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 distributions and test of hypothesis

A manufacturer of a certain type of breakfast cereal claims to produce packets which contain on average 500 grams of cereals. Ten packets were selected at random and the cereals content of each ...
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13 views

Statistics Sampling Type

My question is on Q7. I can't seem to figure this one out. I thought it was a random statified cluster sample because it is breaking down the schools into subsections and then pulling 3 homerooms ...
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6 views

What is the importance of the inclusive range in Reservoir with Random Sort?

I am reading the Reservoir with Random Sort page on Wikipedia, and the algorithm says (copied): ...
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18 views

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 Monte-Carlo Markov ...
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2answers
60 views

Is the Dikin Ellipsoid actually a ball?

I have the inequality (row wise): $Ax \leq b$ The Dikin ellipsoid centered at $x_0$ with radius $r$ is: $$\{z \quad | \quad (z-x_0)^T(z-x_0) \leq \frac{r^2}{H(x_0)}\}$$ where, $$H(x_0) = \sum ...
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Is the alias method “stable”?

The alias method is an algorithm for sampling from a discrete distribution. Let me describe it briefly. First there is a setup phase. You have $N$ values and associated probabilities. You introduce ...
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24 views

Sampling distribution for sample mean and sample variance in case of Normal population [closed]

Let $X_1,X_2...,X_n$ be a random sample drawn from $N(\mu,\sigma^2)$ popultion. Further let $\bar{X}$ and $S^2$ be sample mean and sample variance respectively. Then Find the joint PDF of ...
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1answer
12 views

Random sampling-level of significance

Random samples of house selling prices are obtained from the north and south regions of a country. The results are summarized below: ...
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35 views

Sampling distribution of $Y = \frac{\ln U_1}{\ln U_1 + \ln (1 - U_2)}$, where $U_i \sim U(0,1), \forall i$

For this problem I have used the fact, $-2 \ln U \sim \chi^2_{(2)}$. But I have doubt on the independence of numerator and the denominator which are $\ln U_1$ and $\ln U_1 + \ln (1 - U_2)$. If they ...
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23 views

How to sample from a convex hull?

Let us consider a couple of points $x^{(i)}\in \mathbb{R}^m$ where $i=1,\dots,n$. Convex hull is defined as $$ C = \left\{\sum_{i=1}^{m} \alpha_i x^{(i)} \mathrel{\Bigg|} (\forall i: \alpha_i\ge ...
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12 views

Error propagation with dependent errors

I have a function $f(x_1,\ldots,x_n)$ where the variables $x_k$ have errors $\delta_k$. If these errors are independent I can add them root mean square: $\delta ...
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1answer
19 views

Comparing percentages of a sample to that of the population.

This might be stupid question, but I'm in this sort of situation: 60% of people in a city have a pet cat, but the national rate is 50%. So, assuming we have the required bits of information about ...
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16 views

How to describe a frequency spectrum with its samples?

I'm not sure if the following would be more physics-related, but since statistics are involved, I thought I'd post this here... To me the question is pretty straightforward, but nevertheless I have ...
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1answer
39 views

How can one use a probability distribution to sample from a population

Let us assume that we have a population and we interested in specific property of each element of this population. Let us assume further that this property follows a normal distribution X ~ ...
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1answer
16 views

Finding a uniform distribution on the output of a multivariable function

Suppose we have a non-invertible continuous function that maps from some continuous interval ${I}^n$ to $\mathbb{R}$ with $n \ge 1$. To take an example, let $f(a,b,c) = a \cdot e^{-bc} - b \cdot ...
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2answers
71 views

Independence of Poisson random variables coming from poissonization trick

Context: Let $x \in \mathbb{R}^n$ be the unknown probability vector of a finite discrete distribution $X$. We are able to sample $X$ and we want to learn $x$. What is the minimal sample size needed ...
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1answer
50 views

How to calculate sampling error?

Given a reservoir of size $S$ with each element taking a value of error or not an error, we attempt to estimate the number of errors inside the reservoir through the following We poll the reservoir ...
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2answers
28 views

Would like some help formulating an optimization problem

I have a function $f$ that takes $n \geq 1$ positive real-valued arguments $\mathbf{a} \in R^n_+$. This function is defined for all amounts of inputs (e.g. $f(1)$ and $f(3, \pi, 17)$ are both valid) ...
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29 views

Picking and replacing balls from a bag until you are relatively certain you have picked each one at least once

Suppose I have an unknown number of balls ($N$), each of a different color, hidden in a bag. How many times must I draw a single ball, make a note the color and return it to the bag in order to be ...
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1answer
23 views

Delaunay Triangulation on Convex Polytopes — Uniform Sampling

My goal is to uniformly sample from a convex polytope. I know that for the simpler case, where I have to uniformly sample from a simplex, I can use Bayesian Bootstrap, discussed in these posts: ...
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1answer
35 views

Simulating Random Vectors Based on Conditioning

I'm working on a project where I need to simulate random vectors $(Y, X_1,\dots,X_n)$ in order to understand the joint distribution $f(y,x_1,\dots,x_n)$. I wish to simulate enough random vectors so ...
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SRSWOR involving Weighting

A Simple Random Sample Without Replacement (SRSWOR) survey is conducted that included too many women and not enough men in the sample In the resulting weighting, each female is given a weight of $1$ ...
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30 views

Gibbs sampling truncation for contrastive divergence

I am following Yoshua Bengio's Learning Deep Architectures for AI and at page 31 there is a phrase that confuses me. Starting by lemma 7.1 in the same page: Lemma 7.1. Consider the Gibbs chain ...
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36 views

How to proceed with this simple proof?

If $$\alpha_k = \sum_l a_l \ \ g((k-l)T-l\Delta T)$$ $$s_k = \sum_l \alpha_l \ \ q((k-l)T+k\Delta T)$$ where $a_l \in \pm1$ and $g(t) = \frac {\sin(\pi t/T)}{\pi t/T}$ and $q(t) = \frac {\sin(\pi ...
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2answers
38 views

Sampling from the von-Mises Fisher distribution?

This topic has already been tackled on this website (here). But, unfortunately, no clear cut answers were given. In (Wood,1994), there is apparently a rejection algorithm for sampling from this ...
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41 views

Sampling with no duplicates

I am sampling a population of unknown size and unknown distribution. The sample will be taken over distinct time intervals, but I have to reject any duplicates in the given time interval. The sample ...
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1answer
22 views

Non-integer $n$ in sample size problem

Setup Consider a sample size determination problem with the maximization of expected utility approach (as in Lindley 1997). Let $\theta$ be the state, $x=(x_1,\dots,x_n)$ a sequence of $n$ iid ...
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39 views

Is there an exact solution for this resampling (synchronization) problem?

I want to know if there is an exact solution for the following problem and how to approach solving it: I have a discrete-time signal where the Nyquist theorem is satisfied: $$ r_k = \sum_i ...
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21 views

Formula for sampling with random replacement

I wonder if the following problem and its analysis is already known? Suppose Alice and Bob play a game where Alice has an urn with N hollow balls, all balls numbered uniquely by some integer number ...
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23 views

find the minimum sampling frequency at which the following signal must be sampled to avoid aliasing

find the minimum sampling frequency at which the following signal must be sampled to avoid aliasing $x(t)=0.5+10\cos(2 \pi t)+20\sin(50 \pi t)$ my work The frequency of the analogue signal can be ...
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33 views

Combining Sample Means and Variances

Let us assume we have several samples of unknown size but known mean value $\mu_{i}(x)$ and known variance $\sigma_{i}^{2}$. Now we want to calculate the mean value and variance for the total ...
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27 views

Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions?

I am wondering whether Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions? Any comments or references would be greatly appreciated.
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25 views

Proving that each element in reservoir have equal probability of been selected in reservoir sampling?

Here is the description of the algorithm and proof of the correctness The algorithm creates a "reservoir" array of size $k$ and populates it with the first $k$ items of $S$. It then iterates through ...
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13 views

distribution of non-central chi random sample

Suppose that $X_1,X_2, \ldots, X_n$ is a random sample from a non-central chi distribution with $1$ degree of freedom. What is the distribution of the sample variance of $X_1,X_2, \ldots, X_n$?
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1answer
27 views

Law of large numbers with correlated variables

My sample is a series of measurements of the variable $x$. Measurement t, $x_{t}$, is correlated with $x_{t-n}$. However, as n tends to infinite the correlation tends to zero. If the sample grows, ...
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1answer
21 views

Sampling distribution of $\frac{\bar{X}}{S}$

Suppose that I have a random sample $X_1, … ,X_n$ from a $N(0,\sigma^2)$ distribution. What is the distribution of $$\frac{\bar{X}}{S}$$ and what is it's standard deviation? Here $\bar{X}$ is the ...
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1answer
56 views

How to calculate a population mean for a normal distribution

This is for homework, but I'm a bit confused on how I can find $E(X_i) = \mu$ given a normal distribution. The question is as follows: In a farm, let $X$ denote the number of fruits harvested in a ...
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1answer
22 views

Sampling with replacement: finding all answers with one specific characteristic

I have a book, and I dont get it. We are talking about the following situation.Lets say you have 5 experiments and the sample space for each is $S = \{0,1\}$. So the total amount of possible outcomes ...
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1answer
21 views

Probability of successful sampling fron a population

Say I have a list of 100 questions from which to study for an examination. In the examination I will be asked 10 random questions from the 100 and I pass the exam if I answer 6 questions correctly. ...
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5 views

Sampling specific unit vectors

Given a unit vector $A\in \Bbb{R}^N$ and an angle $\theta$, the unit vector $P$ needs to satisfy $\left<A,P\right>=\cos\theta$. How to sample $P$ uniformly? For example: If $A = ...
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13 views

Sampling in the time domain vs. sampling in the frequency domain

If I have a sample rate of 2 seconds on a 128 second time window (64 samples total) and then I do a Fourier transform, what is my sample "rate" in the frequency domain? Will I end up with 64 ...
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23 views

Random Variables and Statistic

I'm studying Statistical Inference by Casella and I'm confused with the definitions of random variable & statistic. So let we have the probability space $(\Omega, F, P)$ where $\Omega$ is the ...
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39 views

How sample size affects confidence interval.

Suppose the weight of n primary one students has sample mean of 20KG. If n = 40, a certain percentage of confidence interval for the population mean is (15.5,24.5). Find the confidence interval if we ...
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28 views

Comparing two populations

I have sampled from two separate populations, and I want to figure out which population is better. Population 1 has an average score of 84.1 and a standard deviation of 11.8. Population 2 has an ...
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32 views

Simulating r.v.'s from a joint density by rejection sampling in R. Continued

I wish to sample variables $v$ and $w$ from the joint density $$(v+w)e^{-\frac{(v+w)^{2}}{2x_{0}}-2\mu v-(\mu -\lambda )w},$$ where $x_0$, $\mu$ and $\lambda$ can be seen as positive constant. Since ...
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31 views

Fourier transform of a 3sinc^2(100πt)

I'm currently studying for an exam, and I'm not sure the textbook's answer for the fourier transform of 3sinc^2(60πt) is correct. For this question, I incorporated the duality property. Below is my ...
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28 views

MSE for the Method of moments estimator of variance

would appreciate some help here please - Question: Find the MSE for the MOM estimator of the variance $\hat{\sigma^{2}} = \frac{n-1}{n}S^{2}$ based on a random sample from a normal distribution. My ...
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23 views

How can I make the mean of samples be approximately equal to the mean of actual continuous signal?

Suppose there is signal f(t) that is continuous and periodic. It is known that this f is T-periodic. (but it's not necessarily a single cosine f(t).( I'd like to make the mean of samples be ...
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1answer
46 views

Probability of occurrence in a subset of a population [closed]

If 100 individuals i.e. 2% of a population totalling 5000 people, have a certain characteristic, what is the probability of two or more persons within a subset of 10 persons, taken from that same ...
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25 views

Sampling from conditional distiribution

I want to sample from condtional distribution given by formula: $$ P[T_{n+1} =t| T_n=t_n] = ...