Questions tagged [sampling]

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. Use this tag along with the tags (probability), (probability-theory) or (statistics).

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Best way to estimate probability of heads of a biased coin

What is the best way of estimating the probability p of getting heads in a biased coin? An intuitive idea is to flip it over an over again, and look at the empirical frequency etc, MLE type ...
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-4 votes
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categorical sampling

I want to understand the proportions of different calling type managed by a customer services. I don't know the type of categories (es: payments, product, services, etc...) and the proportions (es: ...
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Expectation using samples from a population

There is a population $\mathrm{X} = \{x_1, \ldots, x_N\}$ and a function $F(X) = \sum_{i=1}^N f(x_i)$. A subset $S$ of $\mathrm{X}$ such that $|S| \le N$ is sampled using simple random sampling ...
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Monte Carlo computation of probability of a subset of samples

I would like to compute the probability for some subset $\omega \subset \Omega$ of events to occur, i.e. $P(\omega) = \sum_{x \in \omega} P(x)$ where I know $P(x)$ for all $x \in \Omega$, which are ...
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Naive Monte Carlo Sampling vs. Importance Sampling

Can someone help me understand this paragraph: The naive Monte Carlo estimator introduced in the last section performs well if the prior and posterior distribution have a similar shape and strong ...
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How to compute minimum sample size of a simple linear regression model with given statistics values

Suppose the statistics values are given as follows: $\sum_{i=1}^{n}x_i, \sum_{i=1}^{n}y_i, \sum_{i=1}^{n}x_iy_i, \sum_{i=1}^{n}x_i^2,\sum_{i=1}^{n}y_i^2$ Firstly, we can compute the regression ...
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Combining non-independent random normal-distributed variables

In the below question part b) involves combining normally-distributed random variables which ARE independent. Part d) involves combining normally-distributed random variables which are NOT independent....
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3 votes
1 answer
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Statistical estimator of expected value of the gradient of an unknown function

Fix a probability space $(\Omega, \mathcal{A}, \Bbb P),$ a continuously differentiable function $f:\Bbb R^n \rightarrow \Bbb R,$ and a random vector $X: \Omega \rightarrow \Bbb R^n.$ Furthermore, we ...
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Random sample from Normal Distribution, why isn't the variance divided by n?

Below is a question containing parts b) and c). Further below is the mark scheme. In part c) they take a random sample from T, which is normally distributed. I was expecting the distribution to change,...
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Maximizing the ratio of expected maximum of n IID and the expected value of the IID for a (WHP) non-negative distribution

I’m looking for a distribution that is non negative , or has good tail bounds (so non negative with high probability) and maximizes the following property: $X_1, X_2, …, X_n$ are n IID samples of the ...
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2 votes
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Do individual Metropolis-Hastings map preserve the target measure?

Consider the probability space $(Q,\mathcal{B}(Q),\pi)$, where $Q \subseteq \mathbb{R}$ is a sample space, $\mathcal{B}(Q)$ is the Borel $\sigma$-algebra on $Q$, and $\pi$ is some probability measure ...
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the mean and standard deviation aren't the same as those of the input data i provided after sampling

have a log-normal mean and a standard deviation. after i converted them to the underlying normal distribution's parameters mu and sigma, I sampled from the log-normal distribution however when i take ...
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Central Limit Theorem "find sample size" question, with modulus inequality

I'm doing the below Central Limit Theorem question. I omitted sub-question a) and b) as they were unrelated. I have included the answer below the question. I half-understand the answer but I don't ...
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Covariance between two sample means sampled successively without replacement.

Lets say i have a population of size N, where N > 1, with the values $y_{1},y_{2},y_{3},y_{4}, ... , y_{N}$. (Unknown Population Distribution) Step 1: A Simple Random Sample without Replacement is ...
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Generating samples from the joint distribution

If I have samples from two correlated random variables $X$ and $Y$, would it be possible to simulate samples from their joint distribution?
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Rigorous proof of the Box-Muller transform method

The Box–Muller transform can be described as follows: Let $U_1$ and $U_2$ be independent uniformly distributed random variables on $(0,1)$ and \begin{align}R&:=\sqrt{-2\ln U_1};\\\Theta:=2\pi U_2.\...
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Sample Space for the following experiment

Given there is a bag having 2 red balls, 3 blue and 4 blue balls. The experiment is to pick one ball, inspect the color and return it to the bag, then pick another one. What is the sample space of ...
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-2 votes
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Unbiased estimator from a random sample [closed]

Problem I really don't understand the answer to the below question. It's only 2 marks, so it should be reasonably simple. Attempt I have expanded out the brackets and got as far as $E(Y) = E(X^2) - E(...
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How Many Ways to Pick Objects : So That No One Picks the Exact Same Objects?

Suppose you have a set of: 5 Hats: H1, H2, H3, H4, H5 10 Pants: P1, P2, P3, P4, P5, P6, P7, P8, P9, P10 6 Shoes: S1, S2, S3, S4, S5, S6 Suppose you have 3 Friends (F1, F2, F3). Each friend must at ...
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Disproportionate Stratified Random Sampling

How do you conduct disproportionate stratified random sampling? Home Office Total Men 100 250 350 Women 120 30 150 Total 220 280 500 An overall sampling fraction of 10% has been decided on. ...
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Analysing Failure probability of random sampling with replacement

I am trying to understand the failure probability of random sampling with replacement. Here is one problem instance. Given a bag with M = 5 numbered balls (1, 2, 3, 4, and 5), from which the balls are ...
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1 vote
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False Negative Probability of a probabilistic black box function

Suppose we have black-box access to a function that returns the roots of some unknown 5-degree polynomial in each invocation. (So there will be 5 roots.) But we also know that the function is ...
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Probability of selecting red balls from a bag

I am trying to solve this problem but getting stuck in the general case. Here is the problem statement: A bag contains a total of N balls. Out of which M balls are red, and N-M balls are blue. Here M &...
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1 answer
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Comparing the sampling distribution of the Maximum Likelihood Estimator to other estimators

Suppose that independent observations $X_{1}$ and $X_{2}$ are taken from Poisson $P(aλ)$ and Poisson $P(bλ)$ distributions respectively, where $a$ and $b$ are known and positive. Here is the maximum ...
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Determine my increments from average

I am trying to find my cost increments in a situation where I know my average cost for 5 situations, but don’t know the values of each situation. My cost is 300 average for past year per tests/...
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What does it mean by "sample from a distribution"?

As far as I understand "Sample from a distribution" means: to select n samples from a population P so that, the samples resemble a specific type of distribution e.g. Gaussian, Boltzman, ...
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Estimate the sample size in proportional stratified random sampling to get the same precision of simple random sampling

I was reading example 3.2 on stratified sampling in Sampling: Design and Analysis by Lohr where a stratified random sampling design is proposed and compared with a simple random sampling design : The ...
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Importance sampling for a posterior mean

Suppose X1, X2, · · · , $X_n ∼ N(0, \theta)$ where $\theta ∼ Gamma(3, 0.5)$, with $$ p(\theta) = \frac{\theta^2 e^{-\theta/2}}{2^3 \Gamma(3)},~~\theta > 0 $$ I have found the posterior distribution ...
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How to apply Cochran's formula to geographic sampling

I need help conceptualizing random selection by geography. I need to select a set of mosques for a mortality survey. I have no list of them, so I have to work up a random selection of them. I've ...
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Mean of a random sample from a Uniform Distribution

The following question: has the following answer: The only part I don't understand is why they put E in front of 2X at the very beginning?
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Uniformly sampling from the cone of feasible directions in case of linear inequalities.

I'm working on an algorithm that is minimizing some loss function restricted to a polytope in $\mathbb{R}^n$: $$ \mathcal{X}= \{ x \in \mathbb{R}^n : \sum_{i=1}^n x_i \leq 1, x_i \geq 0, \ i=1,\dots,n\...
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Lower bound of the performance of sampled optimization

Hi I am considering an optimization problem that requires a condition to hold over a continuous range, for example: \begin{equation} \begin{aligned} \max_{\boldsymbol{x}, \gamma} & \ \...
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2 votes
1 answer
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Difference between $S^2$ and $s^2$ and $\sigma^2$?

It's in mentioned in my professor's slides that $S^2$ is the population variance and $S^2$ = $$\sum_{i=1}^N Y_i^2 - \frac{(\sum_{i=1}^N Y_i)^{2}}{N} \over {N-1} $$ which is estimated by $s^2$ which ...
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Derivation of $V(\bar{y})$ using random component of the sample $\tau$ for SRS

I am trying to derive this formula: but I do not understand the parts underlined by red (I do not get from where they were derived) Here's my attempt (sorry it's handwritten, it would take me so long ...
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3 votes
2 answers
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why do we need a sequence of random variable, isn't one function sufficient?

In sampling, we have so many situations involving a sequence of random variables, what I am confusing is why do we need a sequence of random variables to describe the process? It feels like each ...
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Minimum Samples to Identify Box (Coupon Collector Extension)

I am thinking on the follow probabilistic problem setting: Say we have $n$ boxes, each of which has an affiliated set of $k$ balls. We aim to distribute these balls across the boxes in the most fair ...
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3 votes
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Fisher Information and Cramér-Rao lower bound problem

Suppose $X_1,...,X_n$ are random samples from $N(\mu, \sigma^2)$, where both $\mu$ and $\sigma \gt 0$ are unknown, and let $\theta = \sigma^p$ for some $p \gt 0$. I want to find the Fisher Information ...
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Using s to estimate $\sigma$ when finding the sample size in confidence interval questions

I am trying to learn sample confidence interval for $\mu$ , in this topic , there is a subtopic which is finding the sample size. I know that if $\sigma$ is given (standard deviation of population) , ...
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Sampling vertices from a graph to maximize the spread

Given a graph with N vertices, I want to sample K of them such that I have the maximum possible spread. A possible application of this is to build shelters in a road network so that they cover the ...
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1 vote
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Purpose of sampling when we already know the exact PDF formula of the distribution

Mabybe this question is duplicate of my other question here : Is PDF always given when we do sampling? So what I've learned is that, when we already know the exact formula of PDF, we can calcuate any ...
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0 votes
1 answer
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How Many Ways 5 Objects Can Be Selected With Replacement

Suppose I have the following set up: There are 5 objects : A, B, C, D, E The probability for each of these objects to be chosen is : 0.2, 0.3, 0.1, 0.3, 0.1 You want to pick 5 of these objects with ...
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1 answer
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About sampling from a multivariate normal distribution

I am trying to find the time complexity of sampling from a multivariate normal distribution with an $n\times n$ covariance matrix $C$. I have a found an answer here and have a couple of questions. ...
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Is PDF always given when we do sampling?

I studied "Inverse CDF method", "Rejection sampling". So far, what I learned is that when we do sampling, the PDF is always given. Is this true for all samping method?
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4 votes
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How close is the median of a uniformly chosen subsample to the median of the full set?

Take $N$ a set of n numbers, sample s numbers from $N$ uniformly and with replacement giving us the set $S$. What is the relationship between the median of $N$ and the median of $S$ ? I want a result ...
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Estimate the distribution variance for small sampling size

In the case of distribution with a large population size $N \ge 10000$, which follows an unusual distribution with stand deviation $\sigma$. By the central limit theorem, repeat sampling with a large ...
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Generating samples that satisfies MAPE(mean absolute percentage error)

Here is the problem: I have a vector a_i (given) and I would like to generate a vector b_i that satisfies the constraints below, sum(i, |a_i - b_i|/a_i) = n %, where n is given also. I know there are ...
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0 votes
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downsampling of measurement data (time series)

I am designing some software to measure data (voltage over time) and send it to a computer. Now, I am having constraints on the transmission, so I would like to compress data before sending it. ...
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Variance of the total in a simple random sample [closed]

In order to estimate the total for a variable of interest, a simple random sample without replacement of size $N/4$ from a population of size $N$ is sought. After some thought, it is decided that a ...
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0 votes
1 answer
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probability of error in sampling to estimate sum of a population

Given non-negative numbers $$\{m_1, m_2,\dots,m_n\}$$ we have to estimate the sum $$s = \sum_{i=1}^nm_i$$ using sampling (with replacement). If we sample k numbers uniformly at random, then I can ...
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0 votes
1 answer
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Sampling from a high-dimensional multivariate Gaussian distribution when low-rank approximation for the inverse of the covariance matrix is available

I want to get random samples from a high-dimensional (say, $10^4$ or more) multivariate Gaussian distribution. Assume a case of zero mean for simplicity, the probability density is given as $$P({\bf x}...
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