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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11
votes
2answers
493 views

What is the distribution of gaps?

Randomly select $n$ numbers from the universe $\{1,2\dots,m\}$ with or without replacement, and sort the numbers in ascending order. We can get a list of number $\{a_1,a_2,\dots,a_n\}$, and then we ...
7
votes
2answers
475 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 ...
7
votes
1answer
253 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 ...
6
votes
4answers
1k views

Sampling from a $2$d normal with a given covariance matrix

How would one sample from the $2$-dimensional normal distribution with mean $0$ and covariance matrix $$\begin{bmatrix} a & b\\b & c \end{bmatrix}$$ given the ability to sample from the ...
6
votes
1answer
947 views

Uniform sampling of points on a simplex

I have this problem: I'm trying to sample the relation $$ \sum_{i=1}^N x_i = 1 $$ in the domain where $x_i>0\ \forall i$. Right now I'm just extracting $N$ random numbers $u_i$ from a uniform ...
6
votes
1answer
3k views

Probability to choose specific item in a “weighted sampling without replacement” experiment

Given $n$ items with weight $w_n$ each -- what is the probability that item $i$ is chosen in a $k$-out-of-$n$ "weighted random sampling without replacement" experiment? Can a closed-form solution that ...
6
votes
2answers
557 views

Collisions in a sample of uniform distribution

Asked at a Microsoft interview: Assume you have a uniform distribution (can be discrete or continuous) of size X and you randomly select a sample of size Y. 1) What is the probability in terms of X ...
6
votes
1answer
404 views

How is a Halton sequence related to a Latin hypercube?

I currently use a Halton sequence to choose parameter sets for a prognostic model (e.g. using metabolic rate and protein content parameters to predict growth rate). From my understanding, both a ...
5
votes
4answers
183 views

Algorithm for randomly choosing learning cards

I'm programming a learning software. It works with question-/answercards. I´m searching for a algorithm that gives me a higher probability for cards that the user has answered wrong. My actual idea (...
4
votes
2answers
119 views

To sample or not to sample?

I have the following issue (I want to predict sports results). Let $X$ be a discrete RV with $m$ possible outcomes, each having probability $p_i$, for $i=1,\ldots,m$. Assume that I have a large iid ...
4
votes
4answers
161 views

Sample: don't confuse measurements with actual values?

In Wikipedia's article on Sample there is the following remark: ''Note that a sample of random variables (i.e. a set of measurable functions) must not be confused with the realizations of these ...
4
votes
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 ...
4
votes
2answers
105 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 ...
4
votes
1answer
94 views

Samples in the convex body vs. samples on the convex surface

Let $K$ be a bounded convex body in $\mathbb{R}^n$. Suppose we have a sampler $\mathcal{S}_1$ that can generate points uniformly distributed in $\mathrm{int}K$, and another sampler $\mathcal{S}_2$ ...
4
votes
1answer
352 views

The radial part of a normal distribution

I am reading a paper that asks me to sample $s_i$ from a distribution like this: $s_i \sim (2\pi)^{-\frac{d}{2}}A^{-1}_{d-1}r^{d-1}e^{-\frac{r^2}{2}}$ "Here the normalization constant $A_{d−1}$ ...
4
votes
3answers
385 views

What is “sampling from a distribution”?

Exercise 4.11.3 of Grimmett and Stirzaker's Probability and Random Processes reads "Use the rejection method to sample from the gamma density $\Gamma(\lambda,t)$ where $t (\geq 1)$ may not be assumed ...
4
votes
2answers
87 views

What does it mean to sample, in measure theoretic terms?

Suppose I have some random variable $X$ defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$. What does it mean, in measure theoretic terms, to draw a sample from $X$? When $\Omega$ ...
4
votes
1answer
102 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 ...
4
votes
2answers
240 views

Monte Carlo estimator of the number of 1's in a very long binary sequence

Preface The question below is related to a problem I am working on, which requires counting the number of times a logic-valued function evaluates "TRUE" given an input value. The size of my input set ...
4
votes
0answers
47 views

Signal processing and algebraic geometry

Signal processing is a pretty huge branch of what I would (maybe wrongly) call electrical engineering. I have heard here and there whispers of interesting connections between signal processing - in ...
3
votes
2answers
110 views

Why is there a difference between a population variance and a sample variance

Sorry if this answer is simple but I was wondering why is there a difference between a population variance and a sample variance? I understand The variance is calculated as: $$\text{Var} = \frac{1}{...
3
votes
2answers
49 views

Determine periodicity from transition matrix?

I have a two part question. Let's say we have a transition matrix T: \begin{bmatrix} 0 & 0.2 & 0.8 & 0 & 0 \\ 0.7 & 0 & 0.3 & 0 & 0 \\ 0.6 & 0.4 &...
3
votes
2answers
125 views

Selecting k distinct numbers from an array with increasing probability distribution

I have to select k distinct numbers from an array such that probability of a number getting selected is more if it is at the end of the array (probability increases linearly). I'm thinking of ...
3
votes
1answer
109 views

Maximum likelihood estimation - why is $\mathcal{L}$ not the joint pdf?

Here's an excerpt from my notes: Define the likelihood function: $$\mathcal{L}(\vec{x};\theta)=\prod_{i=1}^{n} f(x_i;\theta)$$ Where $f$ is the pdf of the distribution we're sampling the $x$'s ...
3
votes
2answers
489 views

Simple random sampling without replacement of huge dataset

For an application I'm working on, I need to sample a small set of values from a very large data set, on the order of few hundred taken from about 60 million(and growing). Usually I use the technique ...
3
votes
1answer
158 views

Finding the population size from the common elements of multiple samples

A coworker of mine has taken a certification exam twice and has seen the same ten questions on both exams. I am curious if it is possible to determine the total number of questions in the question ...
3
votes
1answer
52 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 ...
3
votes
1answer
845 views

How can I sample a bivariate Gaussian distribution using Gibbs sampling?

I'm trying to sample a bivariate Gaussian distribution using Gibbs sampling, but I think I don't have the correct conditional probabilities. According to this lecture slides, the conditional ...
3
votes
2answers
224 views

Determining sparse frequency distribution via discrete Fourier transform

Consider the function $$f(t) = 2 \sin(t)+\sin(2t)+25 \sin(400t)$$ (for example). In this case, how many samples of this function would I have to take, and at what sampling frequency, to determine the ...
3
votes
1answer
131 views

Adaptive sampling for 2D functions with discontinuities

I have a 2D function, $f(x,y)$, which I can compute on a computer. The function is expensive to calculate, so I would like to use an adaptive sampling method to plot it in a region. That is, I want ...
3
votes
1answer
68 views

Integrating an infinite series of the Dirac function

I am given the following sampling signal function, where $\delta$ is the Dirac delta function, t is time, and Ts is the sampling period. First, I am asked to plot the signal. I do not understand ...
3
votes
1answer
94 views

Is it possible to sample the Dirac delta function?

The Dirac delta function can be a probability measure with the unit/Heaviside step function as its cumulative distribution function. Is it possible to sample such a distribution? If a random variable ...
3
votes
1answer
875 views

unbiased estimator of sample variance using two samples

I have a couple questions, I'm hoping someone can help! Let $X_1...X_n$ is a random i.i.d. sample from a $N(\mu,\sigma^2)$ distribution, and $Y_1...Y_m$ is a random i.i.d. sample from a $N(2\mu,\sigma^...
3
votes
0answers
46 views

Uniformly sampling points from inside a region of cube

Let the dimension n=200 be fixed. The problem I am interested in is sampling points in n-dimensional Euclidean space uniformly from the region $$ \sum_{i=1}^{n} x_{i}\leq 1, $$ where $0\leq x_{i}...
3
votes
0answers
67 views

Intuition of the Hessian of the Log Barrier Function

I have a convex polytope defined by $\mathbf{Ax \leq b}$ (row-wise) The log-barrier function is defined as: $$\phi(x) =-\sum{\log(b_i - a_ix)}$$ The Hessian of the log-barrier is : $$\nabla^2\phi(...
3
votes
0answers
52 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)\...
3
votes
0answers
286 views

Relationship between DFT and FFT/DTFT

Question 1: Assume we already know $x(t):[0,2\pi]\to\mathbb{R}$'s Fourier series $x[n]_{n=-\infty}^\infty$. Perform DFT on $x[n]_{n=0}^N$. How is the result connected to $x(t)$? WHY? Question 2: ...
3
votes
0answers
127 views

Problems sampling from a $pdf$ over $SO\left(3\right)$

I have a probability density function over $SO\left(3\right)$, which I am trying to sample from. The $pdf$ is given as a generalized fourier series: $$ f\left(\omega,\theta,\phi\right)=\sum s_{\...
3
votes
0answers
792 views

Notation for sampling random variate

Is there standard notation for sampling a value from a probability distribution? Like, if I had a random variable $X$, setting $x$ to whatever value I happened to sample from $X$ on this occasion? I ...
2
votes
4answers
148 views

Is it true that $\mathbb E[{\frac{X}{Y}]}={\frac{\mathbb E[X]}{\mathbb E[Y]}}$?

If $X$ and $Y$ are both random variables, does it hold $$\mathbb E\left[\frac{X}{Y}\right]={\frac{\mathbb E[X]}{\mathbb E[Y]}}$$ ??
2
votes
2answers
162 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). ...
2
votes
2answers
97 views

unbiased estimator in a random sample

I Have a statistic statement here which I need to decide if it's true or false Statement: "When the sample size is random, there is no way to get an unbiased estimator for the population average." ...
2
votes
3answers
318 views

Sample of a subset of a plane

I have the equation of a plane $ax+bx+cx+d$ and a point $(x_0, y_0, z_0)$ on that plane. I defined the neighborhood of that point on that plane as the set of points satisfying $(x-x_0)^2 + (y-y_0)^2 +...
2
votes
2answers
61 views

Two-sided confidence intervals and tests

From a sample of 1751 army hospitals, estimate the mean expenses for a full time equivalent employee for all US army hospitals using a 90% confidence interval given x = ...
2
votes
2answers
57 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
votes
1answer
371 views

calculating an incoherence property

With respect to Matrix Completion and Compressive Sampling (CS) I'm trying to understand how to calculate an incoherence property μ between two bases Φ and Ψ. Getting this incoherence is important ...
2
votes
2answers
25 views

Probability and with replacement sampling

I'm reading the book "Model Assisted Survey Sampling" from Särndal et al. In chapter 2, there's a section about Sampling with replacement. I'll put this into context: We have $m$ independent draws, ...
2
votes
1answer
84 views

Correlation between probability of events

Suppose there are two events $A$ and $B$ and that $P(A|A\cup B)P(B|A\cup B) = P(A\cap B | A \cup B)$. Then I am asked to find if $A$ and $B$ are independent, positively or negatively correlated. My ...
2
votes
1answer
53 views

How to find the variance of a random sample with exponential distribution?

This will seem like a very simple question to many of you; but I cannot understand part of the solution, so to give context I have had to unfortunately resort to typing the whole thing out, apologies. ...
2
votes
1answer
92 views

How to calculate error margin on measured number of occurd events when sampling

We are measuring the number of times a certain event happens. We do this with the help of sampling, so that we only report events with a probability p. For example p=0.01 would result in about 1/100 ...