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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82 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 ...
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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
124 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 ...
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2answers
71 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
94 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
52 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: ...
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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
84 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
46 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
45 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
545 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
34 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
22 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 ...
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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 ...
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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 $S$ ...
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1answer
122 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 ...
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1answer
211 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? ...
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1answer
18 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 ...
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2answers
230 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
95 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
57 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
70 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 ...
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1answer
66 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 ...
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1answer
57 views

How to approximate L^1[0,1] functions?

Do functions on a uniform grid with n points in the interval $[0,1]$ approximate $L^1[0,1]$ functions, as $n \to \infty$? I want to sample functions in $L^1[0,1]$ space numerically and I want to be ...
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1answer
84 views

Is Sample Covariance Tied to a Specific Distribution

In many sources on data analysis, the author(s) talk about calculating covariance of the data, and the formula is given as such $$ \Sigma = cov(X) = E[(X-E[X])(X-E[X])^T]$$ This formulation is given ...
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21 views

Sampling distribution with large sample size

As the sample size $n$ of a sampling distribution of sample means increases, the distribution becomes more normal. But if $n$ were the same size as the (finite) population, the "sampling" distribution ...
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1answer
193 views

The radial part of a normal distribution

I am reading a paper that asks me to sample $s_i$ from a distribution like this: $s_i \sim (2\pi)^{-\frac{d}{2}}A^{-1}_{d-1}r^{d-1}e^{-\frac{r^2}{2}}$ "Here the normalization constant $A_{d−1}$ ...
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0answers
101 views

Statistics - Uniform sample vs. Representative sample

I have a question concerning two different samples, with the first being more uniform that the second. a) Chance errors are likely to be smaller... using the first set of subjects using the second ...
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2answers
100 views

Sampling Distributions: Sample size of 1 vs Sample Size of m

I saw this example from a website Suppose there is a jar containing many gumballs, each with a unique number on it. The numbers range from 0 to 32 and there is an equal number of gumballs with ...
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19 views

Correct sampling methods for this data set / requirements

I am just looking for a push in the right direction as to what kind of sampling methods I can use to fulfill this set of sampling criteria. I had thought stratified sampling, but I'm not sure if ...
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1answer
49 views

Sampling random numbers with a certain condition.

I want to randomly sample three variables that are conditioned by $$x_1 \le x_2 \le x_3$$ and $x_1\in [0,\, \ell]$, $x_2\in [0,\, \ell-\ell_1]$ and $x_3 \in [0, \,\ell-\ell_1-\ell_2]$. I have only ...
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1answer
186 views

calculating an incoherence property

With respect to Matrix Completion and Compressive Sampling (CS) I'm trying to understand how to calculate an incoherence property μ between two bases Φ and Ψ. Getting this incoherence is important ...
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1answer
86 views

The mean and variance of the sample median

The population and the median of a sample sized $2k+1$ should have the same mean and variance. Why is that? Will the result still be so tidy for a sample sized $2k$?
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1answer
19 views

How do the Fourier Transform of sampling and the Frequency-domain convolution match?

The Fourier Transform(FT) is $X(\upsilon) = \int_{-\infty}^{\infty}x(t)e^{-2{\pi}i{\upsilon}t}dt$. The impulse train is $\delta_1(x)=\sum\limits_{k=-\infty}^{\infty}\delta(x-k)$, and its FT is ...
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1answer
54 views

Sampling without replacement

I've recently started studying statistics again, and I've just come across sampling without replacement. My book states that if we have a elements of type I and b elements of type II, then the ...
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1answer
139 views

Uniformly distributed points over the surface of the standard simplex

I would like to generate points that are uniformly distributed over the SURFACE of a standard $k$-simplex ($k$ dimensions, $k+1$ vertices). One way to efficiently generate points that are uniformly ...
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58 views

Minimum sample size to obtain population mean?

Knowing the average of a population, how could I evaluate an expected sample average based on k_sample? Data Info: Variance and standard deviation: $\sigma^2= 0.18500975256337462$ ...
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1answer
75 views

Sets and expectations

Imagine two sets $A = \{1, 2, \dots, a\}$ and $B = \{1, 2, 3, \dots, b\}$ with $a \leq b$. Let $f$ be a uniformly independently distributed random map $f:A\rightarrow B$ and $F = \bigcup_{i=1}^a ...
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16 views

Design effect due to survey weights

I have a quick question on design effects due to survey weights. I would like to ask help since I am stuck in some particular parts though. Here it is: Show that the following expressions for design ...
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1answer
28 views

How can i simplify the following term to get the right side?

$$\sum_{h=1}^{L}\frac{W_h^2S_h^2}{n_h}=\frac{1}{n}\sum_{h=1}^{L}{(W_hS_h)}^2$$ where, $n_h=\frac{n}{\sum_{h=1}^{L}N_hS_h}N_hS_h$ $\quad\text{and}\quad$ $W_h=\frac{N_h}{N}$ $\quad\text{and}\quad$ ...
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178 views

Proportional random sampling from a set of variable combinations

I have this table of variables, they can have the following values: $v_1 \in \{1,2\}$, $v_2 \in \{1,2,3\}$, $v_3 \in \{1,2,3,4\}$, I need to generate all the possible combinations $v_1v_2v_3$ of ...
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0answers
110 views

Central Limit Theorem Clarification [duplicate]

The Central Limit Theorem states that the sampling distribution of the sample mean: Converges in distribution to a normal distribution. Has an expected value (mean of the sampling distribution of ...
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1answer
21 views

Deriving Statistical quantites from other quantities

Studying for a statistics exam and I am stumped on the following: For a set of sample data, the following values were found. Fill in missing values Variable: X N: ? Mean: ? SE Mean: 2.05 StDev = ...
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1answer
106 views

Sample size from population?

This is probably very rudimentary maths, but given a strict population size ($N = 20$ for example), is the sample size any number $<N$? For use in calculation of confidence intervals using a ...
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22 views

Gibbs sampler for local linear trend model

Question: Consider the local linear trend model given by: \begin{align*} y_t = \mu_t + \tau \varepsilon_t \ \cdots \ \text{Observation equation} \\ \mu_{t+1} = \phi \mu_t + \eta_t \ \cdots \ ...
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174 views

unbiased estimator of the area of the circle

the radius of a circle is measured with an error of measurement which is distributed normal with mean $0$ and variance $\sigma^2$,$\sigma^2$ unknown.Given $n$ independent measurements of the radius , ...
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4answers
58 views

Can a sampling based method estimate how many species exist?

I've got in to a bit of a debate online and I'm hoping some people here can help clear it up. The position I'm arguing against is "It's impossible even come up with a ballpark estimate for how many ...
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1answer
196 views

What proportion are above x of sample size n where X ~ N(0,1) Homework

I have a homework question that I'm not quiet sure of. It follows as so: Consider a random variable $X$ that has a standard normal distribution with mean $\mu=0$ and standard deviation $\sigma=1$. ...