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).
1,381
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A doubt regarding sampling with and without replacement
For a positive integer $N$, let $[N] = \{1,\cdots,N\}$ and $\binom{[N]}{n}$ denotes the set of all $n$ sized subsets of $[N]$. Supose we want to pick $(n+1)$-sized sample of $[N]$ uniformly randomly (...
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How to sample this subset of $\mathbb{Z}^6$ uniformly
Let $X \subset \mathbb{Z}^6$ be the set of all $(a_1, a_2, a_3, a_4, a_5, a_6)$ with all $0 \leq a_i \leq 63$ and $\sum a_i \leq 127$. The constants $63, 127$ are mostly arbitrary.
I am trying to come ...
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Conditions and techniques for discretizing a real function
Sampling theorem states:
If a function $x(y)$ contains no frequencies higher than $B$ hertz, it is completely determined by uniform samples taken at less than $1/2B$ units apart.
Original sampling ...
user947420
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Statistical significance in context of financial data?
I understand statistical significance in the general sense: we take a sample from a population and compute some parameter from the sample to infer what is the propulsion parameter to some degree of ...
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What are the necessary and sufficient conditions for Gibbs Sampling of discrete random variables?
Given a sample $\left\{ X_i: i=\overline{1,n} \right\}$ with values from $\mathcal{X}$
and their conditional distributions $\mathbb{P}\left( X_i \mid X_t, t \neq i \right)$,
I want to find their joint ...
- 497
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Distribution of maximum of sample means
Let $X_1, ..., X_n$ be a sample from $N(\mu, 1)$. Fix $1 \leq m<n$ and define $$T_i= \frac{1}{m}\sum\limits_{j=i}^{i+m-1} X_j,$$ for $i \in \lbrace 1, ..., n-m+1 \rbrace$. We have the test that ...
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Question about multivariate normal and univariate normal
Suppose I have $\hat{\mu}, b \in \mathbb{R}^d, B \in \mathbb{R}^{d \times d} $.
And sample $\tilde{\mu} \sim \mathcal{N}\left(\hat{\mu}, B^{-1}\right)$ (from multivariate normal distribution).
Then ...
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How to perform Gibbs sampling for this distribution?
I tried to sample this equation by Gibbs sampling.
$
P\{X=i,y \le Y \le y+dy,N=n\}\propto C^n_iy^{i+\alpha-1}(1-y)^{n-i+\beta-1}e^{-\lambda}\frac{\lambda^n}{n!}dy
$
I know I should generate X given $...
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Designing a sampling scheme for a weird problem
Suppose I have a sequence of objects in a line, one by one. Each time, I must draw 5 objects (or any integer less or equal to the total number of objects) next to each other, i.e., a window of size 5.
...
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Find $P(2\bar{X}-\bar{Z}>3\bar{Y})$
Let $(X_{1},X_{2},X_{3},...,X_{8})\sim N(8,16)$ is independent random sample , $(Y_{1},Y_{2},Y_{3})\sim N(1,9)$ is independent random sample and $(Z_{1},Z_{2},Z_{3})\sim N(6,10)$ is independent random ...
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Sampling with replacement with multiple attempts
Let us consider sampling over $N$ objects in a bowl. $Q$ sampling attempts are made, each attempt consisting in picking $K_i\leq N$ objects, with $i = 1,\dots, Q$. After each attempt, the objects are ...
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OFDM IDFT implementation....
i have some doubts with the implementation of the IDFT in OFDM systems.
The question concerns the expression of the IDFT of the OFDM signal.
During the symbols period $T_s$ we have the following base-...
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Sample size vs number of samples used to get variance of sampling distribution of sample means
I am new to statistics and got following confusion:
Suppose there is a population (with variance $ \sigma^2 $) and we draw 10 samples each of size 7 from this population.
For example:
\begin{align*}
...
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Measure-Theoretic Importance Sampling: Do we need equivalence of measures?
Let $\pi$ and $\mu$ be the target and proposal measures on $(X, \mathcal{X})$ respectively, with $\pi \ll \mu$. Suppose $\lambda$ is the reference measure on $(X, \mathcal{X})$ and that $\pi\ll \...
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Is there a combinatorial structure easy to count while difficult to sample?
In the theory of complexity,
the counting problem is to compute the number of specified structures, e.g., count the triangles in a given graph.
the sampling problem concerns how to generate uniformly ...
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How to sample random variables X and Y from a joint distribution.
I have the following joint distribution function $f(x,y)$:
$$f(x,y)=\begin{cases}
\frac{1}{30}xy+\frac{x}{y^2} & \text{ for } 1\le y\le 4,\ 1/2\le x\le 3/2\\
0 & \text{ otherwise.}
\end{cases}$...
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Explanation of Delta Dirac Function
when we turn a continuous signal into a time discret signal we use this function:
$\hat{s}(t)=s(t) \sum_{n=-\infty}^{\infty} \delta\left[t-n T_{a}\right]$, with $\delta\left[t-n T_{a}\right]=\left\{\...
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Sampling/determine dependencies in a nonlinear space
I hope you can help me with your expertise and breadth of knowledge for a problem that I will try to articulate as best as possible :)
Context: For an object, I can calculate n (in my case 9) features ...
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Determine sample size when the population distribution and variance is unknown in interval estimation?
How can I determine sample size in interval estimation?
The oridinary way of determining sample size $n$ would be like the following:
Let $n$ be big enough to the extent of Central Limit Theorem.
...
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Expected value of function of vectors from the p-variate normal distribution - Data Analysis Sample Covariance
I am proving that the sample covariance matrix $\mathbf{S}=\frac{1}{n-1}\mathbf{X}^T\mathbf{X}$ is an unbiased estimator for the covariance matrix $\mathbf{\Sigma}$, where $\mathbf{X}$ is a mean ...
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Algorithm for sampling items fails the tests
I was looking into a coding exercise which asks the following:
Given an array of positive integers $w$ where each $w_i$ describes the
weight of the $i^{th}$ element, implement an algorithm that ...
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Draw samples from binary, mutually dependent joint distributions
I am drawing a mental blank right now:
I have three quadratic matrices $A_{00}$, $A_{01}$ and $A_{11}$ of size $n$. Those represent binary, joint bivariate distributions, i.e.
$$(A_{00})_{i,j} = P(x_i ...
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Is there an efficient estimator for $\theta$?
Given a simple random sampling with density function:
$$f_{\theta}(x) = \frac{\ln(\theta)\theta^x}{\theta - 1} I_{(0,1)}(x)\,, \quad\theta \gt1$$
Is there an efficient estimator for $\theta$?
I know ...
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Find the distribution of the statistic defined by $T = \frac{S_{n-1}^2/\sigma_{1}^2}{S_{m-1}^2/\sigma_{2}^2}$
Let $X$ and $Y$ random variables:$\phantom{3}X \sim N(\mu_1, \sigma_1)$, $Y \sim N(\mu_2, \sigma_2)$
Given 2 simple random sampling of size $n$ and $m$ from $X$ and $Y,$ I'm trying to obtain the ...
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Sampling as adding random variables, especially binomial RVs
Is sampling equivalent to adding random variables? I'm a bit confused because as we can see that the binomial distribution becomes more and more shaped like a normal distribution as $n$ increases. We'...
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How population size affects the variance of the distribution of sample mean
We know the variane of the distribution of the sample mean is $\sigma_{m}^2=\sigma^2/n$, where $n$ is the sample size and $\sigma^2$ is the population variance.
Say we have two populations -- all male ...
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error in calculating the posterior in basic urn sampling problem
As I'm reading E.T. Jaynes' Probability Theory and while trying to do some calculations by myself (in this particular case, re-derive equation 6.15) I'm running into a problem where I just can't find ...
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Sampling without replacement from a population with known fractions of duplicates, triplicates, etc.
Suppose I have a population of a known size. For this population, p_1 + p_2 + .... + p_n = 1, where p_1 is the fraction of the population that has no duplicates, p_2 is the fraction of the population ...
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What is probability that more than 30% of owners pay to walk their dog?
This equation comes from Edgenuity's course of Statistics, and I am taking the course as a high school senior. I understand how to find the z-score and the standard deviation of the sampling ...
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How do I calculate this sampling error?
If I sample individual integer values $0$ to $100$ from a distribution $100000$ times and record the counts for all integer values between $0$ and $100$, how do I calculate the $95$% error bars for ...
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Estimate Ratio of Normalizing Constants from two datasets
Suppose I have a non-negative function $f:\mathbb{R}^N \to [0, +\infty)$ that defines two different (unnormalized) probability densities on two separate subsets $A, B \subset \mathbb{R}^N$ with $A \...
user318854
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Random samples and the distribution w.r.t the population
I have read a couple of hours about random sampling and the distribution and I guess that I have figured it out, but I am not 100 % sure. So, maybe one could cross-check my claims :-)
Assume we have a ...
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Can the definite integral of zero valued function be non-zero?
I am trying to understand the Importance Sampling Technique (IS) for rare event simulation. I use the this tutorial. On page 5, it is written:
"...That is, let g be another probability density ...
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Can one permute samples in Metropolis-Hastings to solve autocorrelation problem?
The wikipedia article on Metropolis-Hastings algorithm suggests that using all of the sample points produced by the algorithm as i.i.d samples from the underlying distribution is bad, since ...
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Repeated samples of multiple draws without replacement between draws or samples
We have a urn containing $N$ marbles of $m$ colours, i.e. $N = \sum_{i=1}^{m} K_i$.
If we draw $n$ marbles from the urn without replacement, the probability that we have a specific distribution $(k_1,...
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Z-Test vs T-Test
A researcher claims the mean score on an agility test will go down after drinking alcohol. The mean score on the test is historically 8.6 for people not under the influence of alcohol. The researcher ...
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Two-Sample T-test
A philosophy professor wants to find out whether the mean age of the men in his large lecture class is equal to the mean age of the women in his classes. He randomly selects 50 men and 50 women. What ...
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Is there tractable way to use Determinantal Point Process for bulky sets?
I want to use a method to select a diverse subset of a bulky set, to which the Determinantal Point Process is a solution. However, based on my basic knowledge of DPP, the first step is making a matrix ...
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how to find the distribution and parameter given normal distribution?
given that $K_{1}$, $K_{2}$, ... , $K_{n}$ is a random variables from a normal distribution with $μ$ = 10, $σ^2$ = 2.5
what is the distribution and parameter for $X$ = $\sum_{i=1}^{10} \left(\frac{K_{...
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Show independence without using Basu theorem
$X_1,...,X_n,i.i.d,X_1\sim N(\mu,\sigma^2)$
prove that $\xi=f(X_1,...,X_n)$ and $\bar X$ are independent, where $f(X_1,...,X_n)=f(X_1+c,...,X_n+c)$ for any constant $c$
I know this can be done by ...
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Generate random samples using Box-Muller
In general, the Box-Muller algorithm samples two independent uniform variates on (0,1) and transforms them into two independent standard normal distributions via
and
Is it possible to use the Box-...
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516
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How to multivariate stratified sampling
How can I implement stratified sampling when each sample holds multivariate information. Also am not sure how to name this problem or where to exactly look for the solution so any indication would be ...
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Frequency of the most frequent element using random sampling
Problem statement
I want to find the frequency $f_{max}$ of the most frequent element (mode) in a list $L$. The element itself is not needed. For example, for $L=[5,3,5,7,7,5,0]$, we have: $$f_{max}=\...
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Random permutation vs uniform sampling
Consider a set $S=\{1,2 \ldots n\}$. I am constructing two random multi-sets $X$ and $Y$. $X$ is a uniformly random permutation of $S$ whereas $Y$ is constructed by drawing n samples uniformly ...
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Why does estimated values for a coefficients in a linear model change when insignificant variables are excluded from the model?
I am trying to determine the illnesses that could contribute to an outcome of a patient ( recovered / dead / transferred to another department) and I'd like to know how big a mistake it is to leave ...
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understand the solution to the unbiased estimator of area of circle when given n independent radius $R$ measurement with error $\sim N(0,\sigma^2)$
$S = \pi R^2$
$E(\bar{X}^2) = Var(\bar{X})+(E(\bar{X}))^2=\sigma^2/n + R^2$
then it states thats an unbiased estimator of $\sigma^2/n = \frac{1}{n(n-1)} \sum_{i=1}^n(X_i-\bar{X})^2$.
However, when i ...
- 561
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45
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Finding the range of expected value of a random variable in both sampling with replacement and without replacement
Let $S$ be a set of $c$ elements sampled uniformly at random from the set $\{a_{1}, a_{2}, \cdots, a_{t}\}$, and $0 \leq \delta \leq 1$.
Let $X$ be a random variable that counts the number of elements ...
- 415
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design a sampling mechanism to guarantee approximated correctness
Given a function $f: X \to \mathbb{R}$ with some discrete domain $X$, $|X| = 2^n$. The cardinality of the domain is expontential.
Our objective is to find $\textbf{argmax}_{x \in X} f(x)$. Which can ...
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205
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Class size paradox vs. length-biased sampling
According to Introduction To Probability by Blitzenstein and Hwang p. 244, one example of length-biased sampling is the following:
"For example, asking randomly chosen mothers how many children ...
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57
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Finding the sample size of students with the confidence level
Here is my question:
How large a sample of students would be needed in order to estimate the population prediction mean within ±2 with 85% confidence?
Given that the sample of wight students ...
