Questions about the statistical process of sampling from a population, in order to obtain information for use in statistical learning, estimation, hypothesis testing about some population or process.

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2answers
55 views

Errors and Residual

Why are errors independent but residuals dependent? As far i know the sum of the residuals within a random sample is necessarily zero, and thus the residuals are necessarily not independent. But also ...
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1answer
85 views

Uniform sampling with replacement item frequency

Suppose we are sampling from $N$ distinct items uniformly with replacement $M$ times. What can be said about the distribution of frequencies of items drawn? For example, if I sort all the frequencies ...
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1answer
109 views

Why would the discrete fourier transform “see” signals like this? What is the origin of spectral leakage?

The discrete fourier transform of $x = (x_{0},\dots,x_{N-1})$ is defined as $\displaystyle X_{k} = \sum_{n=0}^{N-1} x_{n}\omega^{kn}_{N}$ where $\omega^{kn}_{N} = e^{-2\pi ik/N}$ and ...
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2answers
81 views

How to pick a random sample of given size from a set of unknown size

I have the following problem: I'm reading huge amounts of data records; when done, I would like to display one or more randomly selected records from the data set. It's easy to do if I can cache the ...
2
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1answer
44 views

Sampling with an “oversampling” factor, in K-Means||

I'm trying to understand K-Means||, a scalable version of K-Means++, which itself is an "improved" version of the clustering algorithm K-Means. Please find here the link to K-Means|| paper ...
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1answer
47 views

Computing P-value

In a book, from a sample they derived Mantel-Haenszel chi-square statistic $$\chi_1^2=1.41$$ And it is written that : this $\chi_1^2=1.41$ is associated with a one-sided P-value between $0.10$ and ...
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1answer
28 views

Estimating the mean and variance of numbers assigned to each person in population of one billion

Problem : Consider people of one billion, and each has one card containing one number. For instance first has card of number $7$. second has card of number $11$ and so on (simply if number means age ...
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1answer
63 views

Variance- covariance matrix

Consider $H$ denotes hat matrix and $e$ denotes residual. In the book Applied regression Analysis by Draper/Smith, it is written that : $\mathbb V(e_i)$ is given ...
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1answer
92 views

Confidence Interval for Regression Coefficient ,$\beta$

In the book 'Applied regression Analysis' by Draper/Smith, it is written that : Obtain individual $100(1-\alpha)\%$ confidence interval for the various parameters separately from the formula ...
1
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1answer
108 views

Random points inside a convex polytope

Given a convex polytope, defined by set of vertices $P = \{\mathbf{x}^{(i)}\}_{i = 1}^n, x^{(i)} = (x^{(i)}_1, x^{(i)}_2, \dots, x^{(i)}_d): \operatorname{conv}(P) = P$. How to generate uniformely ...
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1answer
62 views

Building histogram of latency using sample of input data

Assume I have input data set consisting of a web page response time. I'd like to build histogram from input data, but for practical reasons I can only use sample of data. Based on histogram I want to ...
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1answer
174 views

Stratified random sampling without replacement

I came across this statement and can't decide if it's true or false. Statement: In a stratified random sampling without replacement, with proportional allocation to the population size, the sample ...
2
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2answers
87 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
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4answers
144 views

Is it true that $\mathbb E[{\frac{X}{Y}]}={\frac{\mathbb E[X]}{\mathbb E[Y]}}$?

If $X$ and $Y$ are both random variables, does it hold $$\mathbb E\left[\frac{X}{Y}\right]={\frac{\mathbb E[X]}{\mathbb E[Y]}}$$ ??
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0answers
41 views

Spherical Sampling of Projected Disk

Given a 2D solid disk centered at some 3D point $\vec{x}$ with radius $r$ and normal $\vec{N}$, I need to compute a random unit vector from the origin that hits the disk. The vector needs to be ...
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1answer
35 views

generating random samples with a PDF

I have the PDF of a distribution from which it is not possible to get a closed from for the CDF or inverse CDF. Is there a technique that would allow me to generate samples from this distribution ...
1
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1answer
68 views

Non-uniform sampling of N-sphere

Suppose I have a unit $N$-sphere from which I want to draw points at random. To obtain uniformly distributed points I do the usual technique of drawing $N$ random variables $x_i$ from a Gaussian ...
0
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1answer
36 views

How to estimate amplitude of a sinusoid from two samples

Given a sinusoid $x(t)=A\cos(t+\theta)$, I can estimate the amplitude $A$ if I take two samples separated by $\frac{\pi}{2}$. If $X_1=x(0+\theta)$ and $X_2=x(\pi/2+\theta)$. Then, $A$ can be estimated ...
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1answer
38 views

Sampling on Axis-Aligned Spherical Quad

Given spherical coordinates on a unit sphere, imagine a spherical quad defined by two ranges $[\phi_0,\phi_1]$ and $[\theta_0,\theta_1]$. If you have a globe, for example, the grid formed by the ...
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1answer
32 views

Confidence Intervals and Inferences

I need help with c, as I have attempted a and b already, but believe they help with context. Suppose you took a random sample of 100 accounts in a large department-store chain, and found that the ...
1
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1answer
19 views

Joint Distribution and Sampling Distribution

There are three different incomes, x, and their proportions, f (x). $ 10,000 0.40 $ 30,000 0.40 $ 50,000 0.20 How do I calculate the joint distribution for X1 and X2, which are a random ...
2
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0answers
23 views

Sampling from a graph

Suppose you have a graph $G=(V,E)$ that is unobservable globally and you wish to take a sample from the vertices of that graph to infer something about its global properties from local properties. ...
4
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2answers
115 views

To sample or not to sample?

I have the following issue (I want to predict sports results). Let $X$ be a discrete RV with $m$ possible outcomes, each having probability $p_i$, for $i=1,\ldots,m$. Assume that I have a large iid ...
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1answer
362 views

Trying to use Cholesky decomposition of covariance matrix to sample error ellipsoid

I'm trying to construct an error ellipsoid from a covariance matrix (which exists for a 3D point) and then sample consistent xyz points in this region. In a previous question when I asked about this ...
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1answer
468 views

Sampling error with weighted mean

I am studying statistics and I am wondering when it comes to standard error or a sampling if the calculation changes when there are weights added. I have a weighted mean: $$\mu_{w} = ...
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2answers
85 views

Why is there a difference between a population variance and a sample variance

Sorry if this answer is simple but I was wondering why is there a difference between a population variance and a sample variance? I understand The variance is calculated as: $$\text{Var} = ...
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0answers
23 views

Convergence of sampling from Brownian motion

For a standard linear Brownian motion $\{B(t)\mid\ 0\le t\le 1\}$, for natural $n\ge 0$ and natural $1\le k\le 2^n$, let $d(n,k)=B\left(k2^{-n}\right)-B\left((k-1)2^{-n}\right)$ be the differences of ...
4
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2answers
75 views

What does it mean to sample, in measure theoretic terms?

Suppose I have some random variable $X$ defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$. What does it mean, in measure theoretic terms, to draw a sample from $X$? When $\Omega$ ...
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1answer
153 views

Consequences of violation of independence assumption in ANOVA test

Can you please explain me what is the consequences of violation of independence assumption in ANOVA test? I couldn't visualize it. Does Cochran Theorem relates to ...
2
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2answers
77 views

Compressive Sensing matrix

I am working with compressive sensing recovery with image and I test with various sensing matrices: Case 1: Sensing matrix A of size MxN is i.i.d Gaussian matrix. Case 2: Sensing matrix A is size of ...
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1answer
97 views

How to mathematically prove that we are sampling from same distributions?

The content of this question is about rigorously proving something which is otherwise considered easily correct intuitively. Let's assume we have a multivariate distribution $g(x_1,x_2,...,x_n)$ over ...
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1answer
56 views

Confidence/Tolerance interval for a percentage of a population

I have a problem I'm not sure how to solve. It goes something like: ...
2
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1answer
108 views

Non-i.i.d Empirical Risk Minimization

I'm not a statisticians so please forgive me if I posed a silly question, but it's a real problem for me in my research. Suppose we have defined risk in a regression problem as $R(f)=\int l(f(x),y) ...
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1answer
91 views

decimation in signal processing

Recently I was struck with the following question: And this is what I think about it: "Downsampling is one of the rare processes that are NOT time invariant. From the very nature of it's ...
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1answer
47 views

Choosing a sample from a sample probability

I am a bit confused about this problem. I understand that you need to pick a sample first, K, and then find the probability of that sample being red, L. The total different combinations of picking a ...
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0answers
48 views

Mean & SD of Sampling Distribution

A population consists of $4$ numbers $\{0, 2, 4, 6\}$. Consider drawing a random sample of size $n = 2$ with replacement. (a) What is the sampling distribution of $\bar x$? Is this a normal ...
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1answer
580 views

Deriving the variance of the sample mean $\mathrm{Var} (\bar{y})=\frac{1}{n}(1-\frac{n}{N})S^2$

For a population of size $N$ with a simple random sample size $n$ derive the formula $$\mathrm{Var}(\bar{y})=\frac{1}{n}\left(1-\frac{n}{N}\right)S^2$$ where $S^2$ is the population variance. Hint: ...
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0answers
36 views

What are incoherent matrices

What does incoherence means in terms of matrices? I am brushing on some compressive sampling theory and I did not find any easy to understand or straight forward answer about what does the word ...
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1answer
24 views

Test for Validity of Artificial Samples

I have a model that actually is learned from some observed samples. Then I use the model to generate several artificial data. My question is: Which test should I use to test if the data is of the ...
2
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0answers
28 views

Variance of a Population of Two Indpendent Random Variables

I have a question regarding a problem I'm looking at out of personal curiosity. Here is the basic setup of the problem: There is a population that contains half of type A, and half of type B. The ...
2
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1answer
40 views

Approaching a desired but infeasible distribution when constructing a sample

Suppose you have $N$ balls in $C$ different colors, and a "desired" distribution of those $C$ colors (eg 20% red, 80% blue). Your task is to build a sample (not really a random sample per se) of $S$ ...
1
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1answer
139 views

Sample uniform direction within cone

My question is pretty much the same as this question below, however I came up with a potential solution to this problem that I didn't see an answer to in the other question and I was wondering if it ...
0
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1answer
227 views

the random heights of north american women

The heights of North American women are normally distributed with a mean of 64 inches and a standard deviation of 2 inches. A random sample of four women is selected. What is the probability that the ...
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1answer
20 views

Generate samples from other samples

Given a family of continuous random samples $(x_i)_{i \in I}$ that approximate some unknown probability distribution. How can I generate more samples that fit to the same unknown distibution? ...
0
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1answer
24 views

Uniform sampling from part of sphere surface

I'd like to pose a question about uniform sampling on the surface of a sphere. I searched this site, and uniform sampling on a sphere surface seems to be quite a common problem. The common solution ...
4
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2answers
233 views

Monte Carlo estimator of the number of 1's in a very long binary sequence

Preface The question below is related to a problem I am working on, which requires counting the number of times a logic-valued function evaluates "TRUE" given an input value. The size of my input set ...
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2answers
100 views

Statistics - Lost with this question

I'm having trouble doing this question because I don't know where to begin. Could someone walk me through this slowly so that I understand the thought process and how to approach questions like this? ...
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1answer
59 views

Estimate correlation coefficient of unknown variable

Consider variable y depends on variable x and z linearly. I have $100$ sample values of $y$ and corresponding $x$ but don't have any values of $z$. The functional model is $$y = \alpha_1x + \alpha_2z ...
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2answers
72 views

Use z confidence interval to estimate population proportion

Which of the following must be true of a sample in order for it to be appropriate to use a $z$ confidence interval to estimate the population proportion? (A) The sample is a random sample from the ...
0
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1answer
69 views

Point estimators via method of moments

Suppose $X_1, X_2, \ldots, X_n$ constitute a random sample drawn from a population which has a probability function given by $$\Pr[X = x] = \frac{1}{\mu} \left( 1 - \frac{1}{\mu} \right)^{x-1}, \quad ...