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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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Sampling distribution output does not look like a continuous function

I made a random distribution using 2 dice rolls, the output usually collects most data around mean: Then I try to do sampling of N sample size. While using sample size of N = 2 it looks more or less ...
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1answer
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Sampling with and without replacement.

Prove that the probability of drawing a unit at any draw from a population of size N, remains same in without and with replacement sampling scheme. I know how to prove that this probability is $\frac{...
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How to sample two values from a random variable X with the lesser to be a random variable Y?

The variable X has pdf $$f(x) = \frac18(6 - x)$$ for $$2 ≤ x ≤ 6$$ A sample of two values of X is taken. Denoting the lesser of the two values by Y, use the cdf of X to write down the cdf of Y. Hence ...
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How can I calculate if my data is diverse

I conducted a survey in which the first question I asked students what year they were in. I want to check that I have a diverse amount of students (from many year groups) so I can say the survey data ...
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Sampling from Geometric distribution in constant time

I would like to know if there is any method to sample form the Geometric distribution in constant time without using $log$ which can be hard to approximate. Thanks.
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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
28 views

Approximate Probability With Beta Distribution

We are given a sample of 7 proportions for the percentage of cloud cover recorded at a set time every day for a week, where the sample mean is $\bar{x}=0.51$ and the sample variance is $s=0.3277$. We ...
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How do I evaluate (prove to myself) that a method for picking uniformly distributed values is correct?

To make this more specific, I show a broken procedure for generating random points in a circle and a correct (hopefully) procedure for generating random dates within an interval. I'd like to be able ...
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34 views

Copula used for pseudo-random generation from continuous variables only?

I know multivariate pseudorandom generation from continuous distributions can be done using copulas and inverse transform sampling. The question is if copula have application in "discrete" pseudo-...
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Finding the conditional distribution from empirical joint CDF

Suppose we are given a table containing the estimates of a discrete bivariate CDF $P(X_1\le x_1, X_2\le x_2)$ and also the known value of a realization of the first variable $X_1 = x_1$. What are ...
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Magnitude spectrum of a sampled time continuous signal

I have a signal $x(t)=\cos(2 \pi 200 t)+2\cos(2 \pi 400 t)+\cos(2 \pi 600 t)$. With sample frequency $F_s=1000 Hz$. How do I draw the magnitude spectrum of the sampled signal $x(n)$? I've tried ...
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What curve represents random sampling sorted by frequency?

I perform a discrete random sample with a uniform distribution (such as drawing colored balls from a bag with replacement). I then plot the frequency that each color has been selected, ordered by ...
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30 views

How many items must be sampled to ensure normal distribution?

I was working on a problem that was dealing with proportions and it asked the following: If a population proportion is believed to be $0.6$, how many items must be sampled to ensure that the sampling ...
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59 views

How can we approximate a function by sampling a distribution proportial to it and making a histogram of samples?

I've read the following (here on page 2): Suppose that you want to approximate a function $f$. One way to do this is to produce a sampling distribution proportional to $f$ and then make a histogram ...
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Statistics - Stratified random Sampling

First, have a look at "Max Ft"'s answer to the following question: Stratified Sampling for Variance Reduction--Need Intuition as to Why it Works His answer makes sense for how stratified random ...
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Determine the number of samples needed with Bayes theorem

I have a simple sample statistic problem, but I am not sure that I solve it properly. I have to enter many student marks to a test in an Excel table. After entering all $N$ marks, I want to know the ...
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Acceptance and rejection sampling.

The standard logistic distribution function F given by $F(x) = 1/(1+e^{-x}) , x$ is a real number. How to generate a random sample of size 10000 using acceptance-rejection sampling using standard ...
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Samples from high dimensional distribution by exploiting the symmetry

I have a discrete 7-element random vector $\vec{X}$ with probability mass function $P_{\vec{X}}(\color{blue}{x_1,x_2,x_3},x_4,x_5,x_6,\color{red}{x_7})$ that has a symmetry in its certain components. ...
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Expected number of tries to choose x unique values

it's been a long time since I've dealt with probability so I thought I would ask here. I'm sampling elements independently and uniformly and with repetition from a population. Given that the ...
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1answer
22 views

Proof Random Sampling Interview

Got this algorithm in an interview. To prove that the algorithm randomly shuffles the array. The random sampling algorithm works as follows: ...
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26 views

N dimension random distribution with constraints

I am trying to draw $M$ random/semi-random numbers in $N$ dimensions applying some constraints. For instants I have a method that "selects" only the right points, but I am sure that it can be done ...
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Divergence between Probability Distributions from Samples via the Chamfer Distance

Suppose I have two probability distributions $P$ and $Q$. I want to compute a divergence/distance between them. I do not have access to their densities, but I can draw samples $x\in D \subset \mathbb{...
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Hogg Intro to Mathematical Statistics Sufficient Statistic Notation Question

When introducing the notation of a sufficient statistic, Hogg uses a peculiar sort of notation at the bottom of page 381 of the 7th edition textbook: given the statistic $$Y_1 = u_1(X_1,X_2,...,X_n)$$ ...
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How to implement a Metropolis Hastings step

I am having trouble implementing a Metropolis Hastings step in a Gibbs sampling problem. The following code was taken from here. Details: It is a capture recapture study, with seven total draws from ...
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1answer
60 views

What is the mathematically appropriate and concise way of writing a constant added to a random number?

Let $x$ be a constant in $\mathbb{R}$. Let $y$ be a random number that is generated according to a certain probability distribution. I want to make the sum $$x + y$$ However, I am not sure how to ...
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31 views

How to sample from a reconstructed pdf

Suppose, that we have a process that generates events. Subsequently these events are recorded and stored in a histogram. Say we have a source that emits particles and subsequently a histogram is build ...
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Hamiltonian monte carlo sampling : Energy Histogram vs Sample Histogram

I'm using HMC to sample from an N-d Gaussian. So the PDF is that of a multivariate normal distribution. HMC requires an energy function and its gradient. The library I'm using, maximizes the ...
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population distribution & sampling distribution [closed]

What is the difference between the population distribution of a random variable X and the sampling distribution of a random variable that is a sample statistic? This is a question I got from my ...
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How to come up with variance for estimators?

I know that the variance is calculated with $Var(X) = \frac{1}{N} \sum^N (X_i - \bar{X})^2$ However, how do I come up with/derive variance formulas for e.g. the mean for different sampling methods. ...
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Hypergeometric distribution with heterogeneous sampling probabilities

I have been trying to find a proper framework for finding the corresponding generalization of the hypergeometric distribution when the sampling probability from the total items $N$ is not uniform. ...
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10 views

Least square estimate for post-stratification sampling

enter image description here I figured out via the normal linear regression method that Beta0 hat = ybar - Beta1 hat xbar. But I am not sure how to find out the least square estimate for Chat. Is ...
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19 views

Finding the bias of an estimator

Consider the following model: $$ y_i = a + b x_i + c z_i + w_i, $$ where $a,b,c$ are unobserved fixed parameters, $x_i$ and $z_i$ are fixed in repeated samples. Assume also $\mathrm{E}[w_i] = 0$ for ...
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probability of extracting predefined k elements out of N in V extractions with replacement.

I have a set of $N$ elements from which I do $V$ extractions with replacement. Given a subset of size $k$. What is the probability that all the $k$ elements are extracted at least once? In other ...
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Uniform sampling of convex polytopes: why is it hard?

There is a surprising number of posts asking about the uniform sampling of convex polytopes (e.g. 1, 2, 3), and an equally surprising number of non-answers (e.g. a, b). From what I have read, this ...
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27 views

Proof on how to sample from a truncated exponential distribution

I understand that if i want a sample from an exponential distribution left truncated at a, i can just take a sample from a regular exponential distribution and add the value of a to every single ...
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Constructing a probability measure on the Hypercube with given moments

Let $H = [-1, 1]^d$ be the $d$-dimensional hypercube, and let $\mu \in \text{int} H$. Under these conditions, I can explicitly construct a tractable probability measure $P$, supported on on $H$, ...
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sampling mechanism for a data pair (X1, X2) where X2 depends on X1

If X2 is dependent on X1, how to generate the random sample of (X1,X2)? One scenario is that we know the prior distribution of X1 and functional relationship between X1 and X2, how to generate the ...
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Meaning of “uniformly sampling”?

Reading an article I came across the following expression: '' $\overline{X}$ is constructed by sampling uniformly along the straight line between the pair of $X$ and $\hat{X}$." I know what a ...
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Probability of having a good set by choosing independently from Universe

Let $S$, $T$ be two disjoint subsets of a universe $U$ such that $|S| = |T| = n$. Suppose we select a random subset $R\subseteq U$ by independently sampling each element of $U$ with probability $p$; ...
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Weighted random sample over continuous data

I'm attempting to write an algorithm which gives a random value x in the domain [0-1) and is weighted according to a function. I don't seem to be able to determine how to do this with continuous data, ...
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13 views

Which shape a sampled point belongs to?

Given a circle and an ellipse, both centered around the origin. The radius of the circle is a, and for the ellipse, the major radius is ...
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Distribution of overlapped sampling

Given $n$ i.i.d. random variables $X_i$ for $1 \le i \le n$. Consider a moving block bootstrap sampling with block size $l$: $B_i = (X_i, \ldots, X_{i + l - 1})$ for $1 \le i \le n - l + 1$, there are ...
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Sampling subsets of positive numbers with sum close to a given number

Given a set of $N$ positive numbers $a_1 \ldots a_N$, I am trying to generate random subsets of this set such that the sum of numbers in subset is close to a given number $M$. Suppose I can only ...
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Finding the most typical individual in a mutivariate population

I have data for a large number of unique populations. In each population each individual is described by a number of categorical and numeric variables. I want to conduct an analysis on each population,...
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How to derive the distribution of mean estimator based on ranked set sampling from normal distribution

I am studying about ranked set sampling from normal distribution. From wolfe,2004 the joint pdf of RSS is $$f_{1,2,...,n:n}(x_1,x_2,...,x_n)=\prod_{i=1}^nf_{i:n}(x_i)$$ where $$f_{i:n}=\frac{n!}{(i-...
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How to fill a large Gaussian with N smaller Gaussians

Lets say I have a multivariate Gaussian. I'd like to "pack" that Gaussian full of smaller Gaussians. Intuitively, it seems like I would just sample N points from the original MV Gaussian for the ...
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How to sample from a multivariate normal distribution in hyperbolic space?

I want to generate samples from a multivariate normal distribution on the hyperbolic space, such that $\forall x \in \mathbb{H},$ $$ \text{pdf}(x) = \frac{1}{c} e^{-\frac{d^2_{\mathbb{H}}(x,\mu)}{2\...
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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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What is the name of this principle?

When generating uniformly-distributed samples from a multidimensional distribution, I believe that sampling each dimension independently produces uniformly-distributed samples from the original ...
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Rate of convergence of unknown distribution on an interval

I have an unknown probability distribution $p(x)$. I don't know what it is, but I know it is well-behaved (smooth and normalised and goes fast to 0 at infinity). I've created an illustrated example of ...