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Questions tagged [sampling-theory]

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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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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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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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1answer
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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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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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Picking path at random in DAG graph with probability equals to path weight.

I'm refering to the following paper: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.329.3653&rep=rep1&type=pdf. There is a lemma states: Let $G$ be a directed acyclic graph with ...
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Let $V_1$ be the variance of the estimated mean from a stratified random sample of size $n$ with proportional allocation.

Let $V_1$ be the variance of the estimated mean from a stratified random sample of size $n$ with proportional allocation. Assume that the strata sizes are such that the allocations are all integers....
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distribution of $\sum_{i=1}^n (X_i-X_{n+i})^2$ where $X_1,X_2,\dots,X_{2n}$ are iid $N(\mu,\sigma^2)$

Suppose $X_1,X_2,\dots,X_{2n}$ are iid $\mathcal N(\mu,\sigma^2)$ random variables. How can I find the distribution of $\displaystyle\sum_{i=1}^n (X_i-X_{n+i})^2$? Should I approach it by taking $Z=\...
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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
32 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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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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2answers
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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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How to randomly sample a social graph to find paths between at least 20% of profiles?

Given a Graph, where we know Total number of nodes (~100,000) Average no of connections per node (~200) Maximum distance between two nodes (~5) How many nodes (and its connections) do we have to ...
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19 views

Inverse transform sampling for a random variable with values in an infinite-dimensional space

I'm currently trying to understand the math used in a paper (page 3) that I'm reading: In light transport simulation we generate a light path from a virtual camera to a light source. A path may be ...
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14 views

Optimal importance sampling distribution

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(E,\mathcal E,\mu)$ be a measure space $f:E\to\mathbb R$ be $\mathcal E$-measurable $p:E\to[0,\infty)$ be $\mathcal E$-measurable ...
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Deriving variance for simple random sampling

Given, a proportion of a certain characteristic: $P = \frac{A}{N}$ I am required to show $Var(P)=\frac{PQ}{n}\bigg( \frac{N-n}{N-1} \bigg)$ $$ S^2 =\frac{1}{N-1} \bigg[\sum_i Y_i^2-\frac{(\sum_i ...
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Detection of certain stochastic process

Hello math stack exchange! Suppose we have arbitrary function of stochastic processes $X_i(t)$, like $$Y(t) = X_1(t) + X_2(t) - X_3(t) + X_4(t) ....,$$ more generally $Y(t) = f(X_1(t), X_2(t), X_3(t)...
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Reconstruction after Unitary Operator on Paley-Wiener space

Background: According to the Sampling Theorem for Reproducing Kernel Hilbert Spaces (RKHS), for any RKHS $H_{rk}$ with reproducing kernel $k(\cdot, \cdot)$, if I can find a set of points $\{t_{n}\}_{...
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Merge weighted random sampled set with different distributions

Here is my problem: I have a set $S_1$ of $N$ items each item i attached to a weight $w_{1_i}$. From this set I sample a subset of m items $(m << N)$ using weighted random sampling with ...
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Can I use random sampling? Or I need simple random sampling?

Context: I have a classification algorithm which works independently on an object. I need to improve the algorithm's accuracy by checking how the algorithm performs ever time I improve the algorithm. ...
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2answers
483 views

How is Welford's Algorithm derived?

I am having some trouble understanding how part of this formula is derived. Taken from: http://jonisalonen.com/2013/deriving-welfords-method-for-computing-variance/ $(x_N−\bar{x}_N)^2+\sum_{i=1}^{N−...
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Standard Error is of Population Total

We have the following data and we are required to obtain the standard error of unbiased estimate of the population total: $N=160,n=64,\sigma^2=4$ My approach We know that: $SE(\bar{X})=\frac{\...
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1answer
80 views

random sampling on random samples

I would like to understand if the sample distribution of the following approaches are the same or not. Setup: population of size $N$, with binomial distribution. required sample size is $n$ ...
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Sampling from a Mixture of Distributions

Sorry, but I give some detail before getting to my question, at the end. Suppose we have a probability distribution for random variable $X$. Let $X$ have the domain $\mathcal{X}$ and be ...
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1answer
99 views

Sampling without replacement with non uniform probabilities

In the article https://projecteuclid.org/download/pdf_1/euclid.aoms/1177704564 they describe procedure how to sample batch of size $n$ with nonuniform probabilities $p_i$, where probability of ...
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42 views

Linking probability measure to classical definition of probability when sampling from finite population

Consider a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ where $\Omega$ is finite, $\mathcal{F}$ is the power set of $\Omega$ and $\mathbb{P}(F)\equiv |F|/|\Omega|$ $\forall F \in \mathcal{F}$....
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215 views

Cardinality of Intersection and Union of Multiple Sets Given Overlap coefficient(s)

I would like to reason about the intersection and union of a number of sample sets and an assumed similarity between them. The calculation doesn't have to be exact, it can be a reasonable estimate. ...
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Which is the relation between between population/probability space/sampling?

I am trying to understand the relation between population/probability space/sampling in Econometrics. My arguments are divided in 3 sub-questions which trace my attempt to link in a logical way the ...
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[data generating process]-[sampling from an infinite population]-[i.i.d.]: some clarifications

I am confused on the relation among [data generating process]-[sampling from an infinite population]-[i.i.d.] Could you tell me whether what I wrote below is a correct interpretation or what is wrong?...
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1answer
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Defining Experiment

I was referring to my probability notes and came upon this note : Mutually exclusive events are choosen when events are raken from same experiment and independency is used when ...
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3answers
444 views

Conditional Probability theorem

Conditional probability is denoted as P(A|B)=Probability of occurrence of A when B occurs OR Probability of Event A when B becomes a sample space. Let us take an example:Let there be 5 white and 4 ...
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2answers
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Proof variance from mean and overall are not covaried?

Let $X_1, X_2,..., X_n$ be independent and identically distributed random variables having variance $\sigma^2$ Show that $Cov [X_i − \bar{X},X] = 0$ So I get that $E[X_i - \bar{X}] = 0$, so all I'm ...
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1answer
61 views

Sample space distinction

Suppose a pair of dice is thrown 5 times then it's space space will be of the form-(123456,345255.....etc).Can we say that it is equivalent to tossing 1O dice simultaneously?If so ,can you tell me ...
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1answer
81 views

How to determine a sample size to get accurate estimates of a given data set?

I have a question with a statistical nature; I think there should be some standard theory about this issue. Suppose I have a large data set of size $N$ items, which has an amount of $K<N$ ...
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1answer
35 views

Integral between max and 2nd max closing up over time?

Suppose you repeatedly sample from continuous distribution F with convex support Let's say you drew 2 4 3 5 in order. Denote the biggest number at $t$th sampling by $b(t)$, second biggest number at ...
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proof of the sample mean being the unbiased estimator of population mean in Simple Random Sampling with Replacement [closed]

Im'm looking for the simplest proof of the sample mean being the unbiased estimator of population mean in Simple Random Sampling with Replacement. I searched for this in literature in my native ...
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Prove a large number of samplings converge to a set of small number of samples

I don't know how to start with this problem. UPDATE: Thanks for the comments from @BruceET and @felipeh, I update the problem as follows: In a $k$-dimensional space, given a set of $n$ points: $X=\{...
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1answer
360 views

How to calculate sample size in a random sampling? [closed]

How one can calculate the sample size in any random sampling? Is it varies with sampling method or it is fixed for all methods? Explain it.
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1answer
139 views

Formula for estimating a continuous random variable in sampling, while most classic theory are discrete

Very often we would like to estimate a continuous variable Y (e.g. mean weight, mean length) in sampling design. However, most of the literature in sampling theory seem to treat the sampling variable ...
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38 views

Given times for athletes to jump is normally distributed, mean = 4, standard deviation = 0.5, what is the sampling distribution?

"A sample of 16 athletes are selected. What is the sampling distribution? Explain clearly and find the mean time and standard deviation" How do I go about answering this question? I know that the ...
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1answer
73 views

Distributed Sampling - Decide whether the item will be in Sample or not only once.

I will be getting a stream of items. I also know the sample size I need. When an item comes, I need to decide whether it will be in will be in the sample or not. I will not get second chance to either ...
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89 views

Stratum sampling cost minimization

I am trying to understand some text from the Sampling: Design and Analysis textbook on stratum sampling. To provide some background information, the variance of the mean is $$V(\bar{y}_{str}) = \...
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1answer
392 views

Bias of ratio mean estimator

I'm having trouble understanding this proof from a textbook for the bias of the mean estimator using ratio estimation. How does $$\frac 1 {\bar{x_u}}[BV(\bar{x})-\operatorname{Cov}(\bar{x},\bar{y}) = ...
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1answer
231 views

Nyquist–Shannon sampling theorem-proof

I don't understand the last derivation. Why there isn't $\pi$ in the sinc function? I would think that it is a mistake , but even in the Wikipedia it appears without $\pi$. https://en.wikipedia.org/...
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1answer
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Inclusion Probability in Simple Random Sampling (SRS) Without Replacement

Imagine that we want to choose a sample of size $ n $ from a population of size $ N $. Let $ j $ be a unit contained within the population of size $ N $. What is the inclusion probability of unit $ ...
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1answer
179 views

sigma squared notation

i wanna to know From where came No. 2 in this: $$E\left( \sum (x_i - \bar x) \right)^2 = E\left(\sum(x_i - \bar x)^2 +2\sum\sum (x_i - \bar x)(x_j - \bar x)\right)$$ and why in this removed $$2\sum\...
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How to sample near optimal solutions?

I have a function $f(x)$ that is strongly convex and the minimum of $f(x)$ is obtained at point $x^*$. I want to sample some $x\in \{x: ||x-x^*||<c \ {and} \ ||f(x)-f(x^*)||<d \}$. Is there a ...
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1answer
71 views

Central limit theorem does not work in this case?

Say we have a dataset that contains 1 million values and Every single value is equal to $ 5$. Say we take take a large number of samples from this set, and work out the mean of the sample each time. ...
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55 views

Generate independent samples from an exponential-like distribution on a n-1 dimensional simplex

The purpose here is to generate independent samples from the following distribution: $$P(x_1,x_2,\ldots,x_n) \propto e^{a_1x_1+a_2x_2+\cdots+a_nx_n}$$ and the support for the distribution is a n-1 ...
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What does it mean if $cov(f(x1), f(x2))$ is positive in the context of LHS sampling?

If cov(f(x1),f(x2)) is positive, does that mean f is close to symmetric along x1 and x2? I am struggling to put this into understandable terms. Edit: The context is equation 6 in this paper: http://...