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).
1,236
questions
1
vote
0
answers
24
views
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 ...
- 140
-4
votes
0
answers
25
views
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: ...
- 1
0
votes
0
answers
10
views
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 ...
- 3
0
votes
1
answer
39
views
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 ...
- 175
0
votes
1
answer
31
views
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 ...
- 13
0
votes
0
answers
24
views
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 ...
- 193
0
votes
1
answer
14
views
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....
- 361
3
votes
1
answer
35
views
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 ...
- 1,785
0
votes
0
answers
16
views
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,...
- 361
1
vote
0
answers
23
views
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 ...
- 21
2
votes
1
answer
42
views
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 ...
- 365
0
votes
0
answers
18
views
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 ...
0
votes
1
answer
25
views
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 ...
- 361
0
votes
0
answers
16
views
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 ...
- 121
0
votes
0
answers
12
views
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?
- 99
0
votes
0
answers
32
views
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.\...
- 12.8k
1
vote
1
answer
17
views
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 ...
- 11
-2
votes
1
answer
38
views
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(...
- 361
0
votes
1
answer
57
views
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 ...
- 533
0
votes
1
answer
25
views
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.
...
1
vote
0
answers
27
views
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 ...
1
vote
0
answers
36
views
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 ...
1
vote
0
answers
43
views
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 &...
0
votes
1
answer
52
views
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 ...
- 41
0
votes
0
answers
8
views
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/...
- 1
0
votes
0
answers
23
views
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, ...
- 1,741
0
votes
0
answers
33
views
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 ...
- 3,785
0
votes
0
answers
22
views
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
...
- 95
0
votes
0
answers
19
views
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 ...
- 103
1
vote
1
answer
40
views
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?
- 361
1
vote
0
answers
37
views
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 $\mathbb{R}^n$:
$$
\mathcal{X}= \{ x \in \mathbb{R}^n : \sum_{i=1}^n x_i \leq 1, x_i \geq 0, \ i=1,\dots,n\...
- 11
1
vote
0
answers
7
views
Lower bound of the performance of sampled optimization
Hi I am considering an optimization problem that requires a condition to hold over a continuous range, for example:
\begin{equation}
\begin{aligned}
\max_{\boldsymbol{x}, \gamma} & \ \...
- 371
2
votes
1
answer
42
views
Difference between $S^2$ and $s^2$ and $\sigma^2$?
It's in mentioned in my professor's slides that $S^2$ is the population variance and $S^2$ = $$\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 ...
- 305
0
votes
0
answers
14
views
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 ...
- 305
3
votes
2
answers
81
views
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 ...
- 787
1
vote
1
answer
32
views
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 ...
- 57
3
votes
0
answers
72
views
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 ...
- 151
0
votes
1
answer
32
views
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) , ...
- 4,358
0
votes
0
answers
26
views
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 ...
1
vote
0
answers
22
views
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 ...
- 119
0
votes
1
answer
26
views
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 ...
- 1,688
1
vote
1
answer
27
views
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.
...
- 607
0
votes
0
answers
36
views
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?
- 119
4
votes
0
answers
40
views
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 ...
- 41
0
votes
0
answers
26
views
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 ...
0
votes
0
answers
7
views
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 ...
0
votes
0
answers
22
views
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. ...
- 252
0
votes
1
answer
67
views
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 ...
- 650
0
votes
1
answer
35
views
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$$ using sampling (with replacement). If we sample k numbers uniformly at random, then I can ...
- 67
0
votes
1
answer
91
views
Sampling from a high-dimensional multivariate Gaussian distribution when low-rank approximation for the inverse of the covariance matrix is available
I want to get random samples from a high-dimensional (say, $10^4$ or more) multivariate Gaussian distribution. Assume a case of zero mean for simplicity, the probability density is given as
$$P({\bf x}...
