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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79 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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1answer
15 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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2answers
42 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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0answers
51 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 0)\...
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0answers
44 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 f=\sqrt{\sum_{k=1}^n\Big(\frac{\...
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1answer
36 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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38 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
93 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 ~ P(M,Sigma)...
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1answer
27 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 e^{\...
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2answers
104 views

Independence of Poisson random variables coming from Poisson sampling

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 $x$. Poissonization: Each observation ...
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1answer
69 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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1answer
32 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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2answers
55 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
39 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: http:/...
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1answer
51 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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0answers
54 views

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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0answers
38 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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2answers
83 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 ...
2
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0answers
54 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
41 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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0answers
46 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 a_i^{(1)}...
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31 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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0answers
36 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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0answers
70 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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0answers
50 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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21 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
44 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, ...
2
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1answer
28 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
105 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
32 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
41 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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37 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 ...
2
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0answers
55 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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0answers
42 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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0answers
60 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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0answers
69 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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0answers
80 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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0answers
24 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
87 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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0answers
26 views

Sampling from conditional distiribution

I want to sample from condtional distribution given by formula: $$ P[T_{n+1} =t| T_n=t_n] = \frac{\alpha_t+m_(t_n)}{\alpha_t+\beta_t+s(t_n)}\prod^{t-1}_{j=0}\frac{\beta_j+r_j(t_n)}{\alpha_j+\beta_j+...
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1answer
49 views

Chi distribution and sample variance

Suppose that the height (in cm) of randomly selected male is distributed according to normal distribution with parameters $\mu = 175$ and $\sigma = 5$. We pick a simple random sample of size $101$ ...
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35 views

Sampling from weighted sum distribution

I have $n$ probability density functions $p_1, \ldots, p_n$, which I am able to sample from. How to sample from a weighted sum of these PDFs $$c_1 p_1+ \cdots+c_n p_n$$ given that $c_1+ \cdots+c_n=1$? ...
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1answer
88 views

Uniform lattice sample inside a particular convex polytope

[update]: hardmath suggests using tools from linear programming. This looks like a good idea indeed. I can now tell that my feasible set is described by: $Set = \{d \in \mathbb{N}^c, -B.d\leqslant ...
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1answer
186 views

Finding the minimum sample size of an opinion poll

I have encountered the following problem in a textbook,: Assume that 28 % of voters favored party A at some point. A later opinion poll gave a result of 30 % of voters favoring party A. What is ...
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2answers
153 views

Understanding Sufficient statistic.

A sufficient statistic for a parameter is a statistic that captures all the information about a given parameter contained in the sample. My question: Is the above sentence correct. (I think it is). ...
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2answers
31 views

Checking whether the given statistic is sufficient

A random sample is drawn from a Bernoulli distribution with $X_i = 1 $ with unknown probability $p$ and zero otherwise. Examine whether the following statistics are sufficient for the parameter $p$ ? ...
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18 views

Random assignments with 95% confidence of at least $K$ duplicate assignments

I'm running an experiment online. Of my $N$ test questions, each participant sees $M$ (randomly selected, $M < N$). If I want a minimum of $K$ responses to each question, how can I calculate a ...
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1answer
62 views

Resampling Probability

I've had a student working on some resampling problems. Here's a question I've posed to her: M consecutive draws of size K are taken from an urn with N balls. Balls are replaced between draws. ...
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831 views

Sampling Distributions

Parts being manufactured at a plant are supposed to weigh 40 grams. Suppose the distribution of weights has a Normal distribution with mean 40 grams and standard deviation 2 grams. Quality control ...
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1answer
60 views

Sampling demonstration using combinatorics

Suppose that I have a finite population of A's and B's, with properties: Population size: $n$ There are $n_1$ A's and $n - n_1$ B's (so that $p = \frac{n_1}{n}$, $q = \frac{n - n_1}{n}$) I'm ...