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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26 views

confused about the proof of Markov chain Monte Carlo

This is the proof from notes I'm confused about the $\pi(x_p|x)$ and $\pi(x|x_p)$ Let's say $X\sim $Bin$(10,0.3)$, so $\pi(x)=\binom{10}{x}0.3^x0.7^{(10-x)}$, so what does $\pi(x_p|x)$ or $\pi(x|...
2
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1answer
71 views

Finding Size-Bias Distributions

For a RV $W$ with mean $\mu$, let $W^*$ denote the $W$-size biased distribution (so that $EG(W^*)=\frac{E(WG(W))}{\mu}$ for all functions $G$ for which the expectations exist). I would like to learn ...
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0answers
47 views

Drawing N balls from an Urn. Probability I have the represenative proportion of balls

I have a urn full of M balls with an unknown mix of white and black balls. I draw N balls out (without replacement) and get a particular proportion of white vs black balls. From the looks of it ...
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7 views

Why aren't these two versions of a two-sample t-test the same?

I'm looking at two versions of a two-sample t-test that appear equivalent to me -- but when I crunch the numbers they don't seem to actually be equivalent. Consider the model $$\mathbf y = \beta_0 + \...
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15 views

Very basic clarifications on sampling processes

Could you help me to clarify some basic notions from sampling theory? Please highlight if anything of what is written below is wrong because I am very confused on the order of logical steps. ...
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1answer
11 views

Sampling Distribution Notations

I'm reading a chapter on sampling distributions of a statistic and I don't seem to have an understanding of the notations used. From probability theory, a random variable is usually denoted by a ...
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0answers
25 views

how many samples needed to obtain an estimate with a given confidence interval

Suppose an urn contains N balls of different colors. I do not know the colors nor the distributions, and I wish to determine the fraction of red balls in the urn, (R/N), to within p% with C confidence....
2
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0answers
164 views

Can we model this set of experiments as an stochastic process and estimate the sample size?

I have an image with the size 5575x9440 and I'm implementing a modified version of the algorithm used in this paper on it, but because the code performance is low ...
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2answers
35 views

How do I compare rates of error between two different sample sizes?

I'm unsure on how to normalize for two different variables. Person A makes 20 pastries total, whereas Person B makes 50. 5 of those pastries, so 25%, are sampled from Person A; 10 for Person B, for ...
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20 views

Importance sampling example

I have a question regarding importance sampling. During a lecture we were told that importance sampling is simply a shift to another PDF to improve point sampling. \begin{equation} I=\int_{a}^{b}dx\ f(...
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1answer
29 views

Extension of Simple Random Sample without Replacement

First, I will show an extension of simple random sample without replacement, and then put forward the question. (1) From Wiki, a simple random sample is a subset of individuals (a sample) chosen ...
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1answer
53 views

General equation for sampling without replacement probability

Looking at a preparatory exam, I'm a little dumbfounded by a question on probability. There are $19$ balls in a box: $5$ red, $3$ white, and $11$ blue. The question is: what is the probability of ...
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0answers
8 views

Multiple surveys with different sample sizes for same population parameter

I don't have a very strong background in statistics but I have seen that results in experiments can be later on found to be statistically unsound. So I just want to make sure that my results are good, ...
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1answer
19 views

Sampling and aliasing of a signal [closed]

I'm given the signal $x(t)=\cos(100*\pi*t)+\cos(200*\pi*t)+\sin(500*\pi*t)$ and I need to find the least sampling frequency in order to reconstruct the $x(t)$ signal from the following of it's samples....
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0answers
6 views

Sparse subgraph preserving distance for n source-destination pairs (s_i,t_i)

We are given a directed graph G with vertex set V(G)={s_1,s_2,..,s_n, t_1,t_2,..,t_n}. So G has a total of 2n vertices. Our aim is to compute sparse subgraph H of G such that distance from s_i to t_i ...
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0answers
17 views

Points in hemisphere over plane defined by a normal vector

I have the following formulas to sample points uniformly on a unit sphere in 3D space: $x = \sqrt{1-u^2} sin\phi$ $y = \sqrt{1-u^2} cos\phi$ $z = u$ where $u \in [-1,1]$ and $\phi \in [0,2\pi]$. ...
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0answers
12 views

Sampling with Clusters

Suppose we have a population of size $N$ where each individual is represented by a vector of the form, $v_i=(\lambda_1^{(i)},...,\lambda_s^{(i)})$ where $\lambda_k^{(i)}$ are finite discrete ...
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2answers
43 views

Application of power series/ binomial theorem in inverse sampling

I have posted this already in other forums. Apologies for cross posting. In order to establish some properties of inverse sampling, Haldane (1945) uses power series and the binomial theorem I assume....
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0answers
92 views

Sampling a sinusoidal signal

Consider the signal $g(t)=\cos(2\pi \lambda t+\phi)$ that is sampled with a frequency $\tau$. Let $g_k$ denote the values of $g$ at the times $t_k=\frac{k}{\tau}$, $k \in \mathbb{N}$. (a) Show that ...
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2answers
31 views

What's the best guess for the parameter of an exponentially distributed sample?

I have a sample of size $N$ values. I know the values are exponentially distributed, i.e. they are distributed according to this probability density function: $$ f(x;\lambda) = \begin{cases} \lambda ...
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2answers
36 views

Probability of rejection (Sampling)

Consider a plant manufacturing chips of which 10% are expected to be defective. The chips are packed 30 to a box for distribution. A sample of size 10 is drawn without replacement from each box. If ...
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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, ...
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26 views

Acceptance Sampling - Random Access Memory Chips

Random access memory chips are packed in batches of $1000$. A sample of size $15$ is randomly selected from each batch and subjected to tests of reliability. The batch is accepted if the sample ...
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1answer
98 views

Is sharing the same support a necessary condition for exchangeability?

I am confused on the meaning of exchangeable random variables. The question is: consider the random variables $X_1,X_2,X_3$ defined one the same probability space $(\omega, \mathcal{F}, P)$; is ...
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1answer
46 views

Help in probability, Difficult Question:// [closed]

Upon testing 80 resistors manufactured by a certain company, it is found that 15 resistors failed to meet the tolerance design specifications a) Construct a 92% two-sided confidence interval for ...
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1answer
24 views

Joint probability with housing stock data

I have a question about probability. Let’s say we have 100 homes of different ages, types and insulation levels, distributed as per the table below. Housing stock data How do I determine how many ...
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0answers
18 views

How to do non-analog sampling from a pdf

I have a distribution given by $$ f(x)=\frac{1}{2}\sinh(\sqrt{2x})e^{-x}. $$ Due to the shape of this distribution, most of my samples will be in the range $x\in(0,2]$. However, I am interested ...
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22 views

How to find sampling distribution S.D in this case

The distribution of the weights of 1000 students is normal with a mean of 55kg and a variance of 25. 100 random samples of size 16 are taken from this population. Determine the mean and standard ...
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34 views

Ways to sample a complicated PDF on an hemisphere

I want to generate samples on the upper real unit hemisphere with the following PDF (it's not really a PDF because I can't guarantee that it integrates $1$) $$\frac{\sum_{i=0}^{n}c_i(\text{ max}(\text{...
2
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1answer
63 views

sampling distribution question

Need clarification on a binomial sample example: we drew a sample of size $100$ from a binomial($m = 2$,$p = 0.2$) distribution and observed $76$ of the $x_i = 0$, $20$ of the $x_i = 1$ and $4$ of ...
1
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1answer
24 views

compute probability density function of a bivariate function without sampling

Suppose $X_1 \sim f_{X_1}(x_1)$, $X_2 \sim f_{X_2}(x_2)$ are random variables with known probability density function. Is there any way to compute the probability density function of a bivariate ...
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2answers
25 views

Simulation methods and generating random variables

Twenty aircraft are sent to bomb a target that is rectangular in shape. It has dimensions 150m by 50m. Each aircraft makes a bombing run along the horizontal x axis and drops one bomb. The point ...
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0answers
22 views

How Much of a Distribution has One Seen After N Samples with Replacement?

I frequently run into this question while modeling processes. I am wondering if there is a general solution or approximation. The question I run into is: For a given distribution with a finite ...
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0answers
13 views

upsampling and plotting a signal in matlab

I want to upsample by 5 a signal in frequency domain, and then plot(stem) it. I figured how to upsample, Fk=(1/5)*upsample(ak_new,5) now this creates a vector ...
1
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1answer
32 views

Basic questions concerning sample means and distributions

I have the following questions: Is The value of the sample mean always the population mean $\mu$, in any sample? I am confused about whether or not it is. Is the sampling distribution of the sample ...
1
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1answer
18 views

hemisphere sampling and vector flip

I am currently implementing a uniform sampling of a hemisphere. Since my hemisphere is oriented around a specific vector N, whatever the sampling algorithm used (uniform/cosine weighted/stratified/......
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0answers
27 views

Bound expected sample mean error

Given $n$ numbers $x_1, \cdots, x_n\in [a,b]$, randomly sample $m$ numbers without replacement $x_{i_1}, \cdots, x_{i_m}$. How can I bound the expectation $$E_{i_1, \cdots, i_m}[\left|\frac{x_{i_1}+\...
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0answers
25 views

Variance of Importance Weights in Importance Sampling

What is the variance of importance weights in Importance Sampling? Is it good to reduce the variance of importance weights and why? And what is the optimum proposal in importance sampling? I assume ...
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0answers
53 views

Drawing successive related samples from a Gaussian distribution

I have a question which describes a system where successive samples are drawn from a Gaussian distribution, and each sample is defined with the following relation. $v_{k+1} = \gamma v_k + \eta_k$ ...
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0answers
21 views

How to do importance sampling if $ f(x)=\frac{\sin x}{\ln x}$?

Let $$ f(x)=\frac{\sin x}{\ln x }$$ In importance sampling, I should choose $g(x)$ to reduce variance. My question is, what kind of function should I choose if $f(x)=\frac{\sin x}{\ln x}$ (like ...
0
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1answer
50 views

$\cos(2\pi f nT +2\pi N_n) =\ cos(2\pi f nT)$ and more, why?

I am studying signals and systems and this came up? Could someone explain why is $\cos(2\pi f nT +2\pi N_n)$ equal to $\ cos(2\pi f nT)$? The book says: "because $N_n$ is an integer" I am ...
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0answers
18 views

Drawing uniform samples from the *range* of a non-invertible function

I am looking for a Bayesian technique to draw samples from a uniform distribution over the range of a non-invertible (that is, there isn’t even a formula) function $\mathbf{f}: \mathbb{R}^N \...
0
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1answer
30 views

Fourier transform of sampling function

Calculate the Fourier transform of $f_{ZOH}$ (the zero-order hold reconstruction of a sampled signal). Where $f_{ZOH} (t)= f(kT), \ \ kT \leq t < (k+1)T,$ and the sampled signal is $$f_s = f(t) \...
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0answers
17 views

Getting a feel for the Normal-Inverse-Wishart conjugate prior to multivariate normal distribution

I am trying to get a feel for the Normal-Inverse-Wishart conjugate prior, which I have started to use, sparingly, in my work, where I am trying to cluster multivariate normal data. As Wikipedia ...
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22 views

Sampling from Irwin-Hall (Uniform Sum) distribution using rejection sampling

Irwin-Hall Distribution is a probability distribution for a random variable defined as the sum of a number of independent random variables,each having a uniform distribution True or False: The ...
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10 views

Sampling from Irwin-Hall distribution using triangular distribution

So I need to sample from the Irwin-Hall distribution using rejection sampling with the triangular distribution. I built 2 functions: The first is d_irwin which receives an $x\in supp(g)$ and the n we ...
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30 views

Generating continuous random variables from a set of Bernoullis

Given a set of $Bernoulli(p_i)$ variables with each having its own arbitrary $p_i$, is there an efficient way to generate continuous random variables sampled from an arbitrary distriubution? To ...
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17 views

Intuition behind sampling distributions – specific case

I'm still trying to understand the basics of understanding the intuition of sampling distributions and calculating the sampling distributions of common estimators. For example, I understand the ...
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25 views

How to approximate a probability distribution vector by sampling

Given a probability distribution vector $\overrightarrow{p}=(p_1,p_2,\cdots,p_n)$ (of course $\sum_{i}^n p_i=1$), assuming we can sample the distribution with obtaining $i$ with probability $p_i$, how ...
1
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1answer
55 views

How to choose new points after grouping/resampling?

I'm resampling a signal (which takes values [0,1]) of N samples (blu points) to one with N/5 samples, where (for each group of 5 samples) I store in two arrays the max and the min values of the ...