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.
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Statistics - Question on Sampling
Here's the question
The scores, $X_1$ and $X_2$, in papers $1$ and $2$ of an examination are normally distributed with means $24.3$ and $31.2$ respectively and standard deviations $3.5$ and $3.1$ ...
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24 views
The relationship between sample and population skewness [on hold]
Get the sample skewness coefficient of a random variable X from a random sample of size n. Explain why it measures asymmetry of the distribution of X. Explain the relation between the sample skewness ...
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1answer
11 views
Compute mean and covariance matrix of $\bar{X}$ from a simple random sample
Given $\{X_\alpha , \alpha =1,...N\}$ a simple random sample obtained from any p-dimensional distribution with mean $\mu$ and covariance matrix $\Sigma$, compute the mean and the covariance matrix of $...
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0answers
22 views
How to continue sampling after a conditional rejection?
I encountered the following (to me) weird problem while trying to do some simple sampling.
I have a system that generates random numbers. Let's for simplicity assume the numbers are 0, 1 and 2 with ...
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10 views
Sample from Poisson process with no collisions.
On a 2D square $[0,\ 1]^2$, I can draw a random configuration of points $c = (n,\ (x_i)_{1..n})$ with interesting independence properties using a Poisson point process: draw the number of points $n \...
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1answer
22 views
Comparing two random variables with monte carlo sampling
Suppose there are two numbers X1 and X2 that are from a random continuous probability distribution with unknown range. You are given the value of X1 and you need to determine whether X1 is less than ...
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19 views
Constructing an upper confidence limit for $σ^2$
Suppose that $X_1, ..., X_n$ form a random sample from the normal distribution with unknown mean µ and
unknown standard deviation σ.
Construct an upper confidence limit U(X1, ..., Xn) for $σ^2$ such ...
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1answer
76 views
Problem of statistical inference Poisson
I am having problems solving this problem of statistical inference and I do not know if it is well done or not, so I would like someone to review it. I just started with inference, so I have my ...
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1answer
17 views
Using Random Sample to Find Estimate
I have to use the Inversion Sampling Method to generate a random sample of 100 from the function $f(x)=\theta x^{\theta - 1}$ if $\theta =5$. Here is my function so far:
...
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1answer
20 views
Showing that an estimator is consistent
Let $X_1,X_2,\ldots,X_n$ be a random sample from $\mathcal{N}(\theta,1)$. Consider the following (randomized) estimator of $\theta$ given a sample of size $n$:
$$
\hat{\theta}_n = \bar{X} + \begin{...
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2answers
19 views
Finding the Expected value of $\hat{\theta}_n$
Let $X_1,X_2,\ldots,X_n$ be a random sample from $\mathcal{N}(\theta,1)$. Consider the following (randomized) estimator of $\theta$ given a sample of size $n$:
$$
\hat{\theta}_n = \bar{X} + \begin{...
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15 views
Showing that a estimator is consistent
Let $X_1,X_2,\ldots,X_n$ be a random sample from $\mathcal{N}(\theta,1)$. Consider the following (randomized) estimator of $\theta$ given a sample of size $n$:
$$
\hat{\theta}_n = \bar{X} + \begin{...
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21 views
How to show that $\sum_{i=1}^m (X_i−X_m)^2$ and $\sum_{i=1}^n(Y_i− Y_n)^2$ are independent
Let $X_1,...,X_m$ be i.i.d. sample with $N(\mu_1,\sigma^2)$, and $Y_1,...,Y_n$ be i.i.d. sample with $N(\mu_2,2\sigma^2)$.
Let $S_x^2 = \sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= \sum_{i=1}^n(Y_i− Y_n)^2$...
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1answer
21 views
Using normal distribution to calculate $P(52.1 < \bar{x} < 53.9)$
I am given the following question in one of the lectures I was looking through.
Suppose a random sample of size $n = 400$ is to be selected from a population of size $N = 2000$. A quantitative ...
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20 views
Is MCMC (or any sampling for that matter) explainable?
Recently, at an interview, I was asked if you use MCMC to build Maximum a posteriori (MAP), and use it for an inference, will the system you create have an explainability?
Now, explainability is ...
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Sampling distribution - approximately normal?
If the population distribution is skewed to the left, the mean is 25,000 and the standard deviation is 2,500 and the randomly selected n=100, would the sampling distribution of the sample mean be ...
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2answers
209 views
Confusing Sampling from observed data
Suppose we are given some small set of data on bundles of electrical wires and increasing voltages run through them, and we note how many of the individual wires fail.
So for example, a large data ...
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11 views
Sampling a conditional joint distribution of continuous random variables using samples from joint distribution and marginal distributions
I am seeking an approach to sampling conditional joint distribution (new to probability). I will put my case in a simple way:
Similar question for discrete variables is asked here but not yet ...
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1answer
18 views
Sample calculation
I have knowledge about the calculating sample, but i am unable to solve this question for the last two hours. Please check this question.
A bank believes that approximately 2/5 of its checking-...
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15 views
The mean value of sample moment with order k
I've this problem of statistics that I can't resolve. So I hope that someone can halp me.
The problem is this:
Let sampling moments $\overline{X_n^k}=\frac{1}{n}\sum_{i=1}^nX_i^k$, where for $k=1$ $...
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1answer
33 views
Determining sample size given true proportion.
I'm attempting to solve a problem from a statistics course in regards to finding the sample size I need to take when given the Margin of Error, Confidence interval, and 'true proportion' (probability)....
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28 views
Working out expectation of a random sample.
I have the problem:
Let $X_1, X_2, X_3, X_4$ be a random sample from a population that has
mean $μ$ and variance $σ^2$.
Find $\mathbb E[(X_1-X_2)^2]$ and hence the value of $k$ such that $T
...
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2answers
23 views
What is the probability that $m$ items drawn from $n$ distinct items contain all $n$ items?
Suppose there are $n$ distinct items to be drawn with replacement for $m$ times, the probability of each item being drawn is assumed to be $\frac{1}{n}$. What is the probability $P(m)$ that $m$ items ...
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9 views
Sampling from joint discrete distribution
I have a set of items $a_1, a_2, \dots, a_n$. My aim is to generate from this set of items, a list of item tuples $\{(a_i, a_j), \dots\}$ such that $a_i\ne a_j$.
The constraints are as follows. The ...
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13 views
Sampling binary values from a discrete probability distribution
Suppose that we have a discrete non-uniform probability distribution X over $\{0,1\}^k$ for n binary noise values.
Let $e = (e_1 ,..., e_m)$ be a vector of independant identically distributed binary ...
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1answer
17 views
Prove the sampling distribution of $S^2$ has the mean $\sigma^2$ and the variance $2\sigma^4/(n-1)$
I would like to ask whether anyone would mind providing me with some direction on how to proceed with this proof.
The question asked me to use the theorem below to prove that, for random
sample of ...
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0answers
22 views
calculating variance and expectation of unknown binomial variables over a window
I have $2^m$ independent random variables. All have binomial distributions, each with $m$ samples. The probability of success for each binomial distribution is somewhere in the range $[0,p]$ (...
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2answers
28 views
Definition of sample mean
I've seen two definitions of sample mean on the internet.
One definition defines it as the average of Random variable other defines it as the average of sample values of a sample.
I'm confused which ...
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0answers
11 views
Taking the expectation of gaps from uniformly chosen points
Suppose I choose $k$ points $x_1, \dotsc, x_k \in \mathbb{R}$ independently with distribution $D$. I'm interested in the value (or an approximation)
$$ \mathbb{E} \left( \operatorname{max}\limits_{i=...
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Sample new point from set of points
I need guidance to solve the following problem, I'm only looking for directions to follow
Given a set of points $X$ laying on an $n$ dimensional surface with an unknown density.
The aim is to sample ...
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12 views
Trying to understand the meaning of Average outgoing quality
I was following an example in my stat's book that had the following information.
Find the AOQ given $N=2000, n=50$, and $c=2$ With a $2\%$ nonconforming. I calculated the $P_a=0.920$, and using the ...
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19 views
sampling from an existing sample
I have an existing data set with 8 000 observations. I want to check an association between two parameters in the data set. I also want to see if maybe for this purpose I do not need 8 000 ...
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18 views
Setting weightage for random sampling to ensure the sample has a certain distribution
I have 165 red, 239 blue and 495 green balls, making a total of 899 balls. I need to pick 150 balls from these through weighted sampling without replacement. In the end I want to have picked approx. ...
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1answer
27 views
Proof/Explanation of Dependence of Two Random Variables (Drawing Cards with Replacement)
"Two Cards are picked from a deck with replacement. Let X= number of aces, and Y= number of kings.
X and Y are both discrete random variables that can take on 0,1 and 2."
I'm trying to show whether ...
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24 views
Sampling Distribution of samples drawn from an arbitrary (non uniform) probability distribution
I am quite new in the probabilistics field, and I am trying to figure out how would drawing n samples from a probability distribution perform.
Suppose We have a non-uniform probability distribution D ...
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How to prove Gradient-based One-Side Sampling in LightGBM
https://papers.nips.cc/paper/6907-lightgbm-a-highly-efficient-gradient-boosting-decision-tree.pdf
In LightGBM article, how to prove the Theorem3.2?
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1answer
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How do I uniformly sample from sorted combinations
I'd like to sample a random sorted combination with replacement.
(Ideally I'd like to do this without rejection, or otherwise in a computationally-efficient manner)
I can start by sampling a ...
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20 views
Size-biased sampling for family size
Imagine taking a survey of the average family size in a certain neighborhood. One way to achieve this is to survey a random sample of women to ask how many children they bore. Let $N$ be the number of ...
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3answers
75 views
Does a data-dependent sampling rule induce correlation?
I'm struggling to understand whether a data stream sliced up in a certain way could produce two quantities that are dependent but uncorrelated.
Suppose I have two iid streams of data that are ...
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18 views
Estimate the number of elements in a ordered set less than x
Suppose U is an ordered set, S⊆U, and x∈U, estimate the number of elements in S ≤ x; Is there a way to sample the S and maintain a structure whose size ≤ log(|S|) from which for any x ...
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0answers
24 views
How to randomly sample a social graph to find paths between at least 20% of profiles?
Given a Graph, where we know
Total number of nodes (~100,000)
Average no of connections per node (~200)
Maximum distance between two nodes (~5)
How many nodes (and its connections) do we have to ...
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19 views
How to find a proper bivariate Gaussian distribution that integrals 95% over a convex polygon
I have a known 2-D convex polygon, and my goal is to find a bivariate Gaussian distribution, that 1) samples from the Gaussian falls into the polygon 95% of the times, and 2) should be reasonably ...
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39 views
How to distribute N approximately equispaced points with a given probability density?
Let $x_i$ be points in $R^D$ space, $i = 0\ ..\ N-1$, where $N$ is fixed.
The problem is to distribute the $N$ points in the space so that their density is equal to given probability density $p(x)$, ...
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Is there a more efficient expected value estimator than the sample average?
I'm wondering if there is a known estimator for expected value that is more efficient than the sample average.
If that is not the case for an arbitrary random variable, then maybe there are examples ...
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2answers
91 views
Sampling from a continuous distribution
The Lebesgue integral of the standard normal pdf over $\mathbb{Q}$ is equal to zero, since the rationals are countable and thus have measure zero. So the probability of "drawing" a rational number ...
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1answer
37 views
Probabilistic subsampling of an Erdős–Rényi graph
Suppose I have an Erdős–Rényi graph ${\cal G}(n,p)$, where $n$ is the total number of nodes and $p$ is the probability of an edge between any pair of nodes (edges are added independently). I subsample ...
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1answer
76 views
Proving $E(\hat s^2)=V(\bar x)$ for finite population
Population is $X_1,X_2,...,X_N$ with large but finite $N$. Sampling indicator $Z_i \in(0,1)~ \forall i$ , such that $\sum^N _{i=1}Z_i=n$ and $Pr(Z_i=1)=\frac {n}{N}$.
Sample mean in this case is $\...
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21 views
Sampling extreme points from Minkowski Sum
I recently stumbled upon the following subproblem: we are given zonotopes $P_1, \dots, P_m$ in $\mathcal{V}$ representation (i.e. we are given the extreme points of each $P_i$). Denote the Minkowski ...
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32 views
calculating confidence intervals for a probability outcome
Suppose you have two possible outcomes, $0$ or $1$. You want to determine the probability of getting a $1$, and have confidence intervals on your answer.
E.g. for $10$ runs you might have
$$ \text{...
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1answer
25 views
How do I work out how many people would be selected into a specific age range by chance?
I have a sample of children that are aged 5 years up to 7 years. In total there are 2000 children from 160 centres. The purpose of this exercise was to give each child an assessment and record the ...
