Questions tagged [sampling-theory]

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Efficiently testing for more zeros than ones in a binary code block

Setting. I have a set $\mathcal{C}$ of binary sequences where each sequence has length $L\in \mathbb{N}$ and the total number of sequences in $\mathcal{C}$ is $N\in \mathbb{N}$. Assume that $N=\exp(O(...
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7 views

Sampling K correlated variables from beta distributions

I have K variables, all correlated to each other, to be sampled N times each from generalized beta distributions. I wonder how I can extend the suggestions for K=2 provided in How to generate ...
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1answer
24 views

Statistical sampling and random variables?

I'm studying statistical sampling and there is a point which is not very clear to me. Let us discuss the following example. Suppose we would like to study the heights of 3.000 students in a given ...
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1answer
52 views

Show that $\operatorname{sinc}(x-y) = \sum_{n \in \mathbb{Z}}\operatorname{sinc}(x+n)\operatorname{sinc}(y+n)$

My problem is: Show that for every $x,y \in \mathbb{R}$ $$\operatorname{sinc}(x-y) = \sum_{n \in \mathbb{Z}}\operatorname{sinc}(x+n)\operatorname{sinc}(y+n)$$ Here is what I did so far: Considering ...
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1answer
22 views

Fourier transform of sampled signal on multidimensional lattice

Suppose $u_c \in L^1(\mathbb{R}^D)$ has the Fourier transform $\widehat{u_c}$, and that we're sampling it on a multidimensional lattice $\Lambda$, with reciprocal lattice $\Lambda^*$. The sampled ...
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14 views

Is sampling from 2 distributions separately and summing the result equivalent to sampling from the sum of the two distributions?

Is sampling from 2 distributions separately and summing the result equivalent to sampling from the sum of the two distributions? for example we have two probability distributions, P1, and P2: ...
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11 views

Find the appropriate grid size to sample porosity across an image

I have a binary image which is composed of black grains and white porosity. I have been using a macro which divides the image into grids and calculates the porosity in each grid by obtaining the ...
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2answers
36 views

Why variance must divide by population?

According to page 2 of this paper, if $X$ is an arbitrary measurement with mean $\mu$ and variance $\sigma^2$, and $\overline{x}$ is the sample mean from random samples of size n, then: $$\...
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12 views

Simple random sampling(expanding square of a sum)

I was doing simple random sampling without replacement and got stuck with this expression. I think on RHS there should be a coefficient of 2 with the second term. $$( \sum _ { i = 1 } ^ { n } y _ { i ...
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1answer
19 views

Is it necessary to have infinite(or near infinite) population size for a binomial distribution?

I was reading about single sampling plan in the book "Introduction to statistical quality control", by DOUGLAS C. MONTGOMERY. The author has mentioned that under the assumption that lot size is ...
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1answer
24 views

How can I efficiently sample a space to find a channel capacity?

I'm investigating Gelfand-Pinsker channels where the capacity formula has been proven to be: $$ C = max_{p(x, u|s)} [I(U; Y) - I(U;S)] .$$ I've written some simulation code that, given the ...
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18 views

How to calculate my sample size to establish my real-WR?

I've been tracking for a period my hour rate on the game of poker and I want to know how much data I've to collect in order to establish my earning per hour. I've been searching online and when it ...
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1answer
64 views

How to determine the sample size for a two sided $z$-test?

Let $X_{1}, \ldots, X_{n}$ be an iid sample from $N(\mu,\sigma^2)$ where $\sigma$ is known. We want to test a hypothesis $$ H_{0}:\mu = \mu_{0} \quad \mbox{versus} \quad H_{1}: \mu \ne \mu_0 $$ Now, ...
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1answer
25 views

Apply formula on random sampling without replacement, but with replacement between each iterations

I am looking for an algebraic solution to explain that I apply a formula on a vector constituted of a random sampling of n elements in a population of size N without replacement. The formula is ...
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1answer
15 views

Algebraic expression of random sampling without replacement

I am new to algeba and I cannot manage to find how to write the algebraic equation to explain that I randomly sample data contained in vectors without replacement. I have e.g. vectors and : Both of ...
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76 views

Are there any generalization of the DKW inequality to the cluster sampling case?

The famed DKW inequality states the following: $$\mathbb{P}\left(\sup_{x\in\mathbb{R}}|F_n(x)-F(x)|>\epsilon\right)\leq 2e^{-2n\epsilon^2}$$. Further using Bahadur's representation, we naturally ...
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1answer
23 views

the sampling distribution of sample means, why are samples taken with replacement?

I have what might be a silly question. I am tutoring a family member in stats and I am trying to help her understand the sampling distribution of sample means, but I don't quite understand why it's ...
2
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1answer
60 views

Expected difference for two largest samples

I'm trying to get an understanding as to how much difference between the two largest samples drawn from some distribution $f(x)$ scales with $n$. That is, as we increase the size of our sample, how ...
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14 views

$\epsilon$ sampling consequences. Lemma 2.1. Tamal's Curve and Surface Reconstruction

I'm trying to strength my background in curve and surface reconstruction. The lemma mentioned in the question is the following Lemma 2.1. (Empty Segment). Let $p \in P$ and $x \in \Sigma$ so that $\...
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36 views

A sample of size $n=80$ is drawn from a population of size $N$. The sample size is divided into two parts .

A sample of size $n=80$ is drawn from a population of size $N$. The sample size is divided into two parts and samples are drawn independently from both the parts.The one-fourth sample is drawn by ...
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1answer
34 views

Determining Normality With Large Samples

I am taking in some files and I must determine if the data sets are normally distributed (yes, within a certain degree of certainty because it cannot be proven only disproven). My data sets are quite ...
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1answer
32 views

Why is sampling distribution of means's standard deviation sd/sqrt(n)?

I'm so used to seeing standard deviations being the sqrt of the entire formula so it's weird to me to see a SD that is defined as sd/sqrt(n). Can someone explain why?
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1answer
36 views

Estimation population size using sampling - real life probability

I have come across similar questions like this previously but I just can't get my head around a correct method Clive catches 50 bees from the beehive and marks each bee with a dye then lets them go. ...
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1answer
174 views

What does it mean that a distribution is hard to sample from?

I'm currently studying about sampling methods, and some terminology that has arisen is that a distribution is "hard to sample from". What does this mean? My current intuition is that the ...
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1answer
25 views

Notations question regarding the equation for sample mean and variance

Let $X$ be the outcome of a chance experiment with $E(X) = \mu$ and $V(X) = \sigma^2$. When $\mu$ and $\sigma^2$ can be estimated by repeating the experiment $n$ times with outcomes $x_1, x_2,...,x_n$,...
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18 views

How to infer the underlying distribution of a statistic (Bayesian inference?)

I have a list of approximately 30,000 venues in a major US city. These venues hold all kinds of events, sports, conferences, concerts etc. I want to know the distribution of the 'capacity' of these ...
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1answer
12 views

Continuous random sampling - random selection of units with average frequency

Scenario: Multiple agents process orders simultaneously. Orders are not shared between agents. Each agent is assigned a sampling percentage. Let say agent A has sampling percentage of 10% and B has ...
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27 views

How to sample the position of many particles?

In Density Functional Theory, there are many types of trail variational wave functions. I have a question from the numerical viewpoint: when I have the 1D density functions, and the $N$-dimensional ...
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68 views

prove that $(n-1)(\frac{1}{n} \sum_{i=1}^{n}\hat{\sigma^2_{-i}}-\hat{\sigma^2})=-\frac{S^2}{n}$ in i.i.d sample(the factor n-1 in the jackknife bias)

let $x=(x_1,\cdots,x_n) $ are n independent samples from unknown distribution. The Jackknife Samples are selected by taking the original data vector and deleting one observation from the set. Thus, ...
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22 views

Sampling distribution of a functional T

While studying the bootstrap method, I came across with the following definition of the sampling distribution of a functional T: Let's say $X_1,...,X_n$ are i.i.d with distribution function $F$, then ...
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1answer
37 views

sampling a connected graph to a smaller connected graph

I have a graph $G$ which has $n$ nodes and $\alpha*(n^2-n)/2$ edges (so the chance of having edge $(i,j)$ is $\alpha$). The graph is connected which means that if we calculate the number of components ...
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2answers
62 views

Rigorous Meaning of “Drawing a Sample” $\omega$ from a Probability Space $(\Omega, \mathcal{A}, \mathbb{P})$

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space. What does it mean (in the most formal and rigorous sense possible) to "draw a sample" $\omega \in \Omega$ from this space? Intuitively, ...
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46 views

Picking path at random in DAG graph with probability equals to path weight.

I'm refering to the following paper: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.329.3653&rep=rep1&type=pdf. There is a lemma states: Let $G$ be a directed acyclic graph with ...
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1answer
50 views

Let $V_1$ be the variance of the estimated mean from a stratified random sample of size $n$ with proportional allocation.

Let $V_1$ be the variance of the estimated mean from a stratified random sample of size $n$ with proportional allocation. Assume that the strata sizes are such that the allocations are all integers....
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1answer
51 views

distribution of $\sum_{i=1}^n (X_i-X_{n+i})^2$ where $X_1,X_2,\dots,X_{2n}$ are iid $N(\mu,\sigma^2)$

Suppose $X_1,X_2,\dots,X_{2n}$ are iid $\mathcal N(\mu,\sigma^2)$ random variables. How can I find the distribution of $\displaystyle\sum_{i=1}^n (X_i-X_{n+i})^2$? Should I approach it by taking $Z=\...
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1answer
18 views

Using Random Sample to Find Estimate

I have to use the Inversion Sampling Method to generate a random sample of 100 from the function $f(x)=\theta x^{\theta - 1}$ if $\theta =5$. Here is my function so far: ...
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1answer
45 views

Showing that an estimator is consistent

Let $X_1,X_2,\ldots,X_n$ be a random sample from $\mathcal{N}(\theta,1)$. Consider the following (randomized) estimator of $\theta$ given a sample of size $n$: $$ \hat{\theta}_n = \bar{X} + \begin{...
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2answers
34 views

What is the probability that $m$ items drawn from $n$ distinct items contain all $n$ items?

Suppose there are $n$ distinct items to be drawn with replacement for $m$ times, the probability of each item being drawn is assumed to be $\frac{1}{n}$. What is the probability $P(m)$ that $m$ items ...
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27 views

How to randomly sample a social graph to find paths between at least 20% of profiles?

Given a Graph, where we know Total number of nodes (~100,000) Average no of connections per node (~200) Maximum distance between two nodes (~5) How many nodes (and its connections) do we have to ...
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36 views

Detection of certain stochastic process

Hello math stack exchange! Suppose we have arbitrary function of stochastic processes $X_i(t)$, like $$Y(t) = X_1(t) + X_2(t) - X_3(t) + X_4(t) ....,$$ more generally $Y(t) = f(X_1(t), X_2(t), X_3(t)...
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45 views

Reconstruction after Unitary Operator on Paley-Wiener space

Background: According to the Sampling Theorem for Reproducing Kernel Hilbert Spaces (RKHS), for any RKHS $H_{rk}$ with reproducing kernel $k(\cdot, \cdot)$, if I can find a set of points $\{t_{n}\}_{...
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40 views

Merge weighted random sampled set with different distributions

Here is my problem: I have a set $S_1$ of $N$ items each item i attached to a weight $w_{1_i}$. From this set I sample a subset of m items $(m << N)$ using weighted random sampling with ...
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2answers
1k views

How is Welford's Algorithm derived?

I am having some trouble understanding how part of this formula is derived. Taken from: http://jonisalonen.com/2013/deriving-welfords-method-for-computing-variance/ $(x_N−\bar{x}_N)^2+\sum_{i=1}^{N−...
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1answer
158 views

Standard Error is of Population Total

We have the following data and we are required to obtain the standard error of unbiased estimate of the population total: $N=160,n=64,\sigma^2=4$ My approach We know that: $SE(\bar{X})=\frac{\...
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1answer
121 views

random sampling on random samples

I would like to understand if the sample distribution of the following approaches are the same or not. Setup: population of size $N$, with binomial distribution. required sample size is $n$ ...
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0answers
167 views

Sampling from a Mixture of Distributions

Sorry, but I give some detail before getting to my question, at the end. Suppose we have a probability distribution for random variable $X$. Let $X$ have the domain $\mathcal{X}$ and be ...
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1answer
255 views

Sampling without replacement with non uniform probabilities

In the article https://projecteuclid.org/download/pdf_1/euclid.aoms/1177704564 they describe procedure how to sample batch of size $n$ with nonuniform probabilities $p_i$, where probability of ...
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48 views

Linking probability measure to classical definition of probability when sampling from finite population

Consider a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ where $\Omega$ is finite, $\mathcal{F}$ is the power set of $\Omega$ and $\mathbb{P}(F)\equiv |F|/|\Omega|$ $\forall F \in \mathcal{F}$....
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317 views

Cardinality of Intersection and Union of Multiple Sets Given Overlap coefficient(s)

I would like to reason about the intersection and union of a number of sample sets and an assumed similarity between them. The calculation doesn't have to be exact, it can be a reasonable estimate. ...
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100 views

Which is the relation between between population/probability space/sampling?

I am trying to understand the relation between population/probability space/sampling in Econometrics. My arguments are divided in 3 sub-questions which trace my attempt to link in a logical way the ...