Questions tagged [sampling-theory]

For questions related to sampling theory. Sampling theory is the field of statistics that is involved with the collection, analysis and interpretation of data gathered from random samples of a population under study.

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Sample from a piecewise exponential distribution [migrated]

Given a distribution $p(x)\propto exp(\min_{i=1}^N [a_i^Tx + b_i])$, where $x$ and $a_i$ are both D-dimensional vectors, $b_i$ is a scalar, and $(a_i,b_i)$ are known. How can I sample from it? This ...
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mathmatical formulation for missing information due to sampling

I want to define a mathematical model (formula) to design missing information from sampling. In my problem, I have real events whose value is binary (0, 1) which might change every second. For a given ...
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Efficient Method for Uniform Sampling from the Space of Increasing Vectors in $[0, 1]$

I am seeking advice on methods for uniformly sampling from the space of increasing vectors within the interval $[0, 1]$. Specifically, I require an efficient algorithm that can handle high-dimensional ...
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Relation T-Distribution between chi-square and normal

Theorem. Let $Z\sim N(0,1)$ and $Y\sim \chi^2(v)$ are two independent random variabels. Then, $$T = \frac{Z}{\sqrt{Y/v}} \sim t-student$$ with degrees of freedom $v$. Is it the converse always true? ...
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What is the intuition behind the single-pass algorithm (Welford's method) for the corrected sum of squares?

The corrected sum of squares is the sum of squares of the deviations of a set of values about its mean. $$ S = \sum_{i=1}^k\space\space(x_i - \bar x)^2 $$ We can calculate the mean in a streaming ...
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Why is the demonstration of the "sample" standard deviation somehow more exact than the classic standard deviation, if $\frac{x}{n} ≠ \frac{x}{n-1}$?

I cannot see why to use the "sample standard deviation" instead of the classic "standard deviation" (the "population") ever, most explanations I find are just prescribing ...
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Doubt about sampling without replacement

A problem consists of five numbers $2,3,6,8 and 11$. Consider all possible sample space of size 2 that can be drawn with and without replacement from this population then find (a) mean of the ...
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Getting my concepts confused with sampling

Let me first set the scene. This is how we learned the variables and their names and symbols. We have the sugar content of 77 different cereal brands. It firsts asks "estimate the true average ...
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Sufficiency in sampling theory

Introduction We have a finite population $U$ with $N$ individuals, namely $U:=\{1,\dots,N\}$. Each individual stores a secret fixed value, so let's write $y_i$ for the value corresponding to the $i$-...
Álvaro G. Tenorio's user avatar
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Two boys paradox

I have a question regarding how to form the sample space in this famous paradox. Usually the sample space is defined as (B,B), (G,B), (B,G) and (G,G). However if I express it as (B1,B2), (G,B) and (G1,...
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An one dimensional sampling version of Penrose tiles in 2D

I was initially interested in aperiodic sampling for signals to address the problem of aliasing in the frequency domain. A design like Penrose tiling in 2D (which is non-repeating) can be very ...
CfourPiO's user avatar
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Sampling a cosine [closed]

If you sample a cosine of fundamental period 0.1 milliseconds with a sampling rate of 10^5 samples per second, how much phase difference is there between two consecutive samples? I can't solve this ...
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Uniform Sampling points on a line using 2 Uniform distributions

I have been struggling with this problem for quite some time but I am not sure how to proceed. So I am given a sampling algorithm : We would like to uniformly sample points on a line between A and B. ...
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Sampling procedure to obtain groups with same categorical distribution

Imagine that we have $n$ (mutually-exclusive) sets of different sizes, $X_1, X_2, ..., X_n$. The total number of elements is $N = |X_1| + ... + |X_n|$. I want to partition these $N$ elements into ...
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Applying Central Limit Theorem to an exponential distribution - how big should sample size be?

For an exponential distribution, in order for the sampling distribution of its mean to be well approximated by normal distribution (via central limit theorem), how big should a "typical" ...
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Brief references on partial coherence

A signal in additive noise is sampled by a receiver with an unstable clock. Maybe the clock has absolute bounds on its frequency drift and sample time aperiodicity, or maybe the clock has some ...
Christian Chapman's user avatar
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Deriving the Oversampling formula

I am familiar with the Shanon Sampling theorem, which states that: Let $f \in L_1(\mathbb{R})$ and $supp(\mathcal{F}f) \subseteq [-B,B]$,then $f(x)=\sum_{n \in \mathbb{Z}} f(\frac{n}{2B}) sinc(2B(x-\...
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Discrepancy between Importance Sampling Estimates and Hoeffding-like Bound?

Consider the following importance sampling estimate: $$\mathbb{E}_{x \sim P}[\ell(x)] = \mathbb{E}_{x \sim Q}\left[\frac{P(x)}{Q(x)}\ell(x)\right] \approx \frac{1}{L}\sum_{k=1}^L\frac{P(x_k)}{Q(x_k)}\...
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Random Sampling vs Stratified Sampling

I am reading the book: Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow by Aurélien Géron and am a little confused by the following segment: So far we have considered purely random ...
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Is metropolis hastings ever more efficient than rejection sampling in 2 dimensions?

so I know that Metropolis Hastings is an MCMC (Markov Chain Monte Carlo) method that is very useful in higher dimensions. The advantages it has over something like simple rejection sampling is that ...
Aditya S's user avatar
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Resample data with given sample (Python, numpy)

I consider two data D1 = (x1, y1) and D2 = (x2, y2), where x1, x2, y1 and y2 are arrays and thus D1 and D2 each describe a graph. Problem: x1 and x2 are different, making it difficult to compare the ...
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Unbiased Variance Estimator for Stratified Two Phase Simple Random Sampling without Replacement

In the context of two-phase sampling for stratification (see Särndall's Model Assisted Survey Sampling pg 350), in the particular case in which we use simple random sampling without replacement (...
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Intuition on random measurement matrix with non-adaptive linear measurements

Recently, I came across the concept of measuring a signal $x \in \mathbb{R}^n$ via a measurement matrix $A \in \mathbb{R}^{m\times n}$, where one has the linear system $y=Ax$ with $y \in \mathbb{R}^m$ ...
Schagomo's user avatar
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Drawing a sample from a measure

I am reading Perfect Simulation, Huber. I'm having some trouble interpreting the following text (Section 2.2): Let $\mu$ be a measure over $\Omega$, and suppose for density $g$ it is possible to draw ...
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What is a distinct feature of an ambiguous result

This question comes from my experience in radar signal processing. As I am going more deep into the theory of sampling, statistical signal processing and estimation theory in general, I have a very ...
CfourPiO's user avatar
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How many samples to be reasonably sure?

I am new to probability, and I am struggling with the right language to ask or google this question. I have a population of size $n$ and everyone is say some color. I want to verify that at least half ...
reesespieces's user avatar
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Sampling from a multi-variate Gaussian distribution restricted to a hypersphere

I consider a Gaussian distribution with density $$p({\bf x}) \propto \exp\left[-\frac{1}{2}({\bf x} - {\boldsymbol\mu})^\top{\boldsymbol\Sigma}^{-1}({\bf x} - {\boldsymbol\mu})\right]$$ where ${\bf x},...
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How to find a lognormal distribution which mimics the shape of a given normal distribution?

Question: If I give you the mean vector and the covariance matrix of a multivariate gaussian distribution (call it G1), is there a way to find a lognormal distribution which mimics the shape of G1? I ...
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Theorems Supporting the Monte Carlo Method

Let $X$ and $Y$ be random variables and $Y=g(X)$. I then drawn $n$ random samples X to get the set $\{x_1, x_2, ..., x_n\}$ according to the CDF of $X$ called $F_X(x)$. I process each sample as $y_i=...
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The bound for the absolute norm of a bandlimited function using the L2 norm

For a finite vector space we know that $\|x\|_{\infty} \leq \|x\|_{2}.$ I am looking for something similar for the topological Paley-Wiener space (bandlimited and $L^2(\mathbb{R})$ ). Specifically, I ...
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If $(X_n)$ is iid and $I=\inf\{n\ge 2:d(X_1,X_n)\ge\varepsilon\}$, what is the distribution of $X_I$?

Let $(E,d)$ be a metric space $\lambda$ be a measure on $\mathcal B(E)$ $p:E\to[0,\infty)$ with $$c:=\int p\:{\rm d}\lambda\in(0,\infty)$$ and $$\mu:=\frac{p\lambda}c$$ $(\Omega,\mathcal A,\...
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monte carlo error induced by a sampling pattern

Assume we want to estimate $$I=\int f(x):{\rm d}x=\hat f(0).$$ This can be done by an estimator of the form $$\tilde I=\sum_iw_if(x_i).$$ Introducing the "sampling function" $$S(x)=\sum_iw_i\...
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Expectation of a sample average at a random point

Let $f(x)=\frac{1}{N}\sum_{i=1}^Nf_i(x)$ where $f_i: \mathbb{R} \to \mathbb{R}$ for $i=1,\dots,N$. Let $f_{B}(x)=\frac{1}{|B|}\sum_{i \in B }f_i(x)$ where $B \subseteq \{1,\dots,N\}$ and $g: \mathbb{R}...
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How can we plot the Fourier transform of a sampling pattern?

In Cook, Stochastic sampling in computer graphics, the author is showing distribution patterns like jittered samples and the corresponding code to produce the output: He's also showing the plot of ...
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Sampling 2 balls of the same colour probability

A bag has 3 red, 3 blue and 3 green balls. 2 balls are picked at random without replacement. What is the probability that they are of the same colour? I am thinking what is the easiest way to do this ...
Astral's user avatar
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2 answers
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Biased Sample Mean

I have the following question in my interview: Suppose I am interested in the average of residents by apartments. So I went to the street and randomly sample people and ask them how many residents are ...
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Deriving the probability of inclusion for a random stratified sample

So I'm a bit stuck on deriving the inclusion probability for a random stratified sample. The question's context is as follows: say we wish to understand household income, but a complete list of ...
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Alternatives to Fuller’s (2009) Sampling Statistics

I’m taking a self-study course in sampling statistics this semester and my professor chose Fuller’s (2009) Sampling Statistics for both the content and the homework questions. However, I am really ...
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Interpolation from unevenly distributed points of function with compactly supported Fourier transform

It is not so hard to prove the Nyquist-Shannon theorem. It basically says a real valued function $f$ having a compactly supported Fourier transform function $\hat f$ can be interpolated precisely from ...
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Prove the variance of Hansen-Hurwitz estimator

The Hansen-Hurwitz estimator of the population total, $Y=\sum_{I=1}^{N} Y_{I}$, is given as: \begin{equation} \mathcal{y}_{\mathrm{HH}}^{\prime}=\frac{1}{n} \sum_{i=1}^{n} \frac{y_{i}}{p_{i}} \end{...
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What is the "best" way to sample a function?

The situation Imagine that you are faced with a box with a numeric display and a knob next to it that can turn in specific increments (that you can specify) and in both directions (increasing values ...
Le Duc Banal's user avatar
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Sample selection from within a certain area

I need to make a random selection of mosques in a large city for a public health survey. I have a map that identifies a thousand of them, but many are unmapped, so I don't know what my 'universe' is, ...
Sebastian's user avatar
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Show that the probability of the sample $S$ is the probability of $S_1 S_2$.

Let $U$ be the population with a size of $N$, and let $n_1$ and $n_2$ be the sizes of two samples $S_1$ and $S_2$, respectively. Let the sample $S = S_1 U S_2$ with size $n = n_1 + n_2$. Show that the ...
user229257's user avatar
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Why is a sample defined as a set of observable random variables?

I have learned that a Random variable is a mapping from sample space to a real number and is denoted as X, Y ,Z for different functions (random variables). Say, in the case of rolling two dice, X can ...
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Sistematic sampling rewritting the variance

If $N = n \cdot a $ where $a$ is an integer, show that the variance $V_{SY}(\hat{t}_\pi)$ given by $$ V_{SY}(\hat{t}_\pi) = N^2 \dfrac{1}{a} \sum_{r=1}^{a} \left( \bar{y}_{Sr} - \bar{y}_U\right)^2 $$ ...
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Subsampling from different distribution data?

I have a simple question. Thanks for helping me Is there any difference between simple random subsampling a set with uniform distribution and a set with normal distribution? How can I subsample in ...
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How to count the specific sampling pairs given several samples of a function?

As illustrated in the figure, we have a function (the solid line) and sample it with a user-defined sampling rate (the blue circles). In addition, we uniformly divide the y-axis into several sub-areas ...
tianhang's user avatar
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Does this discrete probability distribution have a name? (Related to Hypergeometric distribution)

Suppose there are $N$ objects, of which $K$ are of type 1 and $N-K$ of type 2. Objects of type 1 are indistinguishable and objects of type 2 are indistinguishable. One is interested in the probability ...
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Nyquist sampling spacing of a 2D Gaussian windowed sinc function

I'm trying to find a proper sampling grid size to avoid aliasing when modeling a point source model given by: \begin{align} pt(x, y) = \exp\left\{-j\frac{k}{2L}(x^2+y^2)\right\}\text{sinc}\left(\frac{...
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Find the sample size $n$ for a stratified sampling method

Let’s say we have a finite population $N=5000$.We have divided the population into 5 subgroups according to a risk characteristic “very low risk”,”low risk”,”medium risk”,”high risk” and “very high ...
Homer Jay Simpson's user avatar