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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62 views

Sampling from a given pdf

I have the following pdf: $$ f(x) = C x^d I_0\left(b \sqrt{- \log\left(\frac{x}{A}\right)}\right)$$ for $0 < x \leq A$, $C$ is a normalizing constant, $b$, $d$ are constants, and $I_0$ is the ...
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1answer
17 views

Simple Random Sample

Suppose a company has 500 employees and wishes to select a simple random sample of 25. Which would be the more appropriate way of doing so? Choose 25 pages of the employee directory, and take the ...
1
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1answer
29 views

Probability that a sample represents between X% and Y% of the population

Really no idea how to go about this. I thought about using a uniform normal distribution law but the answers I got made no sense. In a country that has a population between 1500000 and 3000000 ...
0
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0answers
32 views

How to estimate a distribution from samples in a histogram

Given a r.v. $\tau$ , I've computed $\Bbb{P}(\tau >a)=e^{-Nx}(e^{Nxe^{-a}}-1) $ , where $N\in\Bbb{N}_{>1} $ and $ x\in \Bbb{R}_{>0} $ are just fixed parameters; say $N = 2 $ and $ x = 1$, ...
1
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0answers
14 views

Arithmetic of sampling

INTRO: Suppose that a large set $Z$ is divided to two subsets $Z^A$ and $Z^B$, such that each element of $Z$ has a probability of 1/2 to be in each subset, independently of the others. By the law of ...
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2answers
28 views

What does one sample average tell me about the average of other samples?

I'm sure this has been asked before but I can't find the exact variant I'm looking for. I have an infinite population of elements. I take a sample of those, say 10000, and make a measurement for ...
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0answers
56 views

subsample order statistics

I am interested in characteristics depending of the $r$-th order statistics $X_r$ of a distribution with unknown pdf. For example, I would like to estimate the Gini mean difference $E(|X_r - X_r'|)$, ...
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1answer
31 views

How to test a collection of samples are sampled with replacement or not?

A box is full of balls with $m$ different colors, and for each color, there are $n$ balls. So the total number of balls is $m*n$. Note that $m$ is unknown, $n$ is already known, and balls can only be ...
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1answer
27 views

Finding the probability of X_Bar with sample variance included?

The question I am asked is $P(\bar{X} > 3 + 0.4984S)$, where I am additionally provided $n = 25, \mu = 3.0, \sigma^2_\text{pop} = 3.0$. $\bar{X}$ is the sample mean and $S$ is the sample variance. ...
4
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46 views

Signal processing and algebraic geometry

Signal processing is a pretty huge branch of what I would (maybe wrongly) call electrical engineering. I have heard here and there whispers of interesting connections between signal processing - in ...
0
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1answer
17 views

comparing probability of different size

I have a set of items. Each item has a probability of correctness calculated from no. of correct/no. of attempts. For example item A has 100% of being correct out of the 10 attempts he did. On the ...
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18 views

Choosing a threshold for the population by using a sample

Can someone please give me some insight about this problem I have come across. Say I have a population "N" for which I want to figure out a threshold "T" of a score(the score can be for anything for ...
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0answers
7 views

How should I set the weights on a linear calibration to make more recent values matter more?

I have a program that interpolates raw values to real values using a linear regression. The linear regression function takes three arrays, X, ...
1
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1answer
36 views

How many 'zero' elements in $\mathrm{A}+\mathrm{B}$ after randomly letting one entry in $\mathrm{A, B}$ be $0$ for $m$ times,respectively?

Both $\mathrm{A}$ and $\mathrm{B}$ are $d$-dimensional vector with all the elements being $1$. Then we randomly and uniformly pick one element from $\mathrm{A}$ with replacement, and let it be $0$. We ...
0
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1answer
31 views

Can random sampling affect view of how much repetition there is in a data set?

$X$ unique individuals take $Y$ actions. If I sample $1\%$ of the actions entirely at random what is the relationship between the number of unique individuals seen in my sample and $X?$ (In my ...
0
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0answers
32 views

Small Sample size Confidence Intervals

Health insurers and the federal government are both putting pressure on hospitals to shorten the average length of stay (LOS) of their patients. The average LOS for men in the US is 5.2 days and the ...
2
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2answers
54 views

Two-sided confidence intervals and tests

From a sample of 1751 army hospitals, estimate the mean expenses for a full time equivalent employee for all US army hospitals using a 90% confidence interval given x = ...
1
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1answer
29 views

$X$ is exponentially distributed $\varepsilon(\theta)$. Using the Method of Maximum likelihood find the best (marking?)Question its centeredness

$X$ is exponentially distributed $\varepsilon(\theta)$. Using the Method of Maximum likelihood find the best (marking?)of sample $n$ for parameter $\theta$ .Question its centeredness and existence. ...
0
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0answers
47 views

Intuition behind rejection sampling proof

I have a quick question about the proof of rejection sampling. Suppose we know how to sample from a distribution with $Y$ pdf $q$, and want to sample from a distribution $X$ with (known) pdf $\pi$. ...
0
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0answers
33 views

Uniform Sampling & CDF inverse

I have a probability exam soon, and our prof told us to study the following question: "Describe a procedure for generating independent identically distributed (i.i.d.) samples of a random variable ...
0
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1answer
57 views

Expectation and variance without replacement

Let $X_{N_1},\cdots,X_{N_n}$ is a sample without replacement from the set $\{1,2,\cdots,N\}$, and let $\bar X_n=\sum_{i=1}^n X_{N_{i}}/n$. Then, how one can find $E(\bar X_n)$, $Var(\bar X_n)$, ...
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0answers
28 views

Question about confidence intervals

I have questions regarding the calculations and computations of confidence intervals basing upon the given accuracy. Here is the description from the textbook: The public health authority wants to ...
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0answers
13 views

sample size calculation for count data

I have a plan to see some treatment effect in several projects in my company where I will compare the average number of errors now and after the treatment. So what I know from current situation is ...
0
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0answers
49 views

A Question dealing with Surveys

I did the following problem but I do not have confidence that I did it right. Could somebody please check it for me? Thanks Bob Problem: In an organization of $1000$ employees, a survey was made ...
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0answers
15 views

Sampling sequences of random vaiables without replacement in a weighted fashion

Let's we say have $n$ random variables $x_1,x_2,\dots,x_n$. Assume for any random variable $x_i$ the set of all possible outcomes is $\left \{a,b,c,d\right\} $with probabilities ...
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0answers
25 views

Cluster sampling: Compare efficiencies

A company operates from 12 branches, and the numbers of cars, $N_i$ and means $\bar{X}_i$ and variances $S_i^2$ of miles driven last year for each brand, are as follows Branch: $N_i$; $\bar{X}_i$ ; ...
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0answers
27 views

Simple Random Sampling: Find the variance

I have trouble answering this simple question. There is a total of 280 trees. The assessed total yield is at 432,6 tons. 25 trees are picked at random and their timber yields are accurately ...
0
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0answers
13 views

Finding probability given a sample from 36 with a given distribution

Let the sample mean and variance be based on a random sample of size 36 from $N(4, 121)$ distribution. Find $P(0 < X < 8, 40 < S^2 < 160)$. Since they are independent it would be ...
2
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1answer
48 views

How to find the variance of a random sample with exponential distribution?

This will seem like a very simple question to many of you; but I cannot understand part of the solution, so to give context I have had to unfortunately resort to typing the whole thing out, apologies. ...
0
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0answers
15 views

Probability of a difference dependent samples, normal distribution but unknow parameters

I have two samples, $X_0, Y_0$, containing the marks in a test before and after training course over the same workers, thus dependent paired samples. X, Y can be considered Normal, with the same ...
0
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0answers
187 views

p-value of uniformity of given distributions,Matlab

Given a vector of real numbers $[a_0,...,a_n]$, how do I find the $p$-value (in Matlab, say) that it is drawn from the uniform distribution over [0,1]? I.e. $H_0$ is the hypotheses that it is drawn ...
3
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0answers
43 views

Uniformly sampling points from inside a region of cube

Let the dimension n=200 be fixed. The problem I am interested in is sampling points in n-dimensional Euclidean space uniformly from the region $$ \sum_{i=1}^{n} x_{i}\leq 1, $$ where $0\leq ...
0
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1answer
36 views

Variance of sample mean difference

I have encountered this a variation of this problem as a test task for a job interview. It's been long since I last encountered probability theory, so I couldn't solve it. Still, it deprived me of ...
0
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1answer
20 views

How to express in an equation, a constant plus a random sample?

Suppose I have a Beta distribution, and want to express in an equation a random sample from this distribution. I could say $x \sim Beta(\alpha, \beta)$. Now, consider the case when $x$ is a constant ...
1
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1answer
28 views

Bacterial Sampling : Dependent event

I have a question regarding distribution of bacteria in wells: I have a tube that contains 200ul of 5000CFU/ml of bacterial solution. If I have 20 tubes to which I will add 10ul of bacterial sample ...
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0answers
12 views

Compute frequencies present in a sampled signal

I have a signal $$ X(t) = \sum^3_{k=1} A_k \cos (2\pi f_k t + \phi_k), -\infty < t < \infty $$ where $E[A_1^2] = 1, E[A_2^2] = 4, E[A_3^2] = 1$ and the phase functions are uniformly ...
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0answers
36 views

Importance of uniform stationary distribution

When I study Markov chain (or sampling) related papers, most of them emphasize "uniform stationary distribution". But, I can't sure why it is important for Markov chain problems or randomized ...
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0answers
62 views

Prove that $ES \leq \sigma$, with S being a random sample

Assume we have a random sample S of (X1,...,Xn) from a population with finite variance $\sigma^2$. Prove that $ES \leq \sigma$ How would you prove this? Is there some way to do this using Jensen's ...
2
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0answers
43 views

Ratio estimator in sampling

Let the population $U=(1,2,3)$. We want to estimate $R=\frac{\mu_y}{\mu_x}$.Consider the estimators $$\hat{R_1}=\frac{\overline{y}}{\overline{x}},\hat{R_2}=\frac{\overline{y}}{\mu_x}$$ where ...
0
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0answers
10 views

How come uncertainty adds up without squaring it?

I'm currently trying to sum measurement with unknown uncertainty. My plan is to make an experiment and try to measure multiple time the same sample and figure the variance of each of my measurements. ...
0
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0answers
25 views

Sampling the Sin(x)/(x) Function

Say we have $$x_c(t) = \frac{sin(\pi t)}{(\pi t)}$$ $x_c(t)$ has a maximum frequency of $\omega_M = \pi$. According to Nyquist's Theorem, this signal can be perfectly reconstructed from its samples, ...
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1answer
76 views

2 simple statistics questions regarding probability and means.

Fire alarms go off in the engineering building an average of 13 times per year. Find the probability of more than one fire alarm going off in the month of December. For this one, I am uncertain on ...
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2answers
22 views

What's the formula to get the x2 in this chi-square table?

I'm just wondering what's the formula to get the x2 answers in this table.
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0answers
10 views

Stratified sampling

In general, is there an optimal number of stratas to use when employing the Monte Carlo variance reduction technique: "Stratified sampling"?
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38 views

Generate Uniformly Random Points on a Transformed Sphere

I have a sphere transformed by an affine transformation (represented as a 4x4 matrix). How should I get uniformly distributed points on the transformed sphere's surface? Note that the obvious ...
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0answers
22 views

Statistics- Why is T being used over Z here?

"In their 1992 study of human internal body temperature, Mackowiak, Wasserman and Levine, their sample mean, x = 98.25 with standard deviation = 0.73, led them to a hypothesis that human internal body ...
0
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1answer
88 views

What does the conclusion with confidence interval mean, non-statistically speaking?

There is a manufacturer who told me that they tested devices in the batch based on 95% confidence level, 5% confidence interval/margin of error and 50% distribution. So from 1500 devices, they tested ...
0
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1answer
22 views

Asymptotically unbiasedness of an weighted estimator

Consider a Markov chain on a state space V with size N, and let $\pi(v_j) = \sum_{v_i \in V} \pi (v_i)P(v_i,v_j)$ be the stationary distribution, where $P(v_i,v_j)$ is the transition probability. ...
0
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0answers
55 views

Sampling from a probability distribution

We are given the probability density functions for the random variables $Y_i$ ($i=1 \ldots N$): $$ f_i(t) = \begin{cases} 0 & t \le 0\\ \frac 1t \cdot (2\pi\Sigma_{ii})^{-1/2} ...
0
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0answers
49 views

General equation for sampling without replacement probability

Looking at a preparatory exam, I'm a little dumbfounded by a question on probability. There are $19$ balls in a box: $5$ red, $3$ white, and $11$ blue. The question is: what is the probability of ...