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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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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 ...
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+50

How do you define the sample points used for Chebyshev approximation/interpolation?

It appears there are somewhat conflicting definitions of the points used in Chebyshev interpolation. Wikipedia and Numerical Recipes define the $x_j^{(n)}$ sample points for $(n-1)^\text{th}$-order ...
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1answer
48 views

Sampling distribution. How large does the sample size need to be?

I am building a data analysis model for clinics visits. Basically, I want to capture the patient arrival distribution for each specific visit type at each specific time. Let's say the previous data ...
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5 views

forward sampling for Bayesian network with continuous variables and equation based causal relationship

I have a physical system which can be represented by the following Bayesian network. It has the following characteristics The encoded variables are continuous variables. The causal relationships ...
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16 views

Sampling from a degenerate multinormal distribution.

Suppose $K \in \mathbb{R}^{n \times n}$ is a symmetric matrix with rank $p < n$ and $y$ is a random variable s.t. $$ p(y| \sigma) \propto \exp(-\frac{1}{2\sigma^2}y^TKy),$$ i.e. $y$ is ...
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11 views

Suggested noninformative hyperprior distributions?

I have a hierarchical model that includes a normal distribution and a beta distribution. For the normal distribution, it has two parameters: $\mu$ and $\tau^2$. However, I want to implement ...
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13 views

How many samples do I need to estimate the mean of a power law distribution?

Take the following power law distribution $$ p(x) = \frac{\alpha-1}{x_{\rm{min}}} \left(\frac{x}{x_{\rm{min}}}\right)^{-\alpha} $$ with $\alpha > 3$ so that the mean and the variance of the ...
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9 views

Is it possible to sample a preference matrix to train and test in recommender systems?

A preference matrix includes $0$ and $1$ elements only. Based on this article, p is constructed from a R matrix in the following way: $$ p_{user,item} = \begin{cases} 1 & \quad \text{...
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2answers
46 views

Rigorous Meaning of “Drawing a Sample” $\omega$ from a Probability Space $(\Omega, \mathcal{A}, \mathbb{P})$

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space. What does it mean (in the most formal and rigorous sense possible) to "draw a sample" $\omega \in \Omega$ from this space? Intuitively, ...
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11 views

sampling n unique items from large dataset

I have a dataset that has N of different unique items and each item appears $A_{i}$ times (every item appears different times). This is mean that I have the probability of each item. I need to find ...
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18 views

Sampling size estimate

I am interested in finding the sample size of a population of transactions in order to get an interesting conclusion on the population. For the estimate of the sample size $n$ in audit I found the ...
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11 views

How to create data out of sample?

I have a device, which measures events 1 out of R, where R is $O(10000)$. I started measurements at time $0$ up to time $T$, where $T$ is large. Now I have a sampled data set, recording event and its ...
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17 views

Expected value of sum from sample without replacement

Suppose we have n numbers, their values are 1 through n, and we sample k times without replacement from these n numbers, the probability of selecting any number is the same, with k < n, let X be ...
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26 views

Integral and inverse CDF of piece-wise Gaussian distortion in polar coordinates

I have a series of radii which are used to describe a series of concentric rings. I also have a piece-wise function which is evaluated on every radii. The function resembles a Gaussian-like distortion ...
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15 views

How to account/penalize for low sample size when computing performance?

I am interested in analyzing user performance at a per user level. Users have the opportunity to engage with an app by answering trivia questions and can choose how many questions they want to answer. ...
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13 views

Intuition behind convergence of MCMC inference methods

I'm studying Gibbs Sampling for inference, a popular MCMC algorithm and I was stunned by its ability to fit a Gaussian Mixture just by sampling. I would like to know the intuition behind it, and why ...
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7 views

Using Propp-Wilson for coalescent Markov chains.

I have a question regarding Propp-Wilson for coalescent Markov chains, please look on page 3 of https://arxiv.org/pdf/1805.09425.pdf for the definition I am using. So, they say with probabilty ...
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1answer
37 views

Sample random vector meeting constraint

I'd like to sample a vector $\mathbf{x}\in\mathbb{R}^k$ such that $\frac{\mathbf{x_i}}{|\mathbf{x}|}\geq c$ for all $i$ where $c \in [0,\frac{1}{\sqrt{k}}]$. Is this possible? How can this be done? ...
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23 views

Optimum sampling strategy for unknown function

this question is derived from a technical context (which I'll describe below so you get where this is aimed), but I think it's more of a math probelm: Assume you want to measure a sensor output which ...
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1answer
22 views

How to find sampling distribution of sample mean

So suppose $Y$ takes values $0$ and $1$ with probabilities $Pr(Y=1)=p=0.78$ and $Pr(Y=0)=1-p=0.22$ I calculated the mean of Y, which is $0.78$ and the variance of Y, which is $0.1716$. I also know ...
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1answer
34 views

What is the relation between $\sum_{i=1}^N x_ix_i^T$ and the covariance matrix?

Suppose $x$ is a random vector in $\mathbb{R}^n$ which is distributed according to $D$. Assume $x_i$ is a sample. What is $\sum_{i=1}^N x_ix_i^T$? How can I relate this to covariance of data $...
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29 views

Multiband sampling

Let's say I have a continuous function $f \in {L^2}\left( { - \infty ,\infty } \right)$ for which it is known that: $$\forall \omega \notin \left[ {0,L} \right] \cup \left[ {3L,4L} \right]{\rm{ }} \...
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33 views

Sampling vector at intersection of hypersphere and hyperplane

I need to sample a vector $(\gamma_1,\ldots,\gamma_k)$ which belongs to a hypersphere with radius $\varphi$ and to the hyperplane orthogonal to the $k$ dimensional vector $(1,\ldots,1)$. This can be ...
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1answer
30 views

Compare the mean of two samplings following normal distribution

The following is a problem for A-level test that I don't know how to solve. The weight of a pig is normally distributed with mean 375 g and standard deviation 22 g. The weight of a rabbit is normally ...
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16 views

Statistics - Question on Sampling

Here's the question The scores, $X_1$ and $X_2$, in papers $1$ and $2$ of an examination are normally distributed with means $24.3$ and $31.2$ respectively and standard deviations $3.5$ and $3.1$ ...
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1answer
21 views

Compute mean and covariance matrix of $\bar{X}$ from a simple random sample

Given $\{X_\alpha , \alpha =1,...N\}$ a simple random sample obtained from any p-dimensional distribution with mean $\mu$ and covariance matrix $\Sigma$, compute the mean and the covariance matrix of $...
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25 views

How to continue sampling after a conditional rejection?

I encountered the following (to me) weird problem while trying to do some simple sampling. I have a system that generates random numbers. Let's for simplicity assume the numbers are 0, 1 and 2 with ...
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19 views

Sample from Poisson process with no collisions.

On a 2D square $[0,\ 1]^2$, I can draw a random configuration of points $c = (n,\ (x_i)_{1..n})$ with interesting independence properties using a Poisson point process: draw the number of points $n \...
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1answer
24 views

Comparing two random variables with monte carlo sampling

Suppose there are two numbers X1 and X2 that are from a random continuous probability distribution with unknown range. You are given the value of X1 and you need to determine whether X1 is less than ...
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22 views

Constructing an upper confidence limit for $σ^2$

Suppose that $X_1, ..., X_n$ form a random sample from the normal distribution with unknown mean µ and unknown standard deviation σ. Construct an upper confidence limit U(X1, ..., Xn) for $σ^2$ such ...
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1answer
78 views

Problem of statistical inference Poisson

I am having problems solving this problem of statistical inference and I do not know if it is well done or not, so I would like someone to review it. I just started with inference, so I have my ...
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1answer
18 views

Using Random Sample to Find Estimate

I have to use the Inversion Sampling Method to generate a random sample of 100 from the function $f(x)=\theta x^{\theta - 1}$ if $\theta =5$. Here is my function so far: ...
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1answer
24 views

Showing that an estimator is consistent

Let $X_1,X_2,\ldots,X_n$ be a random sample from $\mathcal{N}(\theta,1)$. Consider the following (randomized) estimator of $\theta$ given a sample of size $n$: $$ \hat{\theta}_n = \bar{X} + \begin{...
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2answers
26 views

Finding the Expected value of $\hat{\theta}_n$

Let $X_1,X_2,\ldots,X_n$ be a random sample from $\mathcal{N}(\theta,1)$. Consider the following (randomized) estimator of $\theta$ given a sample of size $n$: $$ \hat{\theta}_n = \bar{X} + \begin{...
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17 views

Showing that a estimator is consistent

Let $X_1,X_2,\ldots,X_n$ be a random sample from $\mathcal{N}(\theta,1)$. Consider the following (randomized) estimator of $\theta$ given a sample of size $n$: $$ \hat{\theta}_n = \bar{X} + \begin{...
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21 views

How to show that $\sum_{i=1}^m (X_i−X_m)^2$ and $\sum_{i=1}^n(Y_i− Y_n)^2$ are independent

Let $X_1,...,X_m$ be i.i.d. sample with $N(\mu_1,\sigma^2)$, and $Y_1,...,Y_n$ be i.i.d. sample with $N(\mu_2,2\sigma^2)$. Let $S_x^2 = \sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= \sum_{i=1}^n(Y_i− Y_n)^2$...
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1answer
21 views

Using normal distribution to calculate $P(52.1 < \bar{x} < 53.9)$

I am given the following question in one of the lectures I was looking through. Suppose a random sample of size $n = 400$ is to be selected from a population of size $N = 2000$. A quantitative ...
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21 views

Is MCMC (or any sampling for that matter) explainable?

Recently, at an interview, I was asked if you use MCMC to build Maximum a posteriori (MAP), and use it for an inference, will the system you create have an explainability? Now, explainability is ...
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2answers
210 views

Confusing Sampling from observed data

Suppose we are given some small set of data on bundles of electrical wires and increasing voltages run through them, and we note how many of the individual wires fail. So for example, a large data ...
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33 views

Sampling a conditional joint distribution of continuous random variables using samples from joint distribution and marginal distributions

I am seeking an approach to sampling conditional joint distribution (new to probability). I will put my case in a simple way: Similar question for discrete variables is asked here but not yet ...
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1answer
19 views

Sample calculation

I have knowledge about the calculating sample, but i am unable to solve this question for the last two hours. Please check this question. A bank believes that approximately 2/5 of its checking-...
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0answers
16 views

The mean value of sample moment with order k

I've this problem of statistics that I can't resolve. So I hope that someone can halp me. The problem is this: Let sampling moments $\overline{X_n^k}=\frac{1}{n}\sum_{i=1}^nX_i^k$, where for $k=1$ $...
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1answer
42 views

Determining sample size given true proportion.

I'm attempting to solve a problem from a statistics course in regards to finding the sample size I need to take when given the Margin of Error, Confidence interval, and 'true proportion' (probability)....
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31 views

Working out expectation of a random sample.

I have the problem: Let $X_1, X_2, X_3, X_4$ be a random sample from a population that has mean $μ$ and variance $σ^2$. Find $\mathbb E[(X_1-X_2)^2]$ and hence the value of $k$ such that $T ...
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2answers
23 views

What is the probability that $m$ items drawn from $n$ distinct items contain all $n$ items?

Suppose there are $n$ distinct items to be drawn with replacement for $m$ times, the probability of each item being drawn is assumed to be $\frac{1}{n}$. What is the probability $P(m)$ that $m$ items ...
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13 views

Sampling from joint discrete distribution

I have a set of items $a_1, a_2, \dots, a_n$. My aim is to generate from this set of items, a list of item tuples $\{(a_i, a_j), \dots\}$ such that $a_i\ne a_j$. The constraints are as follows. The ...
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0answers
19 views

Sampling binary values from a discrete probability distribution

Suppose that we have a discrete non-uniform probability distribution X over $\{0,1\}^k$ for n binary noise values. Let $e = (e_1 ,..., e_m)$ be a vector of independant identically distributed binary ...
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1answer
18 views

Prove the sampling distribution of $S^2$ has the mean $\sigma^2$ and the variance $2\sigma^4/(n-1)$

I would like to ask whether anyone would mind providing me with some direction on how to proceed with this proof. The question asked me to use the theorem below to prove that, for random sample of ...
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24 views

calculating variance and expectation of unknown binomial variables over a window

I have $2^m$ independent random variables. All have binomial distributions, each with $m$ samples. The probability of success for each binomial distribution is somewhere in the range $[0,p]$ (...
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2answers
30 views

Definition of sample mean

I've seen two definitions of sample mean on the internet. One definition defines it as the average of Random variable other defines it as the average of sample values of a sample. I'm confused which ...