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0
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0answers
21 views

Kernel distribution estimation

Due to an assignment I need to implement a algorithm based on KDE to schedule an input data in different servers. So far, I studied statistics in my bachelor but we did not go that far and they did ...
1
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0answers
57 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
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1answer
197 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
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1answer
73 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
votes
1answer
80 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
52 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? $$\sum_{n=0}^\infty a_n \approx \sum_{n=0}^m a_n$$ ...
1
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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 ...
0
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2answers
261 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 ...
1
vote
2answers
40 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
votes
1answer
124 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
57 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
35 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
vote
3answers
124 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 ...
2
votes
0answers
109 views

Cramer-Rao bound for $\chi^2$ distribution parameter estimates.

I've stuck in unpleasant problem with noncentral $\chi^2$ distribution. I work with random variables, distributed as $\chi^2_{\nu}(\lambda)$, where $\nu$ is the degree of freedom and $\lambda$ is ...
1
vote
1answer
159 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 $$ ...
0
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1answer
31 views

Estimate growing graphs

Lets make my scenario not generic just so that i could use particular terms Say i have a graph of population per year of someplace over some decades Lets say the graph is like this How can i ...
0
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1answer
66 views

Non-Linear regression

Imagine that I have a function $ f(x,y) $ to model a physical phenomenon. I believe that functions is defined by $$ f(x,y) = A*x + B*y + C*x*y$$ I have many values for $ (x,y,f(x,y)) $, how can I ...
1
vote
1answer
55 views

How to establish the estimate?

I consider the inequality as follows: let $a>0,b>0$, and satisfy $$2a^2-b^2\leq C(1+a).\tag{1}$$ If we assume that $b\leq a$, then there exist a constant $C_1>0$ such that $$a\leq ...
0
votes
2answers
49 views

Correlation bound

Let x and y be two random variables such that: Corr(x,y) = b, where Corr(x,y) represents correlation between x and y, b is a scalar number in range of [-1, 1]. Let y' be an estimation of y. An ...
1
vote
1answer
129 views

Interpret the terms in Strang's second lemma

The second lemma of Strang states that for a certain choice of $V_h$, $a$, $u$ and $f$ there exists a $c>0$ such that $$||u-u_h|| \leq c (\inf_{v\in V_h} ||u-v|| + \sup_{v\in V_h} ...
-1
votes
1answer
95 views

Estimating component variance for a sum of random variables

Say I have two zero mean single variate independent random variables $X$ and $Y$, and a third variable $Z = X + Y$. I can draw samples $z_i$ with $i = 1..n$ from $Z$ and I know $Var(Y)$. How can I ...
1
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1answer
369 views

Sufficiency and UMVUE for Poisson distribution

I need to show that $\hat\lambda = \bar X$ is a sufficient estimator for a Poisson distribution iid $X_1...X_n$, show that $\hat\lambda$ is the UMVUE for $\lambda$ and that $\hat\lambda$ is a ...
0
votes
1answer
47 views

Parameter estimation with GMM

I have estimated the parameters of normal distribution with GMM and got the following results: $mean = -0.01168 , p-value = 0.83519, Sd = 1.77 , p-value = 0.00000.$ I'm bit confused in ...
2
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1answer
210 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
228 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 ...
5
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2answers
263 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 ...
0
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1answer
37 views

estimation of limit / reducing of limit

I am trying to recalculate an exam and the solution for my problem is shown in the picture. How ever ...
0
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0answers
37 views

how to compute Wald Statistic for $\beta_2=0$ and $\beta_2-\beta_1=0$

$\sqrt{T}(\hat{b}-b)\sim N(0,E)$, where $E$ is the matrix $\begin{pmatrix}1&0.5\\0.5&3\end{pmatrix}$ and $b= (b1, b2)$. Q: what is the distribution of AX? Q2: what is the asymptotic ...
0
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1answer
25 views

How to obtain estimate of covariance matrix that will be guarantee to be semi-positive define?

How to obtain estimate of covariance matrix that will be guarantee to be semi-positive define ? (Is CrossValidated better place for this question ?)
2
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1answer
39 views

Multi-dimensional MLE Guassian

I wonder that what is the mu and sigma formula MLE(maximum likelihood estimates) for a 3 dimension guassian ? It is the same form as 1 and 2 dimension (+ 1 mu and sigma for the new vector) ?
1
vote
1answer
77 views

How do I use student's-t distribution without the sample size?

Here is my question (homework obviously): A sample from a normal population produced variance 4.0. Find the size of the sample if the sample mean deviates from the population mean by no more than 2.0 ...
2
votes
1answer
20 views

Estimation for large $k$.

I have a function $f(k)$ defined on the set of natural numbers and I managed to show that $f(k)>n-\binom n k(1-n^{-2/3})^{k(k-1)/2}$ for all integers $n\ge k$. I am hoping to get a further ...
2
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1answer
61 views

Some estimate concerning hyperbolic functions

I want to show that $|\sinh(az)|\leq|\sinh(z)|$ for all $z\in\mathbf{C}$ (or at least for all $z\in\mathbf{H}$, the upper half plane), provided that $0<a<1$ However, I am not even certain ...
2
votes
1answer
231 views

Inverse Laplace transform and Jordan's Lemma

I'm trying find the inverse Laplace transform $f(t)$ where I have $F(p)=\dfrac{9}{p(p+3)^2}$. I know $f(t)$ already to be $1-3te^{-3t}-e^{-3t}$. I have the integral $$f(t)=\dfrac{9}{2 \pi ...
2
votes
1answer
82 views

How to asymptotically estimate a lower bound of this function?

The function is given as $$f(x)\geq \sum_{i=1}^{[x/2]}f(i)+1$$ The boundary condition is $f(0)=0$. What I can get is this function grows faster than any polynomial function, and grows slower than ...
3
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1answer
46 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 ...
1
vote
1answer
89 views

arguing away - complex analysis

Probably a trivial question but I can't understand how to argue away the value of integrals in complex analysis. I am trying to find the inverse Laplace transform of $F(s)=\frac{1}{s(s+1)}$. The ...
2
votes
1answer
85 views

How many citations to read before convergence?

So I have the following question assuming I start with N academic papers, though I was thinking to make this simple I start with one academic paper. And say it has C citations, and each one of these C ...
1
vote
1answer
59 views

Some kind of trace inequality

What is the trick, to prove $\| u\|_{L^2(\Gamma)} \leq k \frac{1}{r}\| u\|_{L^2(\Omega)} + r \| \nabla u\|_{L^2(\Omega)} $ ? $\Gamma$ is one side of $\Omega:= [0,r] \times [0,r] $. I tried partial ...
1
vote
1answer
38 views

machine learning for a rule

I have data in the following form , where x is an integer and r is 0 or 1. I know that if $x < C$, then $r = 1$, if $x\geq C$, then $r= 0$. How can I automatically estimate the value of $C$? ...
1
vote
1answer
86 views

Inverse estimate of gradient of Sobolev function

I need an estimate for $\| \nabla w\|_{L^2{(\Omega \subset \mathbb{R}^n)}}$, such that it is $< c\| w\|,\ w \in H_0^1(\Omega)\ $. Is this possible?
2
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1answer
120 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 ...
1
vote
1answer
107 views

Estimating a sample mean using sub-sample means

we collect a lot of data on a daily basis via an API, and part of this data includes fields that represent sample means. Specifically, we get provided with a sample mean and the sample size, lets call ...
0
votes
1answer
107 views

How can I make estimates on large powers and logarithms such as $e^{10}$?

Just wondering, are there any useful tricks to make estimates of large powers or logarithms just by hand such as for $e^{10}$? Any such ways to get an error less than 1?
1
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1answer
61 views

How can I quantify the amount of space required to store all possible 128kilobit mp3s?

Somone has suggested that Within, say, a collection of every possible 30 second long MP3 file encoded at 128kbps, I'd probably be infringing on a few thousand copyrighted works. 128kilobits per ...
1
vote
1answer
91 views

Real Positive Zeros of Equation

During my research on physical problem, I faced the following simple equation: $r^{2k+1}+ab\,r-a=0$ With: $-1\leq k\leq1\:,\:0<r\:,\: a,b\in\mathbb{R}$ I need to put bounders on $a,b,k$ such ...
-1
votes
1answer
40 views

Estimate: $|f^{(3)}(i/3)|$.

Suppose $f:D(0,1)\longrightarrow \mathbb{C}$ is holomorphic, where $D(0,1)=\{z\in\mathbb{C}∣|z|<1\}$, and assume the maximum $|f(z)|\leq 2$. Estimate: $|f^{(3)}(i/3)|$. I just don't understand how ...
1
vote
2answers
80 views

Independent tests bound. (Chernoff/Azuma?)

I have a series of N Bernoulli tests (p, 1-p). I need to calculate a probability of passing more than N/2 tests, depending on N and p. The obvious solution is Chernoff bound: $\varepsilon \leq ...
2
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3answers
128 views

Precision with Taylor Expansions

when you take a 1st order taylor expansion of a function, so: $$f(a) + f'(a)(x-a)$$ does that mean that if the result is only accurate to one decimal place? so for a value a.bcd, d would be the ...
0
votes
1answer
50 views

System of equations: Can I solve this system of equations?

I want to ask you which field of mathematics contains following kind of problem. Suppose that following equations are given $\alpha\times x_{1}=C_{1}$ $\alpha\times x_{2}=C_{2}$ $\alpha\times ...