For questions about estimation and how and when to estimate correctly

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1answer
17 views

Estimation in maths which money involved [on hold]

Mrs smith sells 22 calculators at £4.24 each she estimates how much money she will collect.write down an estimation she could use to calculate the amount of money she will collect.
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0answers
7 views

Conditional Likelihood estimation

I'm reading a book called "Bayesian Reasoning and Machine Learning" and I have come across a question that gives me some problems. Unfortunately I neither have solutions, nor anyone else who I know ...
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2answers
43 views

Estimating integral $\int_0^{0.5} \ln(1+\frac{x^2}{4})$

Estimate the definite integral $\int_0^{0.5} \ln(1+\frac{x^2}{4})$ with an error of at most $10^{-4}$, using the Alternating Series Estimation Theorem. My approach is as follows: I found the ...
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1answer
22 views

How to prove Markov's inequality $P_X(X\geq t) \leq \frac{\mathbb{E}[X]}{t}$?

As the subject states, how can Markov's inequality $P_X(X\geq t) \leq \frac{\mathbb{E}[X]}{t}$ be proven? Is the proof distribution-dependent or there is a general way to prove it?
2
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0answers
29 views

Simulation Velocity of a harmonic oscillator system

I am write a simulation for get true Velocity of a harmonic oscillator system as Where P=[p1 p2;p2 p3] can find using Rung-Kutta Integration method with P(0)=[1 0; 0 1] This is code to find p Now, ...
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0answers
12 views

Simulation Position of a harmonic oscillator system

I am write a simulation for get true Postion of a harmonic oscillator system as Now, I want to write matlab code to get the true postion z of the system. However, ...
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3answers
31 views

Estimations in a ordered field?

My Problem: I am stuck with a proof strategy on the following: So i have got an ordered field $ (K,+,*,<) $ given. I also have $x,y\in K$ and $0\le y < x$ I have to proof that, for every n ...
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0answers
40 views

What exactly is a true value of a parameter?

I am currently studying the properties of the Maximum Likelihood Estimator. One of these properties being the asymptotic normality, I found the following equation: $$\sqrt{n}(\hat{\theta} - ...
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0answers
15 views

Fitted function - Which is better to use?

So I have some data for program running time, that follows a power law relation aN^b. I log-log plotted the data and saw that it became a straight line, so I calculated the slope of this line to get ...
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0answers
7 views

How to calculate the maximal error of a solution of a physical problem found numerically?

Assume we throw a body from a height h with the velocity v0 in some arbitrary direction. Beside the weight ...
0
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0answers
6 views

Sensitivity analysis of paramaters and input variables

I am trying to perform a sensitivity analysis of an optimization problem $f(x,\alpha)= \min_{ Q} {g(x,\alpha , Q)}$ where $x$ is an input variable for our function, and $\alpha $ is a parameter. ...
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0answers
21 views

Estimation in Multiplicative and additive Noise problem

I have unsolved problems below for random process homework. Consider the problem of estimating a random vector $\underline{S}(u)$ from an observation $\underline{X}(u)$, where ...
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1answer
28 views

Question on probability and approximation

Okay I think you are all familiar to YouTube videos and some facts are: to comment, like and dislike on a video you need a Google account. when someone views the video the view count of the video ...
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0answers
21 views

doubled estimation of a series 1/k^2

how can one show, that: $\frac{5}{4}\leq\sum_{k=1}^{\infty}\frac{1}{k^2}\leq\frac{7}{4} $ I can show by the estimation $\frac{1}{n^2}\leq\frac{1}{n(n-1)}$, that $\sum_{k=1}^{\infty}\frac{1}{k^2}\leq2 ...
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0answers
43 views

Maximum Likelihood Estimation with Laplace Distribution

I want to estimate the parameters $a$ and $b$ of the model $y_i = ax_i + b + \varepsilon_i, i=1,...,n $ via Maximum Likelihood. The $\varepsilon_i$ are assumed to be Laplace-distributed with density ...
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0answers
16 views

Least Square estimator: estimating a parameter from a simple signal model

Assume the following signal model: $b_{i,j} = c_i\;d_j \\ b_{j,i} = c_j\;d_i$ where $i,j = [1,2,...,M]$ with $i \neq j$. Both $c_i, d_i \; \forall \; i$ are mutually independent complex random ...
1
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1answer
39 views

Proof about Simple random walk in $\mathbb{R}^{d}$

I have read something about random walks in $\mathbb{R}^{d}$. The random walks is assumed to be started at origin. There is a theorem said that For $d =1$ or $2$, the random walk is recurrent. (i.e. ...
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1answer
13 views

Prevalence estimates based on randomized sample of clinical data

This is probably one of the more straight forward questions on here but here it is: I want to use a random number generator to sample X number of charts to look for the # occurrences of Y event. So ...
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0answers
9 views

Efficient estimator with exponential noise

Let us consider the model : $x(n) = \theta + b(n)$ with $\theta$ a constant to estimate and $b(n)$ a noise with an exponential density of probability : $p(b(n)) = \lambda e^{-\lambda b(n)}$ so ...
2
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0answers
17 views

Risk in density estimation: grasping the definition

When generalizing estimators to an entire function what is the space in which we perform the integral to obtain the expected value (with respect to this function)? For example, when estimating ...
1
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0answers
31 views

The Hessian Matrix I calculate is twice as much as it should be. Why?

I have a function "fkt." In this example, let it be as simple as $y=a \cdot x+b$. I have a real dataset with values obeying to the model. After regression of the points to the model, I find the ...
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1answer
64 views

Estimation of $\int_x^\infty \frac{e^{-t}}{t}$

How can you show $$\int_x^\infty \frac{e^{-t}}{t} dt\geq \log(1/x)-1 \text{ for }x>0$$? It's quite straigtforward to prove the estimation in the other direction for log(1/x)+1, but I fail ...
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0answers
35 views

Accelerometer data integration (MMSE)

Based on the raw accelerometer measurements use simple integration on the raw $X$ and $Y$ axis data to determine the velocity and position. If we assume a linear model $Y = aX + b$ for determine the ...
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0answers
19 views

asymptotic unbiasedness of weibull mle

It's known that the MLEs of the two-parameter Weibull distribution scale and shape parameters are not available in a closed form. It is, however, known that they do exist, are unique, and moreover, ...
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0answers
19 views

Estimation with logarithm

I have to prove: $q$ an integer $\geq 2$, $\tau$ a real number $\geq 2$ and $\sigma \leq 1- \frac{1}{\log q\tau}$, $M\geq 0$. Then $M^{1-\sigma }\leq (q\tau )^{-\sigma }$ and $(M+1)^{-\sigma }\leq ...
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0answers
21 views

Uniform Distribution Min & Max unbiased estimators?

I have $n$ samples ($iid$) from a uniform distribution over $[a,b]$ with unknown $a$ & $b$, let call them $x_i$. Let $Min\{x_i\}=\hat{a}$ and $Max\{x_i\}=\hat{b}$. What are unbiased estimators ...
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1answer
15 views

Maximal value of $\vert r^2-n\vert$ with a special condition

let $M,n\in \mathbb{N}$ and $R=\lbrace r\in \mathbb{N} \mid \vert r- \sqrt{n}\vert <M<2\sqrt{n}\rbrace $. I have to show that the maximal value of $\vert r^2-n\vert $ for $r\in R$ is at most ...
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0answers
16 views

estimation question (I should be able to solve it but no, I failed)

Given: $ a = \frac{r+i}{r-i} $ $ b = \frac{r+j}{r-j} $ $ 1 < a < b \le 2a << r $ $ 0 < i < j << r $ How to estimate r given a and b?
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0answers
43 views

Calculating the Z-multiplier and Standard Error in Confidence Intervals

If someone can explain the process of working out the z-multiplier of the z-table. I mean, how do we actually calculate instead of looking up on the table? (my primary question) Also how do you ...
1
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1answer
34 views

asymptotic normality and unbiasedness of mle

Suppose $\hat{\theta}_n$ is the MLE for some parameter $\theta$. Suppose also that the MLE is such that the Cramer regularity conditions are fulfilled, and $\hat{\theta}_n$ is asymptotically normal ...
2
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3answers
128 views

Estimating the series: $\sum_{k=0}^{\infty} \frac{k^a b^k}{k!}$

Any idea on how to estimate the following series: $$\sum_{k=0}^{\infty} \frac{k^a b^k}{k!}$$ where $a$ and $b$ are constant values. Greatly appreciate any respond.
2
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1answer
23 views

Find the largest value for $x_1$ in (0,1) such that $f(0.5)-P_2(0.5) = -0.25$ (interpolation)

I'm not really sure where to go with this problem and I'm hoping you can help. The problem states: Let $f(x) = \sqrt{x - x^2}$ and $P_2(x)$ be the interpolation polynomial on $x_0 = 0, x_1$, and ...
0
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1answer
23 views

linear regression model beta estimate

Suppose we want to estimate $\beta$ by minimizing $L(\beta)=\sum_{i=1}^n(y_i-\beta x_i)^2+\lambda|\beta|$, where $\lambda$ is a fixed positive constant. Calculate the estimate. How would I ...
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1answer
36 views

How this integration is solved?

Can anyone explain how this integration has been performed? This is a Bayes estimator for uniform prior assuming quadratic loss function. Thanks in advance
1
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1answer
42 views

Lagrange interpolation: Getting a bound and finding the error

I am struggling to understand this: The problem asks me to find the lagrange error of the polynomial approximation given the nodes $x_0 = 1, x_1 = 1.25, x_2 = 1.6$ with $x = 1.4$ The function I am ...
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1answer
38 views

$\sum _1 ^n |z_j| \ge 1 \Rightarrow | \sum _1 ^k z_{j_m}| \ge C$

Prove that there exists $C > 0$ such that the following implication holds: If $\{z_1, ..., z_n \} \subset \mathbb{C}$ are such that $\sum _{j=1} ^n |z_j| \ge 1$, then there exists $ \{z_{j_1}, ...
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0answers
24 views

Recursive Bayesian Estimation, $p(C_k|x)$ as likelihood

I''ve been struggeling with this problem for the last couple of days. The main goal is to use the probabilistic classification output $p(C_k|x)$, from for example a logistic regression, to enhance ...
1
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1answer
77 views

how to apply non-linear least square

I'm trying to implement the example of estimating an angle between a target $\textbf{x}$ and a sensor $x_{p}$. I'm using the example in this book. There are three available measurements of the angle ...
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1answer
22 views

Subset of samples has any effect on sufficiency of the statistic?

If we have the following iid samples $$ X_1, ..., X_n \sim N(\mu, \sigma^2) $$ where only $\mu$ is unknown. We know one sufficient statistic is the following: $$ T = \frac{1}{n} \sum_{i=1}^n X_i $$ ...
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0answers
20 views

How do I determine a sample size for Poisson distributions?

An average ball player scores 20% of his penalties. How many penalty results would I need to receive before I can say that a player scores at a rate of 15% requires coaching on his penalty shots? ...
0
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1answer
50 views

Sample size required to estimate population proportion with given precision

It plans to conduct a study on the percentage of homeowners who have at least two TVs. What should be the sample size if we want to ensure that $95\%$ of estimation error is less than $0.01$? ...
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0answers
17 views

Polynomial Expansion Proof Estimation

1-(1-x)^n where x is a value between 0 and 1 and n is a large value. This estimates to around x*n. I am having trouble with the polynomial expansion. According to Pascal's triangle, the first few ...
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0answers
30 views

Normalisation of Monte Carlo overlap

Background: In quantum mechanics you are sometimes required to compute the overlap (inner product) of two wave functions (square integrable complex functions) $\Psi(x)$ and $\Phi_i(x)$ as $$ ...
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0answers
28 views

Continuous RV - minimizing absolute deviation

We try to find c value minimizing E[|x-c|], "expected value of absolute deviations", for a continuous random variable X. E[|x-c|]=Integral(-inf,inf)[|x-c|]f(x)dx ...
3
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1answer
81 views

Estimate large covariance matrix using few samples.

Let $\mathbf{x}$ be a random vector in $\Bbb{R}^n$, such that $\mathbf{x}\sim N(\bar{\mathbf{x}}, \Sigma)$. $N$ observations of $\mathbf{x}$ are available, say $\{\mathbf{x}_i, i=1,\ldots,N\}$. The ...
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0answers
21 views

“Interpolating between estimates”?

the headline reproduces the whole problem. What is meant by saying "Interpolating between the estimates (A) and (B), we finally obtain..."? For beeing mor specific I'll give the concrete estimates ...
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1answer
28 views

Is $-\log(x^2) \geq 2(1-x)$ for $x \in (0,1]$?

I want to show that $-\log(x^2) \geq 2(1-x)$ for $x \in (0,1]$, where $\log$ is the logarithm to base 2. How can I do that? I tried to make an estimate by first bringing the minus to the other side ...
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1answer
29 views

what is the probability that the contractor's estimate will be within 5 weeks of the true mean

A contractor uses sample mean lifetime $x'$ of $250$ compressors as her estimate for population mean lifetime m of all new compressors. If this brand of compressors has a standard deviation of $35$ ...
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0answers
16 views

Show that the found value is the MLE

Let $ X_1, ... X_n$ i.i.d with pdf $$f(x;\theta)=\frac{x+1}{\theta(\theta+1)}\exp(-x/\theta), x>0, \theta >0$$ It is asked to find the MLE estimator for $\theta.$ The likelihood function is ...
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0answers
16 views

Unbiased estimator for maximum

Assume $n$ independent random variables with unknown distributions $\{X_1,X_2,...,X_n\}$. Multiple "samples" or observations for each of these variables are given (not necessarily with the same ...