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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23 views

recovering bits from distorted signal

I have a signal: $r(t) = \sum_k a_k g( lT/2 -k(T+\Delta T) )$ where $a_k \in \lbrace\pm 1\rbrace$, $g(t)=\frac {\sin(2\pi t/T)}{2\pi t/T}$, $T/2$ is the sampling period, and $\Delta T$ is a ...
0
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0answers
28 views

Sampling combinations (from a binomial coefficient) without replacement

The total number of combinations of $k$ items out of $n$ total is $n \choose k$, or a binomial coefficient. This can be a very large number even for pretty small $n$. The binomial coefficient ...
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2answers
136 views

Probability of a normal random variable added to a number being greater than another normal random variable, and distribution of average

$X$= random height of a male $Y$= random height of a female $X$ and $Y$ are independent of each other For $x$, $\mu=180\text{ cm}$ and $\sigma^2= 16\text{ cm}^2$ For $y$, $\mu=170\text{ cm}$ and ...
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0answers
12 views

Reconstructing symbols from another set of symbols

I have a discrete-time signal as: $r_l = r(lT/2) = \sum_m h_m \alpha_{l-m}$ where $\alpha_l = \sum_k a_k g(lT/2 -k(T+\Delta T))$ where $a_k \in \lbrace\pm 1\rbrace$, $g(t)=\frac {\sin(2\pi t/T)}{2\pi ...
1
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1answer
61 views

Expected number of distinct nodes visited in a directed bipartite graph

Let $G = (V,E)$ be a directed bipartite graph with $V = \{I \cup O\}$ where $\left\vert{I}\right\vert = n$ and $\left\vert{O}\right\vert = m$. All the edges start from a vertex in $I$ and end on a ...
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0answers
22 views

Expected number of distinct nodes visited in a directed bipartite graph

Let $G = (V,E)$ be a directed bipartite graph with $V = \{I \cup O\}$ where $\left\vert{I}\right\vert = n$ and $\left\vert{O}\right\vert = m$. All the edges start from a vertex in $I$ and end on a ...
0
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0answers
24 views

Sampling from a random distribution with a margin of error

A survey of a population is taken using sampling. It is determined that 70% prefer option A and 30% prefer option B with a margin of error being 5%. Typically when simulating the process with the ...
2
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0answers
26 views

Reconstructing a signal at twice the bit-rate

I have a discrete-time signal as: $\alpha_l = \sum_k a_k g(lT/2 -k(T+\Delta T))$ where $a_k \in \lbrace\pm 1\rbrace$, $g(t)=\frac {sin(2\pi t/T)}{2\pi t/T}$, $T$ is the bit period, and $\Delta T$ is ...
1
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1answer
69 views

Distinguishing between two weighted dice (or discrete distributions)

In a bag, there are two dice, each with sides weighted differently. I know the weighting of the two dice. I reach into the bag and pick one out with equal probability. I want to know how many rolls it ...
0
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0answers
190 views

Box Muller Transform - Proving that Z is Normal Distribution

I'm studying the Box Muller transform and I cannot see how Z0 and Z1 represent standard normal distributions. I've looked at the wikipedia page for the box-muller function but they don't seem to have ...
2
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1answer
359 views

Fast fourier transform and nyquist frequency

Trying to figure out how to use Matlab to calculate the nyquist frequency of a signal. Given a function, lets say $y = 5\sin (2t + \pi /3) + \sin (t + \pi /2)$ for $t > 0$. How do we use a fft in ...
2
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1answer
20 views

Can I use one float random number to generate two random numbers, one discrete, one continuous?

I need two random numbers. The first one, u, is discrete and takes 70% of the time the value 0 and 30% of the time the value 1. The second one, v, is continuous and takes values uniformly inside [0, ...
0
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0answers
16 views

Find function of error in sampling

I was given a teaser that I can only figure out half of. Imagine there is a dartboard that is centered at (0,0). Darts are thrown and the coordinates are modeled ...
1
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0answers
25 views

Number of samples with replacement to reach expected coverage of population under non-uniform sampling

I am interested in finding the number of times $n$ I need to draw with replacement from a population of size $N$ such that the expected proportion of the population seen is at least $P$. From this ...
0
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1answer
35 views

Paths of Nearest Neighbours

I'm working on a project about sampling points, where the next point to be added to sample is the closest point to the current point. Furthermore, each point can only appear once in the sample. ...
0
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0answers
7 views

Compute mean of a set from a biased sample

I want to compute the average of an unknown set $S$ containing real numbers. I can take arbitrary large number of samples from $S$, but the samples are not uniform, meaning that some numbers are ...
0
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1answer
36 views

antithetic sampling

I am reading a book on antithetic sampling.It is said that the idea of antithetic sampling can be applied when it is possible to find transformation of $X$ that leave its measure unchanged (for ...
0
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0answers
12 views

Sample from a distribution using the log of the pdf?

I am reading about slice sampling and I understood (that as Gibbs sampling and other algorithms) you can use it when you do not know the exact pdf of the distribution, but rather a proportional ...
0
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1answer
36 views

Interpreting confidence interval of regression coefficient.

In a Simple Linear Regression analysis, independent variable is weekly income and dependent variable is weekly consumption expenditure. Here $95$% confidence interval of regression coefficient, ...
0
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2answers
56 views

Population estimate from sample

This seems very basic but I can't find a clear statement of it. Suppose I have a population of N balls which are red, white, and blue in some proportion. If I take a sample of S balls (S << N) ...
0
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0answers
11 views

Calculating average with special summary statistics

We want to compute the mean of a data set D. The data set is not accessible. Instead we repeatedly gain access to average and size of a subset of D, which contains a data point d $\in$ D and some ...
0
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0answers
8 views

Motivation for definitions of Frame, Sampling Set

In a Hilbert Space $\mathcal H$, a frame $\mathcal F=\left\{ f_n \right\}$ is a sequence of vectors that satisfy $$\forall f\in \mathcal H : A\|f\|^2\leq \sum_n | \langle f,f_i \rangle |^2 \leq B\| ...
2
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0answers
66 views

Stratified Sampling: Total and mean estimate of population

Question: In a sample survey designed to estimate the total number of cattle, the universe of 2072 farms was stratified into 5 strata on the basis of the total acreage of farms. In the hth stratum (h ...
0
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0answers
30 views

Is there a two-dimensional method to optimally allocate N sampling points on a continuous function with derivatives?

I am looking for a method to optimally allocate sampling points. I have read some papers on this topic that discuss one-dimensional allocation using chebyshev points, but I haven't found a good ...
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2answers
37 views

Does the sampling distribution coincide with the population distribution if every possible sample is taken?

Say you have a population. You take random samples repeatedly, and the distribution of all the means of those random samples is the sampling distribution. Right? So does that mean, that if you take ...
0
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2answers
51 views

Difficult Survey Sampling question

Question: A Secretary of State wants to survey the primary owners of motorcycles registered in the state to estimate the proportion who want the license plates redesigned. (Primary owner means that ...
0
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1answer
99 views

The distribution of sample proportion for given population proportion and sample size

If the population proportion is 0.90 and a sample of size 64 is taken, what is the probability that the sample proportion is more than 0.89? (4dp) work: $n=64$, $\hat p=0.89$, so $X=n \hat p ...
0
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1answer
20 views

Random sampling, need some help and guides

i was asked to do an assignment on examining the Body Mass index of students. I have to select at least 50 students from my school, and i was asked to describe how I ensure the randomness of the ...
0
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0answers
153 views

simple random sampling without replacement proof

For simple random sampling without replacement, starting with the expectation of $\sum_1^n(y_i-\bar Y)^2$, show that $V(\bar y)= (1 − f )S^2/n$ this looks very hard i tried to simplify the right ...
0
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0answers
36 views

Ergodic Versus non-Ergodic Processes

Besides time averaging not carrying over to the ensemble average (in the limit), what are the pros and cons of ergodic and non-ergodic processes? Suppose you were in an engineering situation and you ...
0
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1answer
60 views

Examining the effect of a quantitative factor on response.

To examine the effect of a quantitative factor temperature on yield,the researcher has a plan to use the following model for the analysis: $$y_{ix}=\beta_0+\beta_1 x+\epsilon_{ix}$$ where $y_{ix}$ ...
6
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2answers
197 views

How does accuracy of a survey depend on sample size and population size?

Which survey is more accurate? Assume the samples are taken perfectly randomly. A sample of 100 people out of a population of 1000 (sample is 10% of population) A sample of 1000 people out of a ...
1
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1answer
32 views

reconstruction through sinc interpolation

I have a discrete-time signal $x_k = \sum_l a_l g(kT - l(T+\Delta T))$ where $g(t) = \frac {\sin(\pi t/(T+\Delta T))}{\pi t/(T+\Delta T)}$. Since the signal $x$ has been sampled at rate $1/T>2 ...
0
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1answer
24 views

sample and population (set or collection)

In my Statistics class they introduced a population as the set of all measurements of interest to the investigator(e.g. height of humans) and a sample as a subset of the measurements selected from the ...
0
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0answers
61 views

Confidence interval of a poisson variable $\hat \lambda$ while using importance sampling to estimate $\hat \lambda$

I want to estimate $\hat \lambda$ by taking $n$ samples from a population $k$. I will sample $n$ items from population $N$ with a sample distribution $P(X)$. Therefore, my best estimate is $\hat ...
3
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1answer
37 views

Determine sample size according to some unknown distribution with given error rate and confidence

Assume $x\in\mathbb{N}$ obey some unknown distribution, and I can sequentially and independently acquire infinite samples of $x$. Now, given an error rate $\epsilon$ and confidence $1-\delta$, can I ...
2
votes
2answers
56 views

Sampling from the diamond: $|x_1|+\ldots+|x_n| \le 1$?

Let $\left(x_1, \ldots, x_n \right)$ be a point in $\mathbb R^n$. Sample uniformly at random from the diamond $$ |x_1|+\ldots+|x_n| \le 1. $$ In $\mathbb R^2$, one way is to sample the square, then ...
2
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0answers
25 views

Is a discrete random process issued from a sampled continuous ergodic WSS process also ergodic?

I have a continuous time process $\{X_t,t\in\mathbb{R}\}$ that is WSS and ergodic for the 1st and 2nd moments. I create a random discrete process $\{Y_n,n\in\mathbb{N}\}=\{X_t,t=nT\}$ by discretizing ...
1
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2answers
54 views

Errors and Residual

Why are errors independent but residuals dependent? As far i know the sum of the residuals within a random sample is necessarily zero, and thus the residuals are necessarily not independent. But also ...
4
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1answer
85 views

Uniform sampling with replacement item frequency

Suppose we are sampling from $N$ distinct items uniformly with replacement $M$ times. What can be said about the distribution of frequencies of items drawn? For example, if I sort all the frequencies ...
0
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1answer
98 views

Why would the discrete fourier transform “see” signals like this? What is the origin of spectral leakage?

The discrete fourier transform of $x = (x_{0},\dots,x_{N-1})$ is defined as $\displaystyle X_{k} = \sum_{n=0}^{N-1} x_{n}\omega^{kn}_{N}$ where $\omega^{kn}_{N} = e^{-2\pi ik/N}$ and ...
0
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0answers
12 views

Gaussian of different standard deviation and sample spacing

For given a set of samples position (e.g. $-2,-1,0,1,2$) I can define the correspondent, normalized, gaussian weights for a gaussian with std. deviation $1.0$. I know that if I scale all the sample ...
1
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2answers
72 views

How to pick a random sample of given size from a set of unknown size

I have the following problem: I'm reading huge amounts of data records; when done, I would like to display one or more randomly selected records from the data set. It's easy to do if I can cache the ...
2
votes
1answer
43 views

Sampling with an “oversampling” factor, in K-Means||

I'm trying to understand K-Means||, a scalable version of K-Means++, which itself is an "improved" version of the clustering algorithm K-Means. Please find here the link to K-Means|| paper ...
1
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1answer
47 views

Computing P-value

In a book, from a sample they derived Mantel-Haenszel chi-square statistic $$\chi_1^2=1.41$$ And it is written that : this $\chi_1^2=1.41$ is associated with a one-sided P-value between $0.10$ and ...
1
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1answer
28 views

Estimating the mean and variance of numbers assigned to each person in population of one billion

Problem : Consider people of one billion, and each has one card containing one number. For instance first has card of number $7$. second has card of number $11$ and so on (simply if number means age ...
1
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1answer
61 views

Variance- covariance matrix

Consider $H$ denotes hat matrix and $e$ denotes residual. In the book Applied regression Analysis by Draper/Smith, it is written that : $\mathbb V(e_i)$ is given ...
1
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1answer
89 views

Confidence Interval for Regression Coefficient ,$\beta$

In the book 'Applied regression Analysis' by Draper/Smith, it is written that : Obtain individual $100(1-\alpha)\%$ confidence interval for the various parameters separately from the formula ...
1
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1answer
103 views

Random points inside a convex polytope

Given a convex polytope, defined by set of vertices $P = \{\mathbf{x}^{(i)}\}_{i = 1}^n, x^{(i)} = (x^{(i)}_1, x^{(i)}_2, \dots, x^{(i)}_d): \operatorname{conv}(P) = P$. How to generate uniformely ...
0
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1answer
62 views

Building histogram of latency using sample of input data

Assume I have input data set consisting of a web page response time. I'd like to build histogram from input data, but for practical reasons I can only use sample of data. Based on histogram I want to ...