# Questions tagged [sampling-theory]

For questions related to sampling theory. Sampling theory is the field of statistics that is involved with the collection, analysis and interpretation of data gathered from random samples of a population under study.

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### Sample from a piecewise exponential distribution [migrated]

Given a distribution $p(x)\propto exp(\min_{i=1}^N [a_i^Tx + b_i])$, where $x$ and $a_i$ are both D-dimensional vectors, $b_i$ is a scalar, and $(a_i,b_i)$ are known. How can I sample from it? This ...
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### mathmatical formulation for missing information due to sampling

I want to define a mathematical model (formula) to design missing information from sampling. In my problem, I have real events whose value is binary (0, 1) which might change every second. For a given ...
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### Efficient Method for Uniform Sampling from the Space of Increasing Vectors in $[0, 1]$

I am seeking advice on methods for uniformly sampling from the space of increasing vectors within the interval $[0, 1]$. Specifically, I require an efficient algorithm that can handle high-dimensional ...
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### Relation T-Distribution between chi-square and normal

Theorem. Let $Z\sim N(0,1)$ and $Y\sim \chi^2(v)$ are two independent random variabels. Then, $$T = \frac{Z}{\sqrt{Y/v}} \sim t-student$$ with degrees of freedom $v$. Is it the converse always true? ...
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### What is the intuition behind the single-pass algorithm (Welford's method) for the corrected sum of squares?

The corrected sum of squares is the sum of squares of the deviations of a set of values about its mean. $$S = \sum_{i=1}^k\space\space(x_i - \bar x)^2$$ We can calculate the mean in a streaming ...
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### Why is the demonstration of the "sample" standard deviation somehow more exact than the classic standard deviation, if $\frac{x}{n} ≠ \frac{x}{n-1}$?

I cannot see why to use the "sample standard deviation" instead of the classic "standard deviation" (the "population") ever, most explanations I find are just prescribing ...
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### Doubt about sampling without replacement

A problem consists of five numbers $2,3,6,8 and 11$. Consider all possible sample space of size 2 that can be drawn with and without replacement from this population then find (a) mean of the ...
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### Getting my concepts confused with sampling

Let me first set the scene. This is how we learned the variables and their names and symbols. We have the sugar content of 77 different cereal brands. It firsts asks "estimate the true average ...
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### Sufficiency in sampling theory

Introduction We have a finite population $U$ with $N$ individuals, namely $U:=\{1,\dots,N\}$. Each individual stores a secret fixed value, so let's write $y_i$ for the value corresponding to the $i$-...
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I have a question regarding how to form the sample space in this famous paradox. Usually the sample space is defined as (B,B), (G,B), (B,G) and (G,G). However if I express it as (B1,B2), (G,B) and (G1,...
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### An one dimensional sampling version of Penrose tiles in 2D

I was initially interested in aperiodic sampling for signals to address the problem of aliasing in the frequency domain. A design like Penrose tiling in 2D (which is non-repeating) can be very ...
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### Sampling a cosine [closed]

If you sample a cosine of fundamental period 0.1 milliseconds with a sampling rate of 10^5 samples per second, how much phase difference is there between two consecutive samples? I can't solve this ...
1 vote
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### Uniform Sampling points on a line using 2 Uniform distributions

I have been struggling with this problem for quite some time but I am not sure how to proceed. So I am given a sampling algorithm : We would like to uniformly sample points on a line between A and B. ...
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### Sampling procedure to obtain groups with same categorical distribution

Imagine that we have $n$ (mutually-exclusive) sets of different sizes, $X_1, X_2, ..., X_n$. The total number of elements is $N = |X_1| + ... + |X_n|$. I want to partition these $N$ elements into ...
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### Applying Central Limit Theorem to an exponential distribution - how big should sample size be?

For an exponential distribution, in order for the sampling distribution of its mean to be well approximated by normal distribution (via central limit theorem), how big should a "typical" ...
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### Brief references on partial coherence

A signal in additive noise is sampled by a receiver with an unstable clock. Maybe the clock has absolute bounds on its frequency drift and sample time aperiodicity, or maybe the clock has some ...
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### How can we plot the Fourier transform of a sampling pattern?

In Cook, Stochastic sampling in computer graphics, the author is showing distribution patterns like jittered samples and the corresponding code to produce the output: He's also showing the plot of ...
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### Sampling 2 balls of the same colour probability

A bag has 3 red, 3 blue and 3 green balls. 2 balls are picked at random without replacement. What is the probability that they are of the same colour? I am thinking what is the easiest way to do this ...
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### Biased Sample Mean

I have the following question in my interview: Suppose I am interested in the average of residents by apartments. So I went to the street and randomly sample people and ask them how many residents are ...
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### Deriving the probability of inclusion for a random stratified sample

So I'm a bit stuck on deriving the inclusion probability for a random stratified sample. The question's context is as follows: say we wish to understand household income, but a complete list of ...
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### Alternatives to Fuller’s (2009) Sampling Statistics

I’m taking a self-study course in sampling statistics this semester and my professor chose Fuller’s (2009) Sampling Statistics for both the content and the homework questions. However, I am really ...
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### Interpolation from unevenly distributed points of function with compactly supported Fourier transform

It is not so hard to prove the Nyquist-Shannon theorem. It basically says a real valued function $f$ having a compactly supported Fourier transform function $\hat f$ can be interpolated precisely from ...
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### Prove the variance of Hansen-Hurwitz estimator

The Hansen-Hurwitz estimator of the population total, $Y=\sum_{I=1}^{N} Y_{I}$, is given as: \mathcal{y}_{\mathrm{HH}}^{\prime}=\frac{1}{n} \sum_{i=1}^{n} \frac{y_{i}}{p_{i}} \end{...
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### What is the "best" way to sample a function?

The situation Imagine that you are faced with a box with a numeric display and a knob next to it that can turn in specific increments (that you can specify) and in both directions (increasing values ...
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### Sample selection from within a certain area

I need to make a random selection of mosques in a large city for a public health survey. I have a map that identifies a thousand of them, but many are unmapped, so I don't know what my 'universe' is, ...
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### Show that the probability of the sample $S$ is the probability of $S_1 S_2$.

Let $U$ be the population with a size of $N$, and let $n_1$ and $n_2$ be the sizes of two samples $S_1$ and $S_2$, respectively. Let the sample $S = S_1 U S_2$ with size $n = n_1 + n_2$. Show that the ...
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### Why is a sample defined as a set of observable random variables?

I have learned that a Random variable is a mapping from sample space to a real number and is denoted as X, Y ,Z for different functions (random variables). Say, in the case of rolling two dice, X can ...
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### Sistematic sampling rewritting the variance

If $N = n \cdot a$ where $a$ is an integer, show that the variance $V_{SY}(\hat{t}_\pi)$ given by $$V_{SY}(\hat{t}_\pi) = N^2 \dfrac{1}{a} \sum_{r=1}^{a} \left( \bar{y}_{Sr} - \bar{y}_U\right)^2$$ ...
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### Subsampling from different distribution data?

I have a simple question. Thanks for helping me Is there any difference between simple random subsampling a set with uniform distribution and a set with normal distribution? How can I subsample in ...
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### How to count the specific sampling pairs given several samples of a function?

As illustrated in the figure, we have a function (the solid line) and sample it with a user-defined sampling rate (the blue circles). In addition, we uniformly divide the y-axis into several sub-areas ...
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### Does this discrete probability distribution have a name? (Related to Hypergeometric distribution)

Suppose there are $N$ objects, of which $K$ are of type 1 and $N-K$ of type 2. Objects of type 1 are indistinguishable and objects of type 2 are indistinguishable. One is interested in the probability ...
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### 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 ...