Questions tagged [sampling-theory]

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33 views

Find the sample size $n$ for a stratified sampling method

Let’s say we have a finite population $N=5000$.We have divided the population into 5 subgroups according to a risk characteristic “very low risk”,”low risk”,”medium risk”,”high risk” and “very high ...
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229 views

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 ...
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11 views

Optimal sample length - optimal experiment planning

I have a measurement signal with a length of, say, 1 million samples. On the signal, I use digital signal processing techniques to obtain information that interests me. I noticed that I don't have to ...
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1answer
20 views

Number of samples in SRSWR and SRSWOR

In a population of N units, if a sample of n units is chosen, the number of possible samples is N ^ n (SRSWR) NCn. (SRSWOR) Taking an example, let there be a population (a,b,c) and let a sample of ...
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47 views

Sampling theory (signal processing): why is multiplication by the dirac comb used to represent sampling?

Wikipedia (and other sources) make the following claim (from https://en.wikipedia.org/wiki/Dirac_comb): The claim is that multiplying any function by a Dirac comb transforms it into a train of ...
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15 views

Problem regarding drawing balls by SRSWOR

A box contains 2016 balls labeled 1, 2, 3, . . . , 2016. Two balls are selected at random by sampling without replacement. Let X and Y be the labels on the first ball and the second ball, respectively....
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47 views

What is the average length of a line inside the unit square?

This question lies in the chapter exercise of chapter 14 Sampling of book: Fundamentals of Computer Graphics by Peter Shirley and Steve Marschner. Although this is not mentioned in the question but I ...
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16 views

Help with a t-Student distribution

In class, the professor has left us with the following exercise: Given two samples $(X_1,...,X_8)$ and $(Y_1,...,Y_{15})$, where $X\sim N(\mu_1,\sigma_1)$ and $Y\sim N(\mu_2,\sigma_2)$, all parameters ...
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21 views

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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15 views

Groups of random binary vectors with small intra-group and large inter-grup similarity

Is there a formula for simultaneously sampling N random binary vectors, each of length M, with the property that the distribution of the M binary values is distinct between groups? The N vectors in ...
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23 views

Variance of stratified sampling proportion estimate?

Page 18 of this lecture notes says that, the variance of the unbiased estimator of the proportion, $P$, of population elements from stratified sampling is: $$ Var(P_{str}) = \sum_{h=1}^H W_h^2 \frac{...
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What is a sampling matrix? How do I construct one?

I am reading this page on Compressed Sensing. I quoted maybe more than I needed to for context. Knowledgeable people might want to skip to the end of the quote and read backwards. You might be ...
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39 views

Applying a Stratified Random Sampling Technique to Real-World Problems

I am currently working on a paper that involves applying a stratified random sampling technique to a real-life problem. Say, I have a population in which I have divided the population into 200 strata ...
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9 views

Formalization of simple case of rejection sampling

I am watching online lectures on rejection sampling, and I would like to play with it a little. For this, I am trying to do a very simple exercise proposed in this nice video. Let's suppose that we ...
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9 views

Question on random sampling of e periodic signal

Suppose we have a function $$ D(t) = \text{const.} + \tilde{D}(t) $$ with $$ \tilde{D}(t) = \sum_{k=1}^K a_k \, \sin(\omega_k t + \phi_k) $$ with $a_k, \omega_k, \phi_k \in \mathbb{R}$ . Let $$ \...
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1answer
59 views

Sampling distribution of sample mean

We are sampling $Y_1,\dots,Y_n$ without replacement from the population of size N with $Y\sim$ Bernuolli($\mu$). So what will the asymptotic distribution of $\bar(Y_n)$ would be? Since CLT only ...
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22 views

How to determine Two-tailed or One-tailed test?

For example while testing difference of two means $\mu_1-\mu_2$ as the numerator in the test statistic, is this left-tailed or right tailed? What if I take $\mu_2-\mu_1$ in the numerator? Will the ...
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6 views

Standard error for difference of standard deviation

Since under the Null hypothesis, $\sigma_1=\sigma_2$, why is this not applied in the standard error formula? Shouldn’t it be $\sigma_1=\sigma_2=\sigma$?
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15 views

How to distinguish between sample and population?

Suppose if we have two big populations with some attribute A, the population proportions of this be $P_1$ and $P_2$ and sample proportions with regards to this same attribute be $p_1$ and $p_2$. If we ...
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69 views

How to decide on the standard error for difference of standard deviations

In this problem the null hypothesis is that $\sigma_1 = \sigma_2$ then why in the standard error a common sigma value is not used? It can he found as $\sigma = ({n_{1}s_1^{2}+ n_{2}s_{2}^2})/({n_{1}+...
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sampling using L1 optimization

As much as I understand the F should be the interval medians (correct me if I wrong), according to the next slide, where is also the Loss function defined: What I don't understand is the next note in ...
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1answer
37 views

Is calculating a mean of sample means a standard practice?

In order to learn about the distribution of a statistic in the long-run, a common practice is to construct the sampling distribution of that statistic across a number of samples, each of a certain ...
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Quantization and Sampling - putting it all together

So after I learned this two topic: quantization and sampling, I'm learning the way to look at both of them and try to optimize the split of a given amount of bit B to N and k, where N is the amount of ...
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17 views

Determining whether or not a population with $n$ observations is approximately normally distributed

I'd like to know whether or not a population with $n$ observations is approximately normally distributed. I'd like to do this by collecting data on multiple samples from the population. (In my ...
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1answer
44 views

FFT sampling: why is Nyquist frequency enough for inverse transform?

I am trying to understand an aspect of the Fast Fourier Transform (FFT) that is not making sense to me, but I'm struggling to find the flaw in my reasoning. Suppose I model a continuous signal at a ...
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20 views

Bounding linear interpolation error for bandlimited functions

Given a bandlimited signal $x$ with $\hat{x}(f) = 0$ for $|f|\geq B$, how can we bound the absolute error resulting from linear interpolation of its samples $x(kT)$, where $\frac{1}{T}\geq 2B$. Using ...
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132 views

Variants of Bernoulli factory problems

I am interested in the following two questions related to the famous Bernoulli factory problems. Bernoulli factory for certain functions: suppose we are given a coin with arbitrary (unknown) head ...
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17 views

how to sample a coordinate from a rectangle

i have 2d space and in that space i have a rectangle sepcified by it lower left corner and upper right corner how can i sample a point from a Uniform distribution over the rectangle ?
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24 views

Convolution with an extra term

It is well known that $\int g(\tau) h(t-\tau)d\tau$ corresponds to $G(\omega) H(\omega)$ in frequency domain. Can I simplify/say anything (subject to certain assumptions on $g$, $h_1$, $h_2$) about ...
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1answer
66 views

Is this the correct way of solve a probability question from a deck of cards?

You draw a card from a standard deck of $52$ playing cards then replace it in the deck and draw a second card. Determine the probability of drawing an eight and then heart My work: their are $4$ ...
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47 views

Sampling in Compressed sensing: How can one sample $y_i$ be a linear combination of the sampled signal $x$

In the article Compressive sensing by Richard Baraniuk ($2007$), measurements are formulated as: $$\mathbf{y} = \Phi \mathbf{x} = \Phi \Psi \mathbf{s} $$ Where $\mathbf{y} \in \mathbb{R}^m$ are ...
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How to show in the time domain that in the frequency domain, multiplication with rect function with larger bandwidth has no effect?

In the frequency domain, $G(f) H(f) = G(f)$ if $H$ is a brickwall filter with a higher bandwidth than $G$. I want to show this in the time domain. My attempt: The time domain representation would be $$...
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23 views

Question about partial PDFs

My profound apologies if this question is ill posed. I am not entirely sure myself. Here is the fact. Often in nuclear physics we encounter partial differential distributions. Let's make it simple. I ...
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1answer
59 views

Sampling a signal with halved frequency

I have this signal $ x(t) = sin ( 2 \pi f_0 t ) $ , with $ T_c = \frac{1}{f_0} $ and I have to find $ x_\delta (t) $ and the reconstruction signal. first I found that $ f_c > 2 f_s $ so the Nyquist ...
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1answer
212 views

Efficiently testing for more zeros than ones in a binary code block

Setting. I have a set $\mathcal{C}$ of binary sequences where each sequence has length $L\in \mathbb{N}$ and the total number of sequences in $\mathcal{C}$ is $N\in \mathbb{N}$. Assume that $N=\exp(O(...
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1answer
30 views

Statistical sampling and random variables?

I'm studying statistical sampling and there is a point which is not very clear to me. Let us discuss the following example. Suppose we would like to study the heights of 3.000 students in a given ...
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1answer
64 views

Show that $\operatorname{sinc}(x-y) = \sum_{n \in \mathbb{Z}}\operatorname{sinc}(x+n)\operatorname{sinc}(y+n)$

My problem is: Show that for every $x,y \in \mathbb{R}$ $$\operatorname{sinc}(x-y) = \sum_{n \in \mathbb{Z}}\operatorname{sinc}(x+n)\operatorname{sinc}(y+n)$$ Here is what I did so far: Considering ...
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31 views

Fourier transform of sampled signal on multidimensional lattice

Suppose $u_c \in L^1(\mathbb{R}^D)$ has the Fourier transform $\widehat{u_c}$, and that we're sampling it on a multidimensional lattice $\Lambda$, with reciprocal lattice $\Lambda^*$. The sampled ...
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37 views

Why variance must divide by population?

According to page 2 of this paper, if $X$ is an arbitrary measurement with mean $\mu$ and variance $\sigma^2$, and $\overline{x}$ is the sample mean from random samples of size n, then: $$\...
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13 views

Simple random sampling(expanding square of a sum)

I was doing simple random sampling without replacement and got stuck with this expression. I think on RHS there should be a coefficient of 2 with the second term. $$( \sum _ { i = 1 } ^ { n } y _ { i ...
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1answer
76 views

Is it necessary to have infinite(or near infinite) population size for a binomial distribution?

I was reading about single sampling plan in the book "Introduction to statistical quality control", by DOUGLAS C. MONTGOMERY. The author has mentioned that under the assumption that lot size is ...
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1answer
29 views

How can I efficiently sample a space to find a channel capacity?

I'm investigating Gelfand-Pinsker channels where the capacity formula has been proven to be: $$ C = max_{p(x, u|s)} [I(U; Y) - I(U;S)] .$$ I've written some simulation code that, given the ...
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How to calculate my sample size to establish my real-WR?

I've been tracking for a period my hour rate on the game of poker and I want to know how much data I've to collect in order to establish my earning per hour. I've been searching online and when it ...
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How to determine the sample size for a two sided $z$-test?

Let $X_{1}, \ldots, X_{n}$ be an iid sample from $N(\mu,\sigma^2)$ where $\sigma$ is known. We want to test a hypothesis $$ H_{0}:\mu = \mu_{0} \quad \mbox{versus} \quad H_{1}: \mu \ne \mu_0 $$ Now, ...
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39 views

Apply formula on random sampling without replacement, but with replacement between each iterations

I am looking for an algebraic solution to explain that I apply a formula on a vector constituted of a random sampling of n elements in a population of size N without replacement. The formula is ...
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1answer
23 views

Algebraic expression of random sampling without replacement

I am new to algeba and I cannot manage to find how to write the algebraic equation to explain that I randomly sample data contained in vectors without replacement. I have e.g. vectors and : Both of ...
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Are there any generalization of the DKW inequality to the cluster sampling case?

The famed DKW inequality states the following: $$\mathbb{P}\left(\sup_{x\in\mathbb{R}}|F_n(x)-F(x)|>\epsilon\right)\leq 2e^{-2n\epsilon^2}$$. Further using Bahadur's representation, we naturally ...
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1answer
35 views

the sampling distribution of sample means, why are samples taken with replacement?

I have what might be a silly question. I am tutoring a family member in stats and I am trying to help her understand the sampling distribution of sample means, but I don't quite understand why it's ...
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1answer
65 views

Expected difference for two largest samples

I'm trying to get an understanding as to how much difference between the two largest samples drawn from some distribution $f(x)$ scales with $n$. That is, as we increase the size of our sample, how ...
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1answer
41 views

Determining Normality With Large Samples

I am taking in some files and I must determine if the data sets are normally distributed (yes, within a certain degree of certainty because it cannot be proven only disproven). My data sets are quite ...