Questions tagged [sampling]

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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survey data- is there a general formula for the probability that two people are at the same place at the same time?

I have survey data of interviews with people at certain outdoor recreation sites. The survey records the amount of time each interviewer spends at a site, and each person surveyed at the given site is ...
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38 views

What sample size is needed to ensure a majority?

The results of a sample of voters showed that $55\%$ voted for a given candidate. It was determined that at a confidence level of $0.95$ that candidate would be the winner (i.e. would receive the ...
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How to correctly sample lines from large file when order and lines count are unknown?

Input: potentially large file with text lines lines count is unknown, naive assumption gives range maybe more than 100K lines, but less than 400K it's unknown if lines are sorted or not it's fine to ...
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Joint sampling distribution

"Sampling from the super-population generates a joint sampling distribution on the quadruple of unit-level variables $(Y_i(0),Y_i(1),W_i,X_i)$, i=1,...,N. More explicitly, we assume the $(Y_i(0),Y_i(1)...
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Sample covariance of scores matrix [closed]

The question is: Let X be a mean centered n × p data matrix and let Z be the corresponding n × p scores matrix. Show that the sample covariance of the scores matrix is diagonal. What is the ...
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Sinusoidal curves forms and linspace, anything to do with Nyquist frequency?

I was playing around with sinusoidal curves and saw this: I defined an array of numbers between 0 and $2\pi$ with 200 points. x=np.linspace(0, np.pi, 200) Then I ...
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1answer
23 views

Error Probability: Can anyone share a detailed solution to the problem?

Seventy data clerks at the Department of Motor Vehicles make an average of 18 errors per day, normally distributed with a standard deviation of 4. A field auditor can check the work of 15 clerks per ...
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Optimal estimators of Gaussian under certain conditions

You can ignore this context but I think it adds a little interest to the question.. In finance pricing information is often proprietary and firms do not want other firms to know their price, but of ...
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25 views

Monte Carlo Sampling with non-uniform distributions?

I'm currently studying Monte Carlo sampling, referencing Veach's "Robust Monte Carlo Methods for Light Transport Simulation". On page 63, he writes: The idea of Monte Carlo integration is to ...
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How to derive the probability density function (PDF) of a continuous random variable from a set of data?

I am interested to derive an expression for the probability density function (PDF) of a continuous random variable from a given set of data. To further explain, let us consider that we have the data ...
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21 views

Show that the covariance between the two sample means is $\frac{- \sum_{i=1}^{N} (y_i - \bar{y})^2}{N(N-1)}$

Consider a population with $ N >1 $ units having values $y_1,y_2,..,y_N $. A sample of size $n_1 $ is drawn from the population using SRSWOR.From the remaining part of the population, a sample of ...
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Computing the bias of the sample autocovariance with unknown mean

In "Introduction to statistical time series" by W. A. Fuller (1976), two definitions of the sample autocovariance with lag $h$ for a signal of length $n$ and unknown mean are proposed, namely $$\...
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Find expected value and variance (stratified sampling)

A city has $n$ blocks of which $n_j$ have $x_j$ inhabitants each $(n_1 + n_2 + \cdot\cdot\cdot = n )$. Let $m = \sum n_j x_j/n$ be the mean number of inhabitants per block and put $a^2 = n^{-1} \sum ...
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Testing whether the two population means are the same using sampling distribution of difference between two means

The problem is Given the above data, can we conclude that the two population means are equal? And my question is, how can I solve this question using the sampling distribution of the difference ...
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Detection probability when there are multiple issues

We are conducting multiple rounds of sampling tests, and I am trying to devise a way to perform limited sampling which can uncover all the issues that existed in the population of the previous cycle (...
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57 views

Estimate number of balls by picking a random bin

This is a tweak to the standard balls and bins problem where we (usually) come up with bounds on the max load or empty bins. I am interested in estimating $M$ when $M$ balls are (uniformly) thrown ...
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Determining the minimum score to pass

I am revising some basic statistics problems for my studies and came across this problem: A population of 10 000 students is taking an exam and the exam score is follows approximately a normal ...
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43 views

Finding the value of a sample statistic using chi-squared distribution?

The problem is Find the variance $S^2$ for random sample of size 21 from a normal population with variance 5. (Hint: Use the fact that the statistic $\frac{(n-1)S^2}{\sigma^2}$ has a Chi-squared ...
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Accurate Mathematical Expression for Sampling Process

I would like to confirm the completeness for the mathematical expressions for sampled functions. Consider a continuous-time function $x(t)$ sampled every t=T seconds, the continuous-time sampled ...
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1answer
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Sampling from a Bivariate Cauchy?

Given the bivariate Cauchy distribution: $$f(x,y; x_0, y_0) \sim \frac{1}{2\pi}\frac{1}{((x - x_0)^2 + (y - y_0)^2+1)^{1.5}}$$ How do you generate samples appropriately? I am aware of inverse ...
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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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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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Proportional allocation in stratified sampling: calculating mean and standard error without knowing stratum sizes

There are 3 strata. The total population size is $N = 400 = N_1+N_2+N_3$. From these, a total of $n = 30 = n_1+n_2+n_3$ units is sampled. Here, $N_1$ refers to the number of units in stratum 1 in the ...
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287 views

Probability of a non-random sample to represent initial Poisson distribution

I'm looking for a way to correctly approach the following noise-signal problem. What I'm doing is a looking for arbitrary structures in a seemingly random input data, akin to a night vision with noisy ...
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Normal-inverse-Wishart distribution

I am flummoxed by this parameterization for the PDF of the Normal-inverse-Wishart (NiW) distribution: $f({\boldsymbol \mu },{\boldsymbol \Sigma }|{\boldsymbol \mu }_{0},\lambda ,{\boldsymbol \Psi ...
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Uniform sampling on some region of the surface of an n-sphere

I have a question on how one might sample a region of an n-sphere. I understand the Muller method of computing n normally distributed coordinates and dividing by the norm to map the point onto the ...
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Can this CDF (containing a Gamma function) be inverted?

I'm trying to invert the following CDF function so that I can sample from it: $$CDF(y) = \int_0^y x^b \exp(c x) dx = \frac{y^b (-c y)^{-b} [\Gamma(b + 1, -c y) - b \Gamma(b)]}{c}$$ (The result of ...
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34 views

Sample size in Confidence Intervals

In repeating confidence interval experiments, are we allowed to take samples of different size every time? Because a confidence interval of 95% means that if the sampling process is repeated infinite ...
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Probability of ruin - calculating an integral after exponential change of measure

Let's consider a process $u - S_n$, where $S_n = Y_1 + \ldots + Y_n$ is a sum of iid random variables with the density $p(y)$. Let: $T_0 = \min\{n: u - S_n < 0 \}$, $T_b = \min \{n: u - S_n >b ...
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1answer
59 views

Can three sticks make a triangle? - statistical method of solving

I was challenged to provide as many solutions as I could to the triangle problem: Given a stick of arbitrary length broken into three pieces of independently random lengths, what is the probability ...
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Sampling exercise basic

I have to resolve the follow problem about basic sampling concepts. I've tried a solution, I want to know if I've solved the problem well, please In planning an office network study, the following ...
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Size of two sided t-test

How do we calculate the size of two-sided t-test? Given a random sample with unknown variance and unknown mean, we hypothesize that the true mean is equal to some $\mu_0$. How do we calculate the size ...
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How would you estimate the total price of a city with probability sampling?

There is a website where I can search for each and every price of a property (mostly houses) and I have list of 272 streets (which also consists of avenue, lanes, boulevard and etc) in the city. I ...
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25 views

Sampling from a Multivariate Gaussian whose covariance form is given by Cholesky

I've read a paper "structured uncertainty prediction networks", and I don't understand how to sample from a multivariate Gaussain in the paper. Here is a sampling method used in the paper. Suppose ...
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calculate sampling error given dispersion and probability

A professor is estimating avg number of hours any student needs to pass his stat course. the dispersion is 2.6, With probability 90% calculate number of hours needed for research if the sample error ...
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Understanding acceptance-rejection method - more general point of view

I am searching for a proper geometrical explanation of the acceptance-rejection method. Usually it is presented in a way such that, to sample $X$ from a distribution $f(x)$ with $x\in(a,b)$, one ...
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34 views

A question about the properties of vertices in graphs

I am reading in Graph theory and applications related to graphs but I am really new to this topic. There is an expression which is too confusing and I don't understand. I would like to understand what ...
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Are there other distributions over functions besides the Gaussian process?

The Gaussian process allows you to sample a continuous function $f(x)$ evaluated at arbitrary points $\vec{x}$, \begin{align} f(x) &\sim GP\left(\mu(x), K(x,x')\right). \end{align} Are there any ...
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A probability (exactly k out of n)

Suppose that I know that 18% of a population is willing to pay more taxes. When I select random 48 people from that population, what is the probability that exactly 8 of them are willing to pay ...
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Wilks equation for sampling

So let's assume you sample independently N times a random variable X, for which you do not know the distribution. Then, the probability $\beta$ that a fraction $\alpha$ of the distribution can be ...
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Which statistical test can I use to find if there is a significant difference between 6 data sets at 3 locations?

I'm working on an investigation about heat loss from buildings of different ages. I've studied 3 buildings, and taken 2 30 metre transects at each. I've measured air and soil temperatures along the ...
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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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Summation and binomial coefficient

I'm studying sampling at the moment, but I can't get the passage below: When $n$ units are sampled from $N$ units without replacement, then each unit of the population can occur with other units ...
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23 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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How Importance Sampling Works

I understand that importance sampling involves sampling from one distribution to estimate the expected value of another. We do this when the distribution of the random variable whose mean we are after ...
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Does the Central Limit Theorem only apply to the sample mean?

My question: Is it only the probability distribution of the sample mean statistic of a sample which is normally distributed according to the Central Limit Theorem, or, will any statistic work like for ...
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Discrepancy and Dispersion of a perfect square

Consider the unit square $[0,1]^2$. Let $R$ be the family of all axis parallel rectangles inside $[0,1]^2$ (not necessarily anchored at the origin). Suppose $n$ is a perfect square. Let $\{P = (\frac{...
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Probabilistic heat diffusion on spherical shell

I'd like to sample random paths to approximate the solution to the heat equation on the unit sphere shell. Specifically, I'd like to solve $u_t = \alpha\Delta u$ with some initial condition $u_0$. ...
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Sampling algorithm for distribution with density function f(x) = min{1, 1/x^4}

How can I generate random samples with distribution density function $f(x) = min\{ 1, 1/x^4\}$ ? I know I may use the composition method as the website described, but I have no idea how to use it. ...
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Digital low pass filter using limited number of coefficent

So I need to create a discrete low pass filter. For continuous-time $x(t) \leftrightarrow X(jw)$ and discrete-time $x[n] \leftrightarrow X(e^{j\omega})$, the problem is not too difficult. But for ...

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