator of ...
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0answers
30 views

Confidence Interval for the Survival Probabiltiy

Let $X_{1},\dots,X_{n}$ be random samples from an exponential distribution with pdf $f(x)=\mu^{-1}\exp(-x/\mu)$ over $x\geq0$ ($0$ otherwise) with $\mu>0$. In this case, $\mu$ is the mean. Find ...
1
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1answer
211 views

Gaussian Curve Fitting - Parameter Estimation

I was redirected here because someone in SO pointed out this is more of a math question than a programming question: I have to fit a Gaussian curve to a noisy set of data and then take it's FWHM for ...
1
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2answers
1k views

Fisher Information for Geometric Distribution

Find the Cramer-Rao lower bound for unbiased estimators of $\theta$, and then given the approximate distribution of $\hat{\theta}$ as $n$ gets large. This is for a geometric($\theta$) distribution. ...
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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 ...
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0answers
68 views

Estimation (Newton potential)

Let $f$ be a $C^2$-function on $\mathbb{R}^n$ with compact support. Let $N$ be the Newtonpotential on $\mathbb{R}^n$ and $u:=N\star f$ (i.e. a solution of the potential equation $\Delta u=f$). ...
0
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1answer
119 views

What is $\sum\ln{(x_i!)}$?

I started learning statistics and in my homework i should find the Maximum Likelihood Estimate. The function is $f_x(x)=e^{-\lambda n}\prod_{i=1}^n \frac{\lambda^{x_i}}{x_i!}$ Now i take the ...
1
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1answer
133 views

Evaluation/Estimation of a Gaussian integral

Is there a closed form expression for the following definite integral: $$ F(u) = \frac{1}{2}\int_{-u}^u e^{-\frac{\alpha^2}{x^2}-\beta^2 x^2}\,dx = e^{-2\alpha\beta} \int_0^u ...
2
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1answer
288 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 ...
0
votes
1answer
59 views

MLE problem - the likelihood function has no maximum.

The probability density function is: $f(x)=e^{\theta -x}, \ 0 \le \theta \le x $ Given an n-element sample, the likelihood function is: $$L(\theta)=\exp \left( n\theta - \sum_{i=1}^n x_i \right)$$ ...
2
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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 ...
1
vote
1answer
174 views

$\mathbb E[\frac{\partial}{\partial\theta}\log f(X;\theta)]^2$ and $\mathbb E[\frac{\partial^2}{\partial\theta^2}\log f(X;\theta)]$

$f(x;\theta)=\frac{1}{\pi[1+(x-\theta)^2]}$; $-\infty<x<\infty,\quad-\infty<\theta<\infty$ $\log f(x;\theta)=\log (\frac{1}{\pi[1+(x-\theta)^2]})$ $\Rightarrow \log ...
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0answers
21 views

Data estimation based on progression

Given a data-set $x$ and $y$. x | y ------------------ 153,000 | 0.058848 332,641 | 0.36352 506,629 | 0.53 If $x$ being the number of database records ...
0
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1answer
41 views

MLE estimation for number of customers.

A clerk in a shop has noticed two customers arrived at the shop between 12:00 and 12:45. Another clerk noticed only one customer between 12:15 and 13:00. Assuming a Poisson distribution on the number ...
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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, ...
2
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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 ...
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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 ...
4
votes
1answer
147 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 ...
1
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1answer
58 views

Statistical inference, estimation, conceptual trouble

I've just begun learning about statistical inference and I'm having a bit of trouble understanding the concepts at hand. The exercises I've done and lectures I've read kind of gloss over the details ...
0
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1answer
980 views

Finding an unbiased estimator for the negative binomial distribution

Consider a negative binomial random variable Y as the number of failures that occur before the r th success in a sequence of independent and identical success/failure trials. The pmf of $Y$ is ...
1
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0answers
59 views

Fast way to estimate cardinal number of subset

I have a large set $S$ of items, but the set is not exactly known. All I know are the cardinal numbers of categories i.e. a number of disjoint subsets, $ \vert{S_1}\vert \dots \vert S_n\vert$ with ...
4
votes
1answer
207 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 ...
0
votes
1answer
74 views

Generalisations of the Gronwall's lemma

Suppose we have the following differential inequality $F''(w)\le \frac{p-1}{p}\frac{(F'(w))^2}{F(w)}$ on $w\in(w_0,w_0+\varepsilon)$, $w_0>0$, $\varepsilon>0$, $p>1$. In addition, ...
2
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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
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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? $$\sum_{n=0}^\infty a_n \approx \sum_{n=0}^m a_n$$ ...
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0answers
22 views

Is there a way to estimate the range of fitting coefficients from only the data?

Considering an approximation $f$ for a set of $N$ data points $(x,y)$ using, for example, $M$ radial basis functions at arbitrary sites in the domain $f_i = \sum_{j=1} ^M c_j\phi(||x_i-x_j||)$ where ...
1
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2answers
322 views

Mental Math - Estimating Logarithms

How can we estimate logarithms with different bases? Take $\log_2 10$ ($1\over\log_{10}2$$\approx3.32192809$) for example. If we convert $10$ to binary, we get $1010_2$. So $\log_21010_2$ can clearly ...
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2answers
41 views

Expected value of total accumulated lifetime (understanding gap in proof)

Problem: I understand the first line $E(T) = ...$ However, I don't get the next two steps. I feel like I almost get it. It's like we are factoring out a $\sum_{j=1}^{20}$ but how did he ...
3
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1answer
139 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 ...
0
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1answer
58 views

Likelihood of the mean of one random variable with unknown parameters greater than another

Assume we have two random variables $X$ and $Y$ that are gamma distributed (or normally distributed, if it makes the math easier) with unknown parameters. We have samples $x_1,x_2,...,x_m$ and ...
0
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1answer
37 views

Approximation question

Cars and buses arrive at a bridge according to the independent Poisson processes at a rate of $3$ cars/minute and $1$ bus/ minute. What is the chance that strictly more buses arrive than cars in a ...
1
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3answers
125 views

Let $ X_1,X_2,…,X_n$ be i.i.d. $N(\theta_1, \theta_2)$, please prove that $E[(X_1-\theta_1)^4] = 3\theta_2^2$

If $X_{1}$, $X_{2}$, ..., $X_{n}$ is sampled from $N(\theta_1, \theta_2)$, how can I prove that $E [(X_{1} - \theta_1)^{4}] = 3 \theta_2^{2}$? I started off this question finding the completely ...
1
vote
1answer
185 views

Numerical calculation of fisher information

I am trying to obtain numerically the fisher information. Given a likelihood function $$ f(X,\theta),$$ with $X \in [0,1]$. The fisher information is given by $$ ...