Mathematics

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).

1402 questions
37
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4 answers

uniform random point in triangle in 3D

Suppose you have an arbitrary triangle with vertices $A$, $B$, and $C$. This paper (section 4.2) says that you can generate a random point, $P$, uniformly from within triangle $ABC$ by the following convex combination of the vertices: $P = (1 -…
dsg
  • 1,491
21
votes
1 answer

Probability to choose specific item in a "weighted sampling without replacement" experiment

Given $n$ items with weight $w_n$ each -- what is the probability that item $i$ is chosen in a $k$-out-of-$n$ "weighted random sampling without replacement" experiment? Can a closed-form solution that depends only on $w_i / w_\cdot$ be derived…
krlmlr
  • 640
17
votes
2 answers

Uniform sampling of points on a simplex

I have this problem: I'm trying to sample the relation $$ \sum_{i=1}^N x_i = 1 $$ in the domain where $x_i>0\ \forall i$. Right now I'm just extracting $N$ random numbers $u_i$ from a uniform distribution $[0,1]$ and then I transform them into $x_i$…
LordM'zn
  • 171
14
votes
2 answers

How does accuracy of a survey depend on sample size and population size?

Which survey is more accurate? Assume the samples are taken perfectly randomly. A sample of 100 people out of a population of 1000 (sample is 10% of population) A sample of 1000 people out of a population of 1000000 (sample is 0.1% of population) I…
Colonel Panic
  • 1,232
14
votes
2 answers

Multivariate Inverse Transformation Sampling

Summary Given a multivariate density distribution, I use inverse transformation sampling to sample points from this distribution. While the first dimension exhibits the correct distribution, all other dimensions contain a slight, stable error.…
Nico Schertler
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12
votes
2 answers

What is the distribution of gaps?

Randomly select $n$ numbers from the universe $\{1,2\dots,m\}$ with or without replacement, and sort the numbers in ascending order. We can get a list of number $\{a_1,a_2,\dots,a_n\}$, and then we can get the difference between two consecutive…
Fan Zhang
  • 1,997
12
votes
3 answers

Analytic center of convex polytope

I have a convex polytope defined by $Ax \leq b$ . I want to know how to find the "analytic center" of my convex polytope, because my goal is to sample from the polytope using Markov chain Monte Carlo (MCMC) methods, and it mixes better if i start…
Andy Yao
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11
votes
4 answers

Sampling from a $2$d normal with a given covariance matrix

How would one sample from the $2$-dimensional normal distribution with mean $0$ and covariance matrix $$\begin{bmatrix} a & b\\b & c \end{bmatrix}$$ given the ability to sample from the standard ($1$-dimensional) normal distribution? This seems like…
user57159
11
votes
3 answers

Bob took a quiz

Bob took a quiz consisting of 500 questions with two options: Yes and No. Bob did not prepare for the quiz. Flustered, he hastily reached for a fair coin in his wallet and started to toss this coin for answers. Whenever the coin landed heads, he…
Loki123
  • 347
10
votes
2 answers

What does it mean to sample, in measure theoretic terms?

Suppose I have some random variable $X$ defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$. What does it mean, in measure theoretic terms, to draw a sample from $X$? When $\Omega$ is finite, things make sense: we might say nature…
zenna
  • 1,519
10
votes
1 answer

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 $H$ , which has $\mu$ as its mean. For my purposes,…
πr8
  • 11,080
9
votes
2 answers

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 seems to be a difficult problem; and known…
Jonathan H
  • 2,393
9
votes
2 answers

A procedure for sampling paths in a directed acyclic graph

Consider a directed acyclic graph $\mathcal{G} = (\mathcal{N},\mathcal{E})$. Assume that the graph is very large (on the order of 10000 nodes and edges). Let there be a set of nodes termed starting nodes denoted by…
jonem
  • 453
8
votes
0 answers

Notation for sampling random variate

Is there standard notation for sampling a value from a probability distribution? Like, if I had a random variable $X$, setting $x$ to whatever value I happened to sample from $X$ on this occasion? I was thinking just $x \gets X$, but that seems…
Sneftel
  • 839
8
votes
1 answer

Probability of Choosing an Item in Weighted Random Sampling Without Replacement

Consider the problem of randomly sampling $k$ distinct items from a population of $n$ items without replacement. If all items have the same weight, then the probability that a specific item is among the $k$ selected items is $\binom{n-1}{k-1} /…
smz
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