For questions about estimation and how and when to estimate correctly

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0
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1answer
28 views

unbiased estimator for sample covariance?

I'm new to statistics and and I need some help: Let $X_1,...X_n$~$N(\mu_x,\sigma^2)$, $Y_1,...Y_m$~$N(\mu_y,\sigma^2)$. All r.vs. are i.i.d and $\mu_x,\mu_y,\sigma$ are unknown I was told that $...
5
votes
2answers
116 views

How can I recover a sequence of numbers given a corrupted version of it?

I have an unknown sequence of real numbers $x_i$ and a known sequence of real numbers $y_i$; $y_i$ is a corrupted version of $x_i$, i.e., $$y_i=x_i+n_i$$ where $n_i$ is a random number distributed ...
1
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0answers
25 views

Compute the limit of the error term of Taylor polynomial of $f(x)=e^{3x}-1$

I wonder how to compute the limit of error term in Taylor polynomial of exponential functions (i.e. $f(x)=e^{3x}-1$). First, I found the Taylor polynomial of $f(x)$, which is $$T_n f(x)=\frac{3^1}{1!}...
0
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0answers
52 views

Show that $P(X=-1)=p$ and $P(X=k)=q^2 p^k$, $k=0,1,2,…$ defines a probability distribution for the random variable $X$.

Let $p \in (0,1)$ and $q=1-p$. (a) Show that $P(X=-1)=p$ and $P(X=k)=q^2 p^k$, $k=0,1,2,...$ defines a probability distribution for the random variable $X$. (proved) (b) Given the single observation ...
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0answers
9 views

How to estimate a time step to obtain required accuracy when simulating linear dynamic system?

Assume the the linear dynamic system is $$\dot{x}=Ax$$ , and the initial state is $x_0$, where $|x_0|<K$, $K$ and the matrix $A$ is known. The state after time $t_0$ will be $e^{At_0}x_0$. The ...
0
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0answers
36 views

Point estimation of expected value - disease spread

Firstly I want to put big disclaimer here. This particular problem is a smaller part of my homework. Since even after discussion with my fellow classmates we are not sure how to handle it we decided ...
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0answers
25 views

Monte Carlo integration and variance

With the monte carlo integration of a function f(x), what do they mean with the variance? Is it the variance of the function we want to integrate? $I = ∫^{\inf}_{inf} f(x)p(x) dx$ (with p(x) some ...
1
vote
1answer
33 views

Understanding the point estimation of the expected value

I am trying to understand this problem, however I can't get past some of the definitions used when estimating the expected value. What I would need is to confirm or disprove my conclusions - I read ...
0
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1answer
64 views

Estimating $\sum\limits _{n=1}^k \sin \frac x n$ in the form $f(k,x) \sin(g(k,x))$

When you plot the function for a reasonably large $k$ ($300$ in this case) you get something like this... This seemed like it could be estimated the way I stated previously. The accuracy of that ...
2
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0answers
238 views

How can all players in the Starcraft 2 Grandmaster league win more than they lose?

Starcraft 2 is a competitive online strategy game where players compete in leagues with other players of similar skill. The most difficult and highest league is the Grandmaster (GM) league, which ...
2
votes
1answer
35 views

Estimating the number of books in the world from randomly chosen overlapping lists

Suppose I have lists $L_1 , \dots , L_n$ of, say, books. Assume further that these are uniformly chosen from the set of all books (probably unrealistic for obvious reasons, and if this assumption can ...
1
vote
1answer
64 views

Evaluate $\int_0^n \frac{|\cos nx| }{2+x^2+\sqrt{|\cos nx|}} dx$ or find good upper bound.

Is it possible to evaluate or at least to estimate the following integrals? $$\int_0^n \frac{|\cos nx| }{2+x^2+\sqrt{|\cos nx|}} dx$$ and $$\int_0^n \frac{|\cos nx| }{2+x+\sqrt{|\cos nx|}} dx$$ I ...
2
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0answers
47 views

Is Stirling's Approximation used here, to prove the asymptotic inequalities?

Define $N_c=[\dfrac{1}{2}n\log n+cn]$ where $[.]$ denotes the greatest integer function, and $c$ is any arbitrary fixed real constant. Also, let $M={n\choose 2}$. Then prove, for large $n$, the ...
2
votes
0answers
53 views

For what $r,s$ exist unbiased estimation of $f(p) = p^{r}(1 - p)^{s}$ for binomial distribution?

We have sample $x_1, ..., x_n$ generated by independent binomial random variables $\xi_1, ..., \xi_n$. We know parameter $k$ but don't know probability $p$. k is number of tests: $\xi_i \sim ...
0
votes
1answer
34 views

Estimation, Upper limits, Lower limits

Two rods of length 2.6 cm and 3.5 cm are measured correct to the nearest 0.2 cm. The two rod are joined together, find the lower and upper limit of the new rod. I get stuck. HOw to do?
2
votes
0answers
59 views

Source estimation for identification of anomalous events

I’m stuck on the following problem. There are two sources $S_A$ and $S_B$ at the ends of a channel. Both are made up of a white noise component $W_i$ plus an impulsive component $I_i$: $S_A = W_A + ...
1
vote
1answer
13 views

derivation of $\theta(x)=\int_{a}^{x}\varphi (y)dy - (\int_{a}^{b}\varphi(y)dy)\psi(x)$ and °L^2$-norm estimation

Let $I=(a,b)$, $u\in L^2(I)$ and $\psi\in C^{\infty}(I)$ such that $\psi=0$ on $(a,a+\epsilon)$ and $\psi=1$ on $(b-\epsilon , b)$ for sufficient small $\epsilon$. Let $\varphi\in C_C^\infty (I)$ and ...
0
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1answer
50 views

Estimate parameters of a quadratic function

Suppose that we have two data points which tell us about the output of some function $f(x)$: $(0, 50)$ $(10, 150)$ We know that the function is quadratic (so it's something like $ax^2 + bx + c$). ...
2
votes
1answer
13 views

Saturating space so that at least two lines are close enough

All lines in what follows pass through the origin. The only reason for the angle $2\pi/3$ below is that this is how I began wondering about these questions. Picture the unit disc $S^2$, by which I ...
0
votes
1answer
43 views

Why integral is equal to zero

I wonder why under assumption that w>>$\frac{1}{T}$ then $\int_{0}^{T} sin(wt)dt$ is approximately zero? Since the integral should be like- $\frac{cos(wt)}{w}$ from $0$ to $T$ and after plugging the ...
0
votes
4answers
116 views

How to estimate numbers like $(19/20)^{30}$

Is it possible to estimate by hand what is the value of expresion like $(19/20)^{30}$? $$19/20 = 0.95$$ but $$(19/20)^{30} \approx 0.2146$$ So it is totally different number.
0
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1answer
32 views

Unbiased estimator problem

Let $X_1, X_2,\dots, X_n$ be a sample of size $n$ from a distribution with unknown mean $−\infty<\mu<\infty$, and unknown variance $\sigma^2 > 0$. Show that the $Y = (X_1 + 2X_2 + 3X_3 +\...
0
votes
0answers
9 views

To find sharp infimum (lower bound) of function with indicator function

Let $(x_\varepsilon,y_\varepsilon)\in[0,1]\times[0,x_\varepsilon)$ be a sequence such that $(x_\varepsilon,y_\varepsilon)\to(x,y)\in[0,1]\times[0,x)$ as $\varepsilon\to0$. Is there an integrable ...
0
votes
1answer
45 views

what is the covariance between $\hat Y$ and$\hat \beta_1$?

I'm having a crisis of faith here, I'm trying to prove that $\beta_0$is unbiased. The formula for $\beta_0$(the parameter) is: $$\beta_0=\mu_Y-\beta_1\mu_X$$ The formula for $\hat \beta_0$(the ...
0
votes
1answer
18 views

How do i calculate the number of subintervals n in Midpoint method?

I want to calculate the least error (o) in order to obtain the exact answer for integration using the midpoint method. However I am having trouble doing so since i was given a functions whose second ...
0
votes
1answer
84 views

Estimating Cable Length on a Reel

I have been searching all areas of the internet to try and find a reliable formula for estimating cable length on a reel, I'm trying to create a faster and more reliable way to estimate cable to ...
2
votes
1answer
47 views

estimation of a series $3^n/( 4^n -1 )$

I am trying to show that the series $$ \sum_{i=0}^\infty \frac{3^i}{4^i-1}$$ is convergent, but do not see how to get rid of the one in order to get a bigger series. Thanks for helping.
2
votes
1answer
41 views

The maximum-likelihood estimators of $\sigma^2$

A sample of size $n$ is drawn from each of four normal populations, all of which have the same variance $\sigma^2$. The means of the four populations are $a+b+c$, $a+b-c$, $a-b+c$ and $a-b-c$. What ...
1
vote
2answers
44 views

Does the contour integral of a rational fraction function in the complex plane vanish in large radius limit?

Let $f(z)=\frac{z^m+az^{m-1}+\cdots+b}{z^n+cz^{n-1}+\cdots+d}$ be a rational fraction function of complex variable $z$, where the integers $n-m\geqslant 2$. Is the following integration limit $$\lim_{...
0
votes
1answer
36 views

Maximum Likelihood Estimator of $\theta$

I have the following question I tried to answer I got answer that same like this answer Is this true answer? (Note that: in the question $0<p<\frac{1}{2}$, but in this answer $...
1
vote
1answer
15 views

Does the correlation between the measurement noise influence the result of the LS?

Consider a linear measurement process with some noise: $$y=Hx+v$$ with $$v \sim \mathcal{N}(0,\Sigma)$$ the covariance matrix $\Sigma$ is not a diagonal matrix. As we know, using LS, the $\hat{x}$ is ...
0
votes
0answers
24 views

Statistics application problem - estimate number of items by weight

An experiment was conducted by weighting 20 sets of items with known quantity and the weight of items in each trail were obtained. We also know the weight of each item is supposed to be $W$ kg (...
2
votes
1answer
43 views

$\frac{1}{x^{\alpha}(\log x)^{\beta}}$ in $L^p((1,\infty))$?

Consider $$f(x)=\frac{1}{x^{\alpha}(\log x)^{\beta}},$$ $\alpha,\beta\in\mathbb{R}$. For which $\alpha,\beta \in \mathbb{R}$ is $u \in L^p((1,\infty))$, $1\le p\le \infty$? For $p=1$ I know how to do ...
1
vote
1answer
42 views

How to Find the Linear Approximation of $\ln(8-4x)$ at $x = 7/4$, and Use it to Estimate $ln(0.99)$

I am trying to determine how to find the linear approximation of $\ln(8-4x)$ at $x = 7/4$, and use it to estimate $\ln(0.99)$. So far, I have made the following steps: 1) Find the derivative of $\ln(...
0
votes
1answer
25 views

Estimation method starting with too big and too low values

I not sure to what field exactly this question belongs, but math/statistics seemed closest to me. So here we go: It is a method of estimating a value that informally goes like this (bear with me). ...
0
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0answers
16 views

Series Reversion for $n$ power series

I have $n$ functions with power series representation as $F_i(X)=\sum_{k_1,\dots k_m}a^{i}_{k_1,k_2,\dots k_m}x_1^{k_1}x_2^{k_2}\dots x_n^{k_n}$, where $X=[x_1,x_2,\dots,x_n]$ and $F(x)=[F_1(X),\dots,...
0
votes
1answer
29 views

MLE of β in the gamma distribution?

So I have the pdf for the gamma distribution, $$f(x) = \frac{1}{\Gamma(\alpha)} \beta^\alpha x^{\alpha - 1} e^{-\beta x} $$ and I'm having trouble getting to the MLE of $\beta$, which should be $\frac{...
0
votes
1answer
52 views

Generalizing Algebraic Problem

Molly went to the store to purchase ink pens. She found three kinds of pens. The first cost 4 dollars each; the price of the second kind was 4 for 1 dollar, and the cost for the third kind was 2 for 1 ...
0
votes
1answer
19 views

MLE for the mean of a distribution?

Let $X_1, \ldots, X_n$ be a random sample of size $n$ from a distribution having the pdf $$f(x)=\frac{2x}{\theta^n}.$$ I need to find the MLE for the mean of the distribution but am not sure how.
1
vote
4answers
66 views

Prove that $x_{n+1}=\frac{x_n}{2}+\frac{1}{x_n}\geq\sqrt{2}$

Let $(x_n)_{n\in\mathbb N}$ be a recursively defined sequence with $x_1=9$ and $$x_{n+1}=\frac{x_n}{2}+\frac{1}{x_n}\text{ for }n\geq 1.$$ Show that $x_n\geq\sqrt{2}$ for all $n$. Because $x_n\...
1
vote
1answer
19 views

estimates on an improper integral associated with normal distributuion

Show that $\int_{x}^{\infty}e^{-\frac{t^2}{2}}dt\geq e^{-\frac{t^2}{2}}(\frac{1}{x}-\frac{1}{x^3}) $ for all positive $x$ Does it require the mean value theorem, or the Taylor series expansion? It is ...
0
votes
3answers
38 views

Initial values of a exponential decay

How can I estimate the initials values ($A$, $B$, $C$) of a exponential decay? I got the function and a set of experimental points. $p(t) = Ae^{-1.5t} + Be^{-0.3t} + Ce^{0.05t}$ $p(0.5)=6,\ p(1)=4.4,...
1
vote
1answer
33 views

MLE of uniform distibution again

I've struggled for hours with a seemingly simple problem, I'm supposed to compute the MLE for $\theta$. We have $(y_1, y_2...y_n)$ obervations with a uniform distribution. The density function is as ...
1
vote
1answer
43 views

Deriviative estimates in finite $L^p$ space

I'm stuck as to how to solve the following exercise: If $U$ is open and bounded with smooth boundary, $1<p<\infty$, $\epsilon>0$, and $u\in C^{\infty}(\bar U)$, show $\exists C$ s.t. $||Du||...
0
votes
0answers
29 views

Exponential Family, derivative of Marginal Likelihood zero at MLE

My question refers to the proof of theorem 6.3 in Lehmann/Casella (1998): Theory of point estimation. We have a Bayes setting: $ X_i \mid \eta \stackrel{ind.}{\sim} f_i(x_i \mid \eta),\quad i = 1, .....
2
votes
0answers
44 views

Ratio estimator in sampling

Let the population $U=(1,2,3)$. We want to estimate $R=\frac{\mu_y}{\mu_x}$.Consider the estimators $$\hat{R_1}=\frac{\overline{y}}{\overline{x}},\hat{R_2}=\frac{\overline{y}}{\mu_x}$$ where $Y=(...
0
votes
0answers
39 views

How to propagate uncertainty into the prediction of a neural network?

I have inputs $x_1\ldots x_n$ that have known $1\sigma$ uncertainties $\epsilon_1 \ldots \epsilon_n$. I am using them to predict outputs $y_1 \ldots y_m$ on a trained neural network. How can I obtain ...
0
votes
0answers
6 views

How to find the partial likelihood estimates of $\rho_1$ and $\rho_2$?

I am having problems with the following exercise: I am given a model $$ X_t = \rho_1 X_{t-1} + \rho_2 V_t + \epsilon_t~~~~for~t\in \mathbb{N}$$ We have the intial value $X_0=0$, $\vert \rho_1 \...
0
votes
2answers
49 views

Numerical approximation of a dx/dy derivative

I have to find numerical approximation of the derivative of dx/dy where y(x)=exp(sin^2(x)+cos(x)exp(x^2)) at the point Xo=0.5. As far as I understand, I have to pick a close point to X0 for example 0....
0
votes
4answers
51 views

How do you determine how many digits of pi are necessary?

It is said that you only need to calculate pi to 62 decimal places, in order to calculate the circumference of the observable universe, from its diameter, to within one Planck length. Most of us are ...