# 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. Use this tag along with the tags (probability), (probability-theory) or (statistics).

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### Skipping CDF for sampling

i am writing a thesis and am sadly stuck... I am trying to sample from a distribution of the form $$a*\exp(-2 \pi^2 x^2)(x^{d-1})$$ Now my instinct was to "simply" calculate the CDF and ...
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### Exercise 2.19 from Cochran's - Sampling techniques

I'm trying to solve the following exercise from Cochran's "Sampling Techniques ": This exercise is another example of estimators geared to particular features of populations. After the ...
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### Quick uniform sampling with replacement

I seek to quickly uniformly sample $N$ items with replacement from a list $[a_1, a_2]$ of length $n=2$, and ideally for larger $n$ as well. For the case of $n=2$, approaching this by summing binomial ...
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### Sampling theory - Can 2 samples drawn from nested frames, and based on 2 different (correlated) stratification variables, be considered independent?

Suppose that we have two samples (say, a and b) that are drawn from two different frames (say, 1 and 2) of the same population. In particular, suppose the frames are "nested" as shown in the ...
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### Exam Resources for Mathematical Statistics [closed]

I am interested in past exam questions with solutions for mathematical statistics (univariate and bivariate RV's, expectations, variance, MGF/CGF/PGF, etc) and inference (CLT, estimations, sampling ...
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### What am I measuring regarding the maximum entropy of a sequence of bounded sets converging to an unbounded set?

Let $A\subseteq\mathbb{R}^{n}$ be an unbounded Borel set, and $(F_r)_{r\in\mathbb{N}}$ be a sequence of bounded sets which has a set theoretic limit of $A$. Also, $\text{dim}_{\text{H}}(\cdot)$ is the ...
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### Does loading items in parallel with different load times cause biased sampling?

This is for sampling images for ML training. If there are multiple threads loading data, but each item can take a different time due to different resolutions or disk speeds. The loaded images will get ...
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### Finding parameters for Fisher's multivariate hypergeometric distribution given means

I want an efficient-to-sample distribution over sets $S\subseteq [n] = \{1,\ldots,n\}$ such that $|S|=k$ and $\Pr[i\in S]=p_i$ for some known $k,p_1,\ldots,p_n$. A natural choice is Fisher's ...
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Say we have a function $f(x_1,x_2,x_3,\ldots,x_n): \mathbb R^n \mapsto \mathbb R$ and $n$ independent random variables $X_1, X_2, X_3, \ldots, X_n \in \mathbb R$. Given a positive number $s \in \... • 2,170 0 votes 0 answers 15 views ### Inverse-Wishart Posterior for Gibbs Sampling I've a question regarding Gibb's sampling where I'm stuck at a particular point. Let B has a prior Inverse-Wishart Distribution IW($aI_3$,b). Now if I've data such that: \mu \sim ... 1 vote 1 answer 64 views ### Finding the Max of a Discrete Probability Distribution [closed] I have a discrete random variable$X$with$N$possible values and some distribution Suppose that there is some$X_m$such that $$P(X_m) > P(X_j),\quad\forall\ j \neq m$$ Suppose that the gap to ... 2 votes 0 answers 37 views ### Calculating confidence intervals for a population proportion? The proportion of European men who are red-green colour-blind is 8%. How large a sample would need to be selected to be 95% certain that it contains at least this proportion of red-green colour-blind ... • 37 0 votes 0 answers 29 views ### What is the optimal sample size when determining the overlap of two subsets? Let's say I start with a set of elements ~ 100M in size. In addition, let's say I can partition the set in a deterministic way. Now given two arbitrary subsets, perhaps few million in size, but ... • 477 -1 votes 2 answers 39 views ### Are$\mu_{\hat{p}}$and$\sigma_{\hat{p}}$considered parameters or statistics? Is$\mu_{\hat{p}}$(the mean of the sampling distribution of$\hat{p}$) and$\sigma_{\hat{p}}$(the standard deviation of the sampling distribution of$\hat{p}$) considered parameters or statistics, ... • 15 0 votes 1 answer 99 views ### Probability of 3 darts landing in the same half of the board [duplicate] Problem: Find the probability of 3 randomly thrown darts landing in the same half of the board. More generally, if$n$points picked uniformly randomly on a disk, find the probability of them lying in ... • 81 0 votes 0 answers 45 views ### Sample a number x uniformly from$\lbrace 1,2,3\rbrace$using two fair coin flips with a finite, deterministic number of steps I want to randomly choose one of the numbers 1, 2 and 3 where they have the same probability. One way of doing this would be to interpret the coin flip result as a binary number, repeating the flip if ... 1 vote 0 answers 35 views ### Is there a way to sample the boundary of the union of spheres? Without rejection Suppose a shape is constructed from a union of spheres,$\bigcup_{i} S_i$where$S_i$are spheres in$\mathbb{R}^3$. Is it possible to efficiently uniformly sample the boundary of this shape without ... • 1,362 6 votes 4 answers 481 views ### Confusion on defining uniform distribution on hypersphere and its sampling problem Fix a dimension$d$. Write$S^{d-1}$for the surface of a hypersphere in$\mathbb{R}^d$, namely set of all$x = (x_1, \ldots, x_d) \in \mathbb{R}^d$such that$|x|^2 = x_1^2 + \cdots + x_d^2 = 1$. I ... • 2,722 0 votes 0 answers 29 views ### Bias of a ratio estimator I am reading the book Sampling and Design Analysis In chapter 4, page 128 there is computation of the bias of the ratio mean estimator which I have hard time understanding Specifically why does $$\ -... • 1,449 0 votes 1 answer 56 views ### How might one estimate the number of samples in a uniform distribution given only the sample range? Let's say we have a fair roulette wheel with 2^256 segments, each printed with a unique integer in the range 0 to 2^256 - 1. Let's say that the wheel is hidden from us and that Trevor spins it ... • 149 0 votes 0 answers 22 views ### Estimating KL Divergence from Multiple Independent Samples I'm working with two discrete probability distributions, (P) and (Q), and exploring different ways to measure the divergence between them using Kullback-Leibler (KL) divergence. Specifically, I'm ... • 11 0 votes 0 answers 10 views ### Sampling a vector of size n which sums to k sequentially Suppose I want to sample a vector X such that every entry is sampled from some distribution p and all entries sum to k (the entries are not independent). I thought I can simply sample X_1 from ... • 256 0 votes 1 answer 30 views ### Drawing random samples from statistical distribution I'm looking for a list of methods to draw random samples from Pareto distributions. I already found : Inverse transform sampling Rejection sampling Metropolis–Hastings algorithm Gibbs Sampling Are ... 1 vote 1 answer 38 views ### How to bound the total number of trials for a Binomial distribution if number of success is given? Suppose I am sampling a random variable that follows a binomial distribution Binomial(n, p). The sample rate p is known. Now, after the sampling, I got m success cases. How do I find the upper ... • 53 0 votes 0 answers 9 views ### Generating Random Numbers using Rising and Falling Exponential Function I am working with this normalized pdf,$$ f(t) = (\tau_d - \tau_r)^{-1}(-e^{-t/\tau_r} + e^{-t/\tau_d}),~~t \geq0 $$And \tau_d > \tau_r. Are there any methods for generating random values t \... • 155 1 vote 1 answer 56 views ### Resampling techniques to obtain uniform random variables. Suppose we have n iid samples X_i from a distribution, say a triangular distribution symmetric about the origin, with support [-h,h], h>0 (denote this density f(x)). I would like to ... • 1,032 0 votes 0 answers 20 views ### Transformation of random variables to get a sample for another probability density function Question: Let \mathcal{D} be a two dimensional unit disk, given by \mathcal{D}=\{(x,y):x^2+y^2\leq 1\}. Using rejection sampling algorithm, I managed to generate a two dimensional random vector (... • 1,728 0 votes 1 answer 67 views ### Why is the variance of sample mean equal \frac{\sigma^2}{n^2} and not \frac{\sigma^2}{n} The variance of a sum \bar{X} of variables X_i for i=1,...,n is the sum of the variances \sigma_1^2 + ... + \sigma_n^2 If the variables all have the same variance then the sum becomes n\sigma^... 1 vote 1 answer 81 views ### The gap distribution of the random variables. Randomly sampling n numbers from the Normal distribution N(\mu, \sigma^2), whose PDF and CDF are f(a) and F(a), respectively.. We can get a list of number \{a_1,a_2,\dots,a_n\}. Sort the ... • 47 0 votes 0 answers 24 views ### Does uniform sampling from a sample set preserve its distribution Given a set of N i.i.d samples \delta_1, \dots, \delta_N, where each \delta_i \sim \mathbb{P} is distributed according to some distribution \mathbb{P}, i.e., (\delta_1,\dots,\delta_n) \sim \... 0 votes 0 answers 37 views ### How can we compute the Expectation of Log of Truncated Gamma over (0, 1]? How can we compute \mathbb{E} \left[ \log \text{Gamma}_{(0, 1]} (\alpha, \beta) \right]? Right now I'm using a Monte Carlo estimator by sampling from the Truncated Gamma distribution over (0, 1] (... • 25 2 votes 3 answers 88 views ### uniformly sample a point in a triangle (1,0,0), (0,1,0), (0,0,1) To choose a point with uniform distribution in a triangle A:(1,0,0), B:(0,1,0), C:(0,0,1), my thought is to project the triangle onto X-Y plane first, and the projected triangle is A:(1,0,0), B:(0,... 0 votes 0 answers 45 views ### entropy of weighted sampling without replacement Let us have k elements of weights w_1,…,w_k. We sample elements with weights without replacement. At the first, the probability of element i to be sampled is w_i / \sum_{l=1}^k w_l. Then, the ... 1 vote 0 answers 26 views ### Exponential distribution and random sampling Statement It's night and you are looking into the sky waiting to see a falling star. After the first hour you still haven't seen anything, so you check online and find two sources s_1 and s_2. ... • 11 0 votes 1 answer 24 views ### Why can we treat samples/permutations as happening one by one? When considering a sample over some set, we often picture picking elements for the sample one by one. This is used implicitly in many problems. For example, in determining how many k-length samples ... • 3 0 votes 0 answers 27 views ### What is the pdf of the sample mean of a distribution So I'm struggling with what I feel like should be a simple passage in a statistical inference book. Given an iid sample (X_1, ..., X_n) and a sample mean \bar{X} = \frac{1}{n}(X_1+...+X_n), if f(... • 21 2 votes 0 answers 21 views ### Mean of samples without replacement is no less concentrated than with replacement? Consider a population \mathcal{C} of N real numbers, possibly with multiplicities. For an integer n\leq N, let A_n be the random variable denoting the mean of n random samples of \mathcal{C}... • 733 0 votes 1 answer 20 views ### Prove that MLEs of two independent samples are independent I am trying to prove that if I have two samples of iid random variables, then MLE's based on these two samples will be also independent. More formally, let$$\mathbf{x} = (x_i)^T_{i = 1} \stackrel{iid}... • 159 0 votes 0 answers 14 views ### number of additional unique samples from a distribution? after drawing N times from distribution P (over a large but discrete set of things), we have K unique things. If we draw N+M times, how many unique things will we have, in expectation? make ... • 141 0 votes 0 answers 34 views ### Does the error of monte carlo integration scale with the number of dimensions or not? In this Wikipedia article, they derive the variance of a Monte Carlo estimator$Q_N$for a function$f: \mathbb R^m \rightarrow \mathbb R$, using$N$samples drawn uniformly over an integration region ... • 125 0 votes 1 answer 60 views ### is there a closed solution to counting pairwise combinations of the elements of two sets without resampling? ... • 218 3 votes 1 answer 85 views ### Sampling from Gaussian with very large covariance matrix in block form I'm interested in sampling from a Gaussian with zero-mean and covariance given by: $$\Sigma = \begin{bmatrix} \Sigma_{11} & \Sigma_{12} & \cdots &\Sigma_{1,100}\\ \Sigma_{21} & \... • 6,126 0 votes 0 answers 6 views ### Shouldn't the product of importance sampling weight and likelihood equal 1 in PSIS? I'm reading the 'Overthinking: Pareto-smoothed cross-validation' in Chapter 7.4 of Richard McElreath's textbook Statistical Rethinking 2nd edition. The author said: Cross-validation estimates the ... 0 votes 0 answers 27 views ### Efficient Method for Uniform Sampling from the Space of Increasing Vectors in [0, 1] I am seeking advice on methods for uniformly sampling from the space of increasing vectors within the interval [0, 1]. Specifically, I require an efficient algorithm that can handle high-dimensional ... 0 votes 0 answers 24 views ### Multivariate sampling for inherently positive data with given mean values and covariance matrix Context : We perform measurements of a set of parameters (p_1,p_2,...) to retrieve the experimental mean of each parameters and the covariance matrix between them. The means of the parameters are ... 0 votes 0 answers 38 views ### An inequality based on conditional independence of resampling Suppose we have i.i.d. samples X_1,X_2,...,X_n and Y_1,Y_2,...,Y_{K_n} comes from resampling from X with replacement, K_n\leq n. Can we say that Y_1,Y_2,...,Y_{K_n} are i.i.d conditional on ... • 123 1 vote 0 answers 47 views ### pythonic way to sample a point from a hyperplane of an euclidean space. Consider the 8-dimensional euclidean space \mathbb R^8. I denote a point in \mathbb R^8 by$$(a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$$I want to sample a point from a convex subset$S$of$\mathbb R^8$, ... • 194 0 votes 1 answer 52 views ### Number of Samples v. Number of Observations Background In the Equation (2.45) Introduction to Econometrics, GLobal Edition by Stock and Watson, it says$E(\overline{Y}) = \frac{1}{n}\sum_{i=1}^{n}E(Y_i) = \mu_Y$where observations$Y_1, \cdots, ...
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I would like to evaluate the mean and variance of the edge density for subgraphs obtained by repeatedly subsampling nodes. Specifically, suppose we have an undirected graph $G$ with $N$ vertices and ...