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1
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1answer
138 views

What proportion are above x of sample size n where X ~ N(0,1) Homework

I have a homework question that I'm not quiet sure of. It follows as so: Consider a random variable $X$ that has a standard normal distribution with mean $\mu=0$ and standard deviation $\sigma=1$. ...
1
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1answer
91 views

Why Gibbs sampling needn't “remixing”

I am generating $\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \dots, \mathbf{x}^{(n)}$ using Gibbs sampling methods. So I want $\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \dots, \mathbf{x}^{(n)} \sim$ some ...
1
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0answers
36 views

Relation with $F$ distribution and $t$ distribution

If $X\sim F_{n,n}$ , then show that $$\frac{\sqrt n(\sqrt X-\frac{1}{\sqrt X})}{2}\sim t_n$$
1
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1answer
30 views

How many words do I know, extrapolated from a sample?

I would like to check my knowledge of foreign words by sampling a N words dictionnary: after checking randomly n words I would find that I know k of them (and do not know n-k). How should I choose n ...
5
votes
1answer
197 views

Uniform sampling of points on a simplex

I have this problem: I'm trying to sample the relation $$ \sum_{i=1}^N x_i = 1 $$ in the domain where $x_i>0\ \forall i$. Right now I'm just extracting $N$ random numbers $u_i$ from a uniform ...
1
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2answers
70 views

While describing sampling theorem, is it $f_s \ge 2 f_m$ or $f_s > 2 f_m$?

I have a doubt regarding sampling theorem. Sampling theorem states that if a band limited signal has to be recovered after sampling, then the sampling frequency $f_s$ should obey $f_s \ge 2 f_m$ ...
0
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1answer
30 views

Equivalent standard error for different populations

So I have population A and population B. The demographics are similar (assumes that the true expected value is the same in both populations). Population B, however, is twice the size of population A, ...
1
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1answer
64 views

Number of draws needed to get a positive element using a *weighted* sampling without replacement

Imagine we have $M$ elements, where some of them ($y$) are positive and the rest, $z=M-y$, are negative. The probability of drawing any of them is given by a distribution. Let's call $p_1$, $p_2$, ...
0
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1answer
80 views

Central limit theorem: question about √n and σ2

Still making (good) progress with my knowledge on statistic. Sorry if I ask lots of questions about this recently but I really like math.stachexchange and really appreciate the quality of the answers ...
1
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0answers
135 views

Interpreting the meaning of sampling distribution

I have asked a couple of questions related to statistics recently as I just started to study the topic again (I ignored my university course on statistics and I now eat my fingers in anger). I asked ...
1
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1answer
122 views

Required sample size to reduce the interval size by half

I have a small doubt regarding a statistics problem and would like a confirmation. If we want to reduce the length by half of the confidence interval (at 95%), we should: Multiply the size of the ...
2
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1answer
76 views

Inverse of the German Tank Problem?

I have a problem that maps to estimating the discrete distance to a goal. The sample space is n discrete positions on a circle labeled sequentially; n is known. A target position is randomly ...
0
votes
1answer
235 views

Gibbs sampling to produce posterior pdf

Suppose we have the following classical normal linear regression model: $$y_i = \beta_1 x_{1i} + \beta_2x_{2i} + \beta_3x_{3i} + e_i$$ where $e_{i} \sim iid.N(0, \sigma^2)$ for all $i = 1, 2, ...
1
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0answers
260 views

Shannon vs dirichlet kernel interpolation method for signal reconstruction

I am currently studying fourier transform, and especially the way that the signal could be reconstructed from its spectrum. In many lectures, I have seen the shannon interpolation method to ...
1
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3answers
133 views

Sample of a subset of a plane

I have the equation of a plane $ax+bx+cx+d$ and a point $(x_0, y_0, z_0)$ on that plane. I defined the neighborhood of that point on that plane as the set of points satisfying $(x-x_0)^2 + (y-y_0)^2 ...
1
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1answer
265 views

probability of a certain event in a repeated sampling with replacement (without ordering)

I have a problem that is bugging me for a couple of weeks now. I have asked some friends etc but the answers were not satisfying at all. So here we go. Suppose we have a set ...
1
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0answers
51 views

Gibbs / MCMC sampling for sum of parameters - how to improve slow mixing?

Suppose I have a hierarchical Bayesian model, where my observational prediction, $y'$, is calculated as the sum of other parameters, ${\alpha_i}$. My observation equation (the likelihood) is: $P(y | ...
2
votes
1answer
329 views

How can I sample a bivariate Gaussian distribution using Gibbs sampling?

I'm trying to sample a bivariate Gaussian distribution using Gibbs sampling, but I think I don't have the correct conditional probabilities. According to this lecture slides, the conditional ...
1
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1answer
40 views

Simple counterexample to sampling theorem

I know this has to be wrong, but can't see what is wrong with it: Take a simple sinusoid. It crosses zero every half cycle. Sample it at double its frequency. If the samples coincide with the ...
2
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0answers
142 views

Notation for sampling a random variable

my question is pretty much the same as the one asked here: Notation for sampling random variate (I did not find a satisfying answer there...): I have a random variable $X \sim U(1, 10)$. I want to ...
3
votes
1answer
85 views

Is it possible to sample the Dirac delta function?

The Dirac delta function can be a probability measure with the unit/Heaviside step function as its cumulative distribution function. Is it possible to sample such a distribution? If a random variable ...
1
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0answers
22 views

Population size and accuracy of expected value

If I have a series of populations, and a set of outcomes for these populations, how can I be certain that the observed proportions are, in fact, credible? I have investigated certain sampling methods ...
0
votes
1answer
37 views

Survey vs interview size

I have a survey of size 120 and 20 interviews. Can I compare the results? Both groups answered same questions. Example: let's say a question was do you like to read novels ? Can I say that 20% of ...
1
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1answer
73 views

symmetric polynomial inequality?

I put $n\ge 2$ balls of various sizes into an urn. I draw two balls (without replacement) from the urn. With each draw, I draw any given ball with probability proportional to its size. Can you ...
0
votes
1answer
51 views

Sample size question / margin of error

I'm a complete novice in this area - so "explaining to me like I'm 5" would be most appreciated. Essentially, I've been tasked with changing the a relevancy algorithm for products displayed on a ...
3
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0answers
190 views

Relationship between DFT and FFT/DTFT

Question 1: Assume we already know $x(t):[0,2\pi]\to\mathbb{R}$'s Fourier series $x[n]_{n=-\infty}^\infty$. Perform DFT on $x[n]_{n=0}^N$. How is the result connected to $x(t)$? WHY? Question 2: ...
2
votes
1answer
266 views

Proof that $\frac{(\bar X-\mu)}{\sigma}$ and $\sum_{i=1}^n\frac{(X_i-\bar X)^2}{\sigma^2}$ are independent

Let $X_i\sim N(\mu,\sigma^2)$ ; where$[i=1,2,\ldots,n]$ $Z_i\sim N(0,1)$ ; where$[i=1,2,\ldots,n]$ Proof that $\bar Z=\frac{(\bar X-\mu)}{\sigma}$ and $\sum_{i=1}^{n}(Z_i-\bar ...
1
vote
1answer
66 views

An Exercise of noncentral $\chi^2$ distribution.

Let $Y_1,\ldots,Y_n$ be independent random variables with $Y_k$ distributed as $N\sim(a_k,\sigma^2)$, and $\bar Y=\sum_{k=1}^{n}\frac{Y_k}{n}$ denote the sample mean, $S^2$ denotes the sample ...
1
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1answer
75 views

cumulants of non-central $\chi^2$ distribution

Cumulant generating function is defined by logarithm of moment generating function. $$K_X(t)=\log M_X(t)$$ Let $X$ be a non-central $\chi^2$ variate with parameters degrees of freedom, $n$ and ...
1
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1answer
64 views

Sampling a Population of Unknown Size

How do you sample a population whose size is not known? For example, I have a day's population from whom I need to select 1000 items from. Each hour (for one day) some number of new items (can be ...
0
votes
2answers
84 views

Probability that I am not selected in any of 2000 samples?

The population contains 100 million adults, which includes myself. Simple random sampling is used to choose a sample of 1000 adults, 2000 times, independently. I need to find the probability that I ...
1
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0answers
77 views

Problem generating random vectors with a randomized linear programming with equality constraints (weird clustering)

Summary For simulation problems, I need to be able to generate large numbers of random lists of numbers, say $x_1, x_2, \dots, x_n$ (where $n \approx 1000$), subject constraints similar to what one ...
1
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1answer
71 views

Calculate each point of my Gauss curve

I'm coding a program which calculates the confidence interval. He takes the population and a sample of the population as parameters, and the percent of people of that sample who answered "yes" to a ...
2
votes
1answer
71 views

Sampling labeled items on a conveyor belts

I have items on a moving conveyor belt. Every item has a label with a number that goes from $1$ to $N$; on the conveyor belt there are more than $N$ items. I have a camera above the items on the belt, ...
1
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0answers
163 views

Discrete approximation to a continuous probability density function

I want to approximate a continuous, finite probability density function, with a specified number $N$ of points, in the following way: If the pdf is 1-dimensional, defined over the section [0,1], then ...
3
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1answer
89 views

Maximum likelihood estimation - why is $\mathcal{L}$ not the joint pdf?

Here's an excerpt from my notes: Define the likelihood function: $$\mathcal{L}(\vec{x};\theta)=\prod_{i=1}^{n} f(x_i;\theta)$$ Where $f$ is the pdf of the distribution we're sampling the $x$'s ...
1
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1answer
792 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 ...
2
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0answers
69 views

Sampling probability

I have a pool of 36 samples (each is a pool itself). If I take a subset of eight samples at random from this pool, how do I calculate the probability of obtaining two of the same type of sample from ...
0
votes
1answer
26 views

What does MSSL sampling refers to?

In a document I came across this sentence "calculation of MSSL using fixed and dynamic methods". The document itself is about sampling techniques and no other explanation is given there. I tried to ...
3
votes
0answers
112 views

Problems sampling from a $pdf$ over $SO\left(3\right)$

I have a probability density function over $SO\left(3\right)$, which I am trying to sample from. The $pdf$ is given as a generalized fourier series: $$ f\left(\omega,\theta,\phi\right)=\sum ...
0
votes
2answers
54 views

Sample $x$ from $g(x)$

I got confused with all this randomness and probability functions. I was trying to implement the rejection sampling method which (apparently) is really simple. I was reading from Rejection Sampling in ...
1
vote
1answer
36 views

Optimal sampling for an arbitratry area

I have a closed area (2D) of arbitrary shape and some number of sample points to take from the area. Now I would like to find optimal distribution of the sample points, so that distances from sample ...
0
votes
1answer
278 views

please prove the following proof related to F distribution.

Suppose $S_1^2$ and $S_2^2$ are two independent unbiased estimate of the common population variance $\sigma^2$ from two random sample of sizes $n_1$ and $n_2$ respectively. Then show that ...
1
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0answers
34 views

Suitable change of measure with importance sampling

I'm working on a project on Importance Sampling. When one tries to estimate the small probability $\alpha:=\mathbb{P}(X \in A)$, one can do a change of measure. There is change of measure which ...
1
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0answers
56 views

Continuous random sampling with replacement.

Construct a set $s\subseteq[0,1]$ by sampling points in $[0,1]$ with uniform probability density $x\leq1$ so that $|s|=x$. Interpret this as a sampling frame during which data is captured. Now, ...
0
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0answers
249 views

Simulate random sampling with replacement

Any ideas on how to approach this problem? (Due to Karp) Consider a bin containing d balls chosen at random (without replacement) from a collection of n distinct balls. Without being able to see or ...
2
votes
1answer
112 views

Monte Carlo Rejection Sampling Method

I have the following passage from a set of lecture notes I am working on that I would like to understand a little better. $\underline{\text{Algorithm for Rejection Sampling}}$: Given two densities ...
1
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0answers
37 views

Sphere on a grid

So, this is a little tricky kind of a question and I'm not totally sure if it's a mathematic question or a more programming one, but I nevertheless hope to find answers. I want to find out the error ...
1
vote
1answer
47 views

variance reduction

Say i have $n$ variables with variances $V_1,V_2,...V_n$. The sum of the variables will have a variance of $V=V_1+V_2+..V_n$ .Now if i am given N total simulations to reduce the variance V, how do i ...
3
votes
2answers
278 views

Simple random sampling without replacement of huge dataset

For an application I'm working on, I need to sample a small set of values from a very large data set, on the order of few hundred taken from about 60 million(and growing). Usually I use the technique ...