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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Help in probability, Difficult Question:// [on hold]

Upon testing 80 resistors manufactured by a certain company, it is found that 15 resistors failed to meet the tolerance design specifications a) Construct a 92% two-sided confidence interval for ...
0
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1answer
21 views

Joint probability with housing stock data

I have a question about probability. Let’s say we have 100 homes of different ages, types and insulation levels, distributed as per the table below. Housing stock data How do I determine how many ...
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0answers
13 views

How to do non-analog sampling from a pdf

I have a distribution given by $$ f(x)=\frac{1}{2}\sinh(\sqrt{2x})e^{-x}. $$ Due to the shape of this distribution, most of my samples will be in the range $x\in(0,2]$. However, I am interested ...
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22 views

How to find sampling distribution S.D in this case

The distribution of the weights of 1000 students is normal with a mean of 55kg and a variance of 25. 100 random samples of size 16 are taken from this population. Determine the mean and standard ...
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21 views

Determine p value of the statistics [on hold]

Given X is a geometric random variable with pmf and cdf $$p(x) = 0.153(1-0.153)^{x-1} $$ and $$F(x) = 1-(1-0.153)^x$$ A statistics S is defined by $S = \min\{X_1, X_2,X_3,\ldots,X_n\}$ If the ...
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26 views

Ways to sample a complicated PDF on an hemisphere

I want to generate samples on the upper real unit hemisphere with the following PDF (it's not really a PDF because I can't guarantee that it integrates $1$) $$\frac{\sum_{i=0}^{n}c_i(\text{ ...
2
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1answer
44 views

sampling distribution question

Need clarification on a binomial sample example: we drew a sample of size $100$ from a binomial($m = 2$,$p = 0.2$) distribution and observed $76$ of the $x_i = 0$, $20$ of the $x_i = 1$ and $4$ of ...
1
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1answer
24 views

compute probability density function of a bivariate function without sampling

Suppose $X_1 \sim f_{X_1}(x_1)$, $X_2 \sim f_{X_2}(x_2)$ are random variables with known probability density function. Is there any way to compute the probability density function of a bivariate ...
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2answers
20 views

Simulation methods and generating random variables

Twenty aircraft are sent to bomb a target that is rectangular in shape. It has dimensions 150m by 50m. Each aircraft makes a bombing run along the horizontal x axis and drops one bomb. The point ...
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0answers
21 views

How Much of a Distribution has One Seen After N Samples with Replacement?

I frequently run into this question while modeling processes. I am wondering if there is a general solution or approximation. The question I run into is: For a given distribution with a finite ...
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0answers
10 views

upsampling and plotting a signal in matlab

I want to upsample by 5 a signal in frequency domain, and then plot(stem) it. I figured how to upsample, Fk=(1/5)*upsample(ak_new,5) now this creates a vector ...
1
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1answer
30 views

Basic questions concerning sample means and distributions

I have the following questions: Is The value of the sample mean always the population mean $\mu$, in any sample? I am confused about whether or not it is. Is the sampling distribution of the sample ...
1
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1answer
13 views

hemisphere sampling and vector flip

I am currently implementing a uniform sampling of a hemisphere. Since my hemisphere is oriented around a specific vector N, whatever the sampling algorithm used (uniform/cosine ...
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0answers
25 views

Bound expected sample mean error

Given $n$ numbers $x_1, \cdots, x_n\in [a,b]$, randomly sample $m$ numbers without replacement $x_{i_1}, \cdots, x_{i_m}$. How can I bound the expectation $$E_{i_1, \cdots, ...
0
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0answers
17 views

Variance of Importance Weights in Importance Sampling

What is the variance of importance weights in Importance Sampling? Is it good to reduce the variance of importance weights and why? And what is the optimum proposal in importance sampling? I assume ...
0
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0answers
53 views

Drawing successive related samples from a Gaussian distribution

I have a question which describes a system where successive samples are drawn from a Gaussian distribution, and each sample is defined with the following relation. $v_{k+1} = \gamma v_k + \eta_k$ ...
0
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0answers
17 views

How to do importance sampling if $ f(x)=\frac{\sin x}{\ln x}$?

Let $$ f(x)=\frac{\sin x}{\ln x }$$ In importance sampling, I should choose $g(x)$ to reduce variance. My question is, what kind of function should I choose if $f(x)=\frac{\sin x}{\ln x}$ (like ...
0
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2answers
37 views

$\cos(2\pi f nT +2\pi N_n) =\ cos(2\pi f nT)$ and more, why?

I am studying signals and systems and this came up? Could someone explain why is $\cos(2\pi f nT +2\pi N_n)$ equal to $\ cos(2\pi f nT)$? The book says: "because $N_n$ is an integer" I am ...
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0answers
17 views

Drawing uniform samples from the *range* of a non-invertible function

I am looking for a Bayesian technique to draw samples from a uniform distribution over the range of a non-invertible (that is, there isn’t even a formula) function $\mathbf{f}: \mathbb{R}^N ...
0
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1answer
23 views

Fourier transform of sampling function

Calculate the Fourier transform of $f_{ZOH}$ (the zero-order hold reconstruction of a sampled signal). Where $f_{ZOH} (t)= f(kT), \ \ kT \leq t < (k+1)T,$ and the sampled signal is $$f_s = f(t) ...
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0answers
10 views

Getting a feel for the Normal-Inverse-Wishart conjugate prior to multivariate normal distribution

I am trying to get a feel for the Normal-Inverse-Wishart conjugate prior, which I have started to use, sparingly, in my work, where I am trying to cluster multivariate normal data. As Wikipedia ...
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16 views

Sampling from Irwin-Hall (Uniform Sum) distribution using rejection sampling

Irwin-Hall Distribution is a probability distribution for a random variable defined as the sum of a number of independent random variables,each having a uniform distribution True or False: The ...
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0answers
8 views

Sampling from Irwin-Hall distribution using triangular distribution

So I need to sample from the Irwin-Hall distribution using rejection sampling with the triangular distribution. I built 2 functions: The first is d_irwin which receives an $x\in supp(g)$ and the n we ...
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0answers
27 views

Generating continuous random variables from a set of Bernoullis

Given a set of $Bernoulli(p_i)$ variables with each having its own arbitrary $p_i$, is there an efficient way to generate continuous random variables sampled from an arbitrary distriubution? To ...
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15 views

Intuition behind sampling distributions – specific case

I'm still trying to understand the basics of understanding the intuition of sampling distributions and calculating the sampling distributions of common estimators. For example, I understand the ...
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0answers
20 views

How to approximate a probability distribution vector by sampling

Given a probability distribution vector $\overrightarrow{p}=(p_1,p_2,\cdots,p_n)$ (of course $\sum_{i}^n p_i=1$), assuming we can sample the distribution with obtaining $i$ with probability $p_i$, how ...
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1answer
55 views

How to choose new points after grouping/resampling?

I'm resampling a signal (which takes values [0,1]) of N samples (blu points) to one with N/5 samples, where (for each group of 5 samples) I store in two arrays the max and the min values of the ...
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2answers
36 views

Probability of inequalities between max values of samples from two different distributions

Given samples from two empirical distributions (not necessarily passing tests of normality, but a solution for the normal case would definitely be useful) what's the probability that the maximum value ...
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1answer
23 views

Do 'Sample from' and 'insert parameter' commute?

The Setting is as follows: We are given random variables $X$ and $\Theta$ but we are not so much interested into $X$ itself as its Distribution needs a Parameter $\theta$ which is produced by ...
2
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1answer
41 views

Equation for estimation of sample size is a quadratic?

The equation for calculation of sample size for a prevalence study happens to be $$\ n= \frac {Z p (1-p)}{e^2}$$ where $Z$ is the $Z$ score, $e$ is the precision we want to achieve and $p$ is the ...
2
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1answer
25 views

Mod of a random variable

I had this problem where I wanted to generate random variables (discrete) in a way that certain numbers were more probably than others (basically geometric) but since I wanted to use this number as an ...
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1answer
33 views

Vertex and barycentric coordinate enumeration for a polytope

Problem 1 Enumerate all the vertices of the polytope defined by: ${\bf Ax}\leq {\bf b}$ where ${\bf A} \in R^{m \times n}$, ${\bf x} \in R^n$, ${\bf b} \in R^m$ and each element of ${\bf x} = ...
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2answers
24 views

Do unbiased estimators have to be exactly equal to the true value of the parameter?

Is it true that for an unbiased estimator, the mean of the sampling distribution is very close to, but not always equal to, the true value of the parameter being estimated? My textbook says that "An ...
6
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1answer
3k views

Probability to choose specific item in a “weighted sampling without replacement” experiment

Given $n$ items with weight $w_n$ each -- what is the probability that item $i$ is chosen in a $k$-out-of-$n$ "weighted random sampling without replacement" experiment? Can a closed-form solution that ...
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0answers
15 views

Multivariate Inverse Transformation Sampling

Summary Given a multivariate density distribution, I use inverse transformation sampling to sample points from this distribution. While the first dimension exhibits the correct distribution, all ...
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0answers
23 views

Padua points or Chebyshev grid for more than two dimensions?

I'm looking for a good point grid in order to sample from a (polynomial) function $f:R^n\rightarrow R$ at discrete points lying in a rectangle. I don't need any weights or interpolating polynomial. I ...
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0answers
24 views

Compare two samples tests

Sampling $1000$ birds from a population which has $10$ types of birds the expected outcome is $100$ birds of each type (this is for the sake of simplicity; general case is each distributed with ...
2
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1answer
82 views

Correlation between probability of events

Suppose there are two events $A$ and $B$ and that $P(A|A\cup B)P(B|A\cup B) = P(A\cap B | A \cup B)$. Then I am asked to find if $A$ and $B$ are independent, positively or negatively correlated. My ...
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2answers
40 views

Determine periodicity from transition matrix?

I have a two part question. Let's say we have a transition matrix T: \begin{bmatrix} 0 & 0.2 & 0.8 & 0 & 0 \\ 0.7 & 0 & 0.3 & 0 & 0 \\ 0.6 & 0.4 ...
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1answer
34 views

Statistically quantifying a variable with a limited number of samples

Regarding a variable which at any time can have one of two values, but which we only have a limited number of samples for, I'd like to be able to make a statement along the lines of: With 95% ...
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0answers
32 views

Sample size for 95% Confidence

If $X_i$ and $Y_j$ are normal distributions where $i = 1,...,n$ and $j= 1,...,n$ with different $\mu$ but same $\sigma^2$, and $\mu_x$ - $\mu_y$ = $\sigma/3$ what is common sample size $n$ needed to ...
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1answer
24 views

Sampling Without Replacement : Draw x Red before Drawing RGB

I am struggling to get my head around how to solve the following problem and can only find solutions to simpler versions Consider a bag containing x Red, y Blue and z Green marbles What is the ...
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0answers
12 views

Determine two sets of samples being independent by fisher exact test method

I am using Matlab to determine two sets of samples being independent by fisher exact test method. Now I have generated 1000 samples for both random variables, and I already used reject-accept sampling ...
0
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1answer
25 views

sampling distribution of mean

Suppose we have a binomial population with parameters $n$ and $p$ so that the mean in $np$. How to find the sampling distribution of mean from the population?
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11 views

How does a DTFT relate to physical frequency?

After performing a DTFT and normalizing the frequency plot I ended up with the following figure The resulting data is correct, as the input signals were of 5kHz and 25kHz frequency. The part I am ...
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1answer
17 views

Sample variance: degree of freedom argument

In sample variance we divide by n-1 and not n. I know a couple of arguments for this - one is that this is sort of a normalization to ensure that the expected value of sample variance is equal to ...
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0answers
10 views

U-statistics and Independent Sum

I have i.i.d. paired samples (X,Y): $(X_1, Y_1), (X_2, Y_2), \dots, (X_n, Y_n)$ I compute the statistics $\sum_{i \neq j} X_i \cdot Y_j$ People have told me that the above is actually a sum of $n$ ...
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0answers
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p-value of probability distribution

Suppose $\{X_{i}\}_{i=1}^{50}$ are independent and identically distributed samples from the following probability distribution: $$(1/\theta)\exp(-x/\theta); \hspace{1mm} x>0.$$ Given ...
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1answer
42 views

Find the mean and standard deviation of the sampling distribution of the restaurants sample mean expense per customer.

A restaurant charges $8.95$ pp. Management finds it's expenses per person has a distribution that is skewed to the right with a mean of $8.20$ and a standard deviation of $3.00$. Q: If $100$ ...
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1answer
58 views

Wanted: Polynomial $P(x)$ with $P(-l(l+1))=1/(2l+1)$, for $l\in \mathbb{N}$

I'm looking for a polynomial $P(x)=a_1+a_3 x+ a_5 x^2+\dots$ (numbering of $i$ in $a_i$ is due to the application of this) with sampling points $P(-l(l+1))=\frac 1{2l+1}$, for $l=1,2,3,\dots$ ...