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1answer
26 views

Variance- covariance matrix

Consider $H$ denotes hat matrix and $e$ denotes residual. In the book Applied regression Analysis by Draper/Smith, it is written that : $\mathbb V(e_i)$ is given ...
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0answers
18 views

Initializing MCMC walkers with ambiguous direction (-/+)

I'm running a sampler program where there are observations given as sample data which are derived from an equal sized population of parameters that are converted to the observations using a known ...
0
votes
1answer
22 views

Confidence Interval for Regression Coefficient ,$\beta$

In the book 'Applied regression Analysis' by Draper/Smith, it is written that : Obtain individual $100(1-\alpha)\%$ confidence interval for the various parameters separately from the formula ...
1
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1answer
27 views

Random points inside a convex polytope

Given a convex polytope, defined by set of vertices $P = \{\mathbf{x}^{(i)}\}_{i = 1}^n, x^{(i)} = (x^{(i)}_1, x^{(i)}_2, \dots, x^{(i)}_d): \operatorname{conv}(P) = P$. How to generate uniformely ...
0
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0answers
11 views

Covariance of independent samples

Let's say that I took a bunch of samples --may be mean age -- from a rural area in Africa and another set of samples from a very different and unrelated population in the US. Moreover, one person ...
0
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1answer
14 views

Building histogram of latency using sample of input data

Assume I have input data set consisting of a web page response time. I'd like to build histogram from input data, but for practical reasons I can only use sample of data. Based on histogram I want to ...
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4answers
124 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 $$\mathbb E\left[\frac{X}{Y}\right]={\frac{\mathbb E[X]}{\mathbb E[Y]}}$$ ??
1
vote
1answer
39 views

Stratified random sampling without replacement

I came across this statement and can't decide if it's true or false. Statement: In a stratified random sampling without replacement, with proportional allocation to the population size, the sample ...
2
votes
2answers
54 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." ...
1
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2answers
622 views

sampling problem;

The values below are the scores (maximum 20) obtained in an aptitude test by a random sample of 11 graduates. It is known that for the non-graduate population the median score is 12. Is there ...
0
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0answers
14 views

Spherical Sampling of Projected Disk

Given a 2D solid disk centered at some 3D point $\vec{x}$ with radius $r$ and normal $\vec{N}$, I need to compute a random unit vector from the origin that hits the disk. The vector needs to be ...
-1
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0answers
21 views

What is sampling with unequal probabilities without replacement

sampling without replacement: i know if my sample size is $n$, and if $i^{th}$ unit is drawn then we will not replace the unit in the population , so that in my next draw i can get any unit except ...
1
vote
1answer
25 views

generating random samples with a PDF

I have the PDF of a distribution from which it is not possible to get a closed from for the CDF or inverse CDF. Is there a technique that would allow me to generate samples from this distribution ...
1
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1answer
40 views

Non-uniform sampling of N-sphere

Suppose I have a unit $N$-sphere from which I want to draw points at random. To obtain uniformly distributed points I do the usual technique of drawing $N$ random variables $x_i$ from a Gaussian ...
0
votes
1answer
23 views

How to estimate amplitude of a sinusoid from two samples

Given a sinusoid $x(t)=A\cos(t+\theta)$, I can estimate the amplitude $A$ if I take two samples separated by $\frac{\pi}{2}$. If $X_1=x(0+\theta)$ and $X_2=x(\pi/2+\theta)$. Then, $A$ can be estimated ...
1
vote
1answer
21 views

Sampling on Axis-Aligned Spherical Quad

Given spherical coordinates on a unit sphere, imagine a spherical quad defined by two ranges $[\phi_0,\phi_1]$ and $[\theta_0,\theta_1]$. If you have a globe, for example, the grid formed by the ...
0
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1answer
19 views

Confidence Intervals and Inferences

I need help with c, as I have attempted a and b already, but believe they help with context. Suppose you took a random sample of 100 accounts in a large department-store chain, and found that the ...
1
vote
1answer
16 views

Joint Distribution and Sampling Distribution

There are three different incomes, x, and their proportions, f (x). $ 10,000 0.40 $ 30,000 0.40 $ 50,000 0.20 How do I calculate the joint distribution for X1 and X2, which are a random ...
2
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0answers
15 views

Sampling from a graph

Suppose you have a graph $G=(V,E)$ that is unobservable globally and you wish to take a sample from the vertices of that graph to infer something about its global properties from local properties. ...
0
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0answers
24 views

Unconditional sampling distribution of regression coefficients

I am trying to find the (unconditional) sampling distribution of a regression coefficient in a simple linear regression. The linear regression $Y = \beta_0 + X\beta_1 + \epsilon$ is conducted for $N$ ...
4
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2answers
90 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 ...
1
vote
1answer
534 views

probability of sample variance lying between given values

Let $X_1,\ldots,X_n$ be a random sample of size $n = 10$ from a population which is Normally distributed with mean $= 48$ and variance $= 36$. What is the probability that the sample variance of such ...
0
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0answers
17 views

Trying to use Cholesky decomposition of covariance matrix to sample error ellipsoid

I'm trying to construct an error ellipsoid from a covariance matrix (which exists for a 3D point) and then sample consistent xyz points in this region. In a previous question when I asked about this ...
0
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1answer
50 views

Sampling error with weighted mean

I am studying statistics and I am wondering when it comes to standard error or a sampling if the calculation changes when there are weights added. I have a weighted mean: $$\mu_{w} = ...
0
votes
2answers
36 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} = ...
0
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0answers
17 views

Kalman Filter, deriving the conditional distribution for covariance matrix

I have a Kalman Filter model where: 1- State space is $x_{1:N}={(x_1,x_2,...,x_N)}$ and observation space is $y_{1:N}=(y_1,y_2,...,y_N)$ 2-$\mu_1$ and $V_1$ are the mean vector and covariance ...
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votes
1answer
20 views

Probability in a Random Sample [closed]

V. The mean monthly rental rate for a two-bedroom apartment in Atlanta is $\$982$ (Elle. September 1998). Assume that the population mean is $\mu = \$982$ and the population standard deviation is ...
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0answers
16 views

Convergence of sampling from Brownian motion

For a standard linear Brownian motion $\{B(t)\mid\ 0\le t\le 1\}$, for natural $n\ge 0$ and natural $1\le k\le 2^n$, let $d(n,k)=B\left(k2^{-n}\right)-B\left((k-1)2^{-n}\right)$ be the differences of ...
3
votes
2answers
59 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 $\Omega$ ...
0
votes
1answer
22 views

decimation in signal processing

Recently I was struck with the following question: And this is what I think about it: "Downsampling is one of the rare processes that are NOT time invariant. From the very nature of it's ...
0
votes
1answer
71 views

Sets and expectations

Imagine two sets $A = \{1, 2, \dots, a\}$ and $B = \{1, 2, 3, \dots, b\}$ with $a \leq b$. Let $f$ be a uniformly independently distributed random map $f:A\rightarrow B$ and $F = \bigcup_{i=1}^a ...
0
votes
1answer
21 views

Consequences of violation of independence assumption in ANOVA test

Can you please explain me what is the consequences of violation of independence assumption in ANOVA test? I couldn't visualize it. Does Cochran Theorem relates to ...
1
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2answers
38 views

Compressive Sensing matrix

I am working with compressive sensing recovery with image and I test with various sensing matrices: Case 1: Sensing matrix A of size MxN is i.i.d Gaussian matrix. Case 2: Sensing matrix A is size of ...
2
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1answer
92 views

Non-i.i.d Empirical Risk Minimization

I'm not a statisticians so please forgive me if I posed a silly question, but it's a real problem for me in my research. Suppose we have defined risk in a regression problem as $R(f)=\int l(f(x),y) ...
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0answers
21 views

Pairwise independent subsets of fixed size

For a given set $\Omega$ of even size $n$, does there exists, in general, a collection of subsets $F \subset 2^\Omega$ such that $\forall A \in F. |A| = \frac{n}{2}$, $\forall \omega \in \Omega. ...
0
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0answers
6 views

“Noisy space-filling” iterator for multiple discrete finite dimensions

Suppose you have a D-dimensional space, where dimension $i$ takes values in the finite set $V_i$. Take as an example $V_1=\{1,2\},\; V_2=\{a,b\}, \; V_3=\{\alpha,\beta,\gamma\}$ One simple way to ...
0
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1answer
18 views

Confidence/Tolerance interval for a percentage of a population

I have a problem I'm not sure how to solve. It goes something like: ...
0
votes
1answer
65 views

How to mathematically prove that we are sampling from same distributions?

The content of this question is about rigorously proving something which is otherwise considered easily correct intuitively. Let's assume we have a multivariate distribution $g(x_1,x_2,...,x_n)$ over ...
0
votes
0answers
9 views

Identically distributed subset of samples selected dependently

Assume we have a set of (vectorial) data point drawn from an unknown distribution $p(x)$. we select a subset of these samples one by one. Selecting a new point is such that make it dependent on the ...
1
vote
1answer
65 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 $A_{d−1}$ ...
1
vote
1answer
37 views

Choosing a sample from a sample probability

I am a bit confused about this problem. I understand that you need to pick a sample first, K, and then find the probability of that sample being red, L. The total different combinations of picking a ...
0
votes
0answers
16 views

Calculating person-time

Suppose that an investigator wants to measure the incidence rate of high blood pressure in the $1997-98$ freshman class of a university. Assume that there were $1000$ entering freshmen and that this ...
1
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0answers
35 views

problem on standard error in sample data

From a population of 20,000 observations, a sample of 500 observations is selected. how to Calculate the standard error of sample mean if the population standard deviation equals 20.
0
votes
1answer
94 views

Deriving the variance of the sample mean $\mathrm{Var} (\bar{y})=\frac{1}{n}(1-\frac{n}{N})S^2$

For a population of size $N$ with a simple random sample size $n$ derive the formula $$\mathrm{Var}(\bar{y})=\frac{1}{n}\left(1-\frac{n}{N}\right)S^2$$ where $S^2$ is the population variance. Hint: ...
1
vote
0answers
24 views

Mean & SD of Sampling Distribution

A population consists of 4 numbers {0, 2, 4, 6}. Consider drawing a random sample of size n = 2 with replacement. (a) What is the sampling distribution of $\bar x$? Is this a normal distribution ? ...
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0answers
22 views

What are incoherent matrices

What does incoherence means in terms of matrices? I am brushing on some compressive sampling theory and I did not find any easy to understand or straight forward answer about what does the word ...
0
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0answers
28 views

stroboscope effect

I have a disc with a line drawn on one of his radius that is turning with frequency $f$, and I want to sample the place of the line to find the frequency of the disc. So we know from the ...
0
votes
1answer
21 views

Test for Validity of Artificial Samples

I have a model that actually is learned from some observed samples. Then I use the model to generate several artificial data. My question is: Which test should I use to test if the data is of the ...
2
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0answers
24 views

Variance of a Population of Two Indpendent Random Variables

I have a question regarding a problem I'm looking at out of personal curiosity. Here is the basic setup of the problem: There is a population that contains half of type A, and half of type B. The ...
2
votes
1answer
33 views

Approaching a desired but infeasible distribution when constructing a sample

Suppose you have $N$ balls in $C$ different colors, and a "desired" distribution of those $C$ colors (eg 20% red, 80% blue). Your task is to build a sample (not really a random sample per se) of $S$ ...