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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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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19 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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9 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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21 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 ...
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50 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 ...
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32 views

Formula to count sample size from given population

I am trying to find out the formula for calculation required sample size from a given population when I know the confidence interval (5%) and confidence level (95%). I found online calculators of ...
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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. ...
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53 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} ...
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46 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 ...
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13 views

bradley-terry generating synthetic data

Im trying to generate synthetic data by Bradley-Terry model perspective. I think I need to steps first generate skill parameters and then generate the matrix showing number of winning of ith player ...
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44 views

Rejection Sampling From Conditional Distribution

I am familiar with rejection sampling in the univariate case, where we have a proposal $h(x)$ (which we can sample from) for the target density $p(x)$ such that $p(x)<Mh(x)$ at all $x$. We sample ...
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9 views

Data filtering VS Sampling Process

I am writing on a write up of how i obtained my sample from a large population. But I am contemplating whether I should categorize them under 'data filtering' or a 'sampling process'. Any idea what ...
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52 views

Integrating an infinite series of the Dirac function

I am given the following sampling signal function, where $\delta$ is the Dirac delta function, t is time, and Ts is the sampling period. First, I am asked to plot the signal. I do not understand ...
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46 views

Question about the expected number of sampling until one item out of k is found

) I have k items (let's say balls) that all have a different color, but I do not know the color until I pick them. I would like to know how many balls I have to pick until I get one with a specific ...
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21 views

Maximum Sample value for a given Sample Size and Distribution

Suppose a sample size of 100 samples are drawn randomly from an Exponential distribution. For a given mean of the exponential distribution (say $\mu$), what could be the maximum possible sample value ...
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60 views

right way of weighting sampling for a distribution?

Recently I have read a paper, in a part of that the authors try to sample from the distribution $p(\alpha|\tau )=P(\tau | \alpha)\times p(\alpha | \alpha_0)$ where $\tau$ is some evidences in the ...
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20 views

Express covariance function of a sampled process as a fourier transform (help with proof)

I'm wrestling with this theorem in my 'stationary stochastic processes' course.It's about sampling of a continous process and a way of rewriting the covariance function of the sample(I'm not entirely ...
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23 views

Sample size and confidence interval

We want to produce a $0.90$ confidence interval for the proportion of vegetarian recipes at one cookbook. We will use simple random sampling without replacement to select a sample of $2311$ ...
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8 views

Include or exclude sample from population?

i have extracted a sample group of funds from a population (i.e. fund universe) through some given criteria,however the question is: when comparing the performance of the sample group of funds with ...
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1answer
81 views

How to calculate error margin on measured number of occurd events when sampling

We are measuring the number of times a certain event happens. We do this with the help of sampling, so that we only report events with a probability p. For example p=0.01 would result in about 1/100 ...
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1answer
14 views

Matching: Calculated data vs. sample from the real world

I have a small set of measured values from the real world. It is a rather small sample (10 from 1000). Actually there is no way for me to increase the sample, it is just possible to use 10 values ...
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77 views

Generation of random variable from a complicated CDF

Suppose I am given a CDF of a distribution, given by $F(x) ∝ \int_0^1 x^y e^{-y} dy.$ Here,'x' ranges from 0 to 1. How do I generate a random variable from this distribution?
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26 views

Goodness-of-Fit tests for Multinomial and Binomial Data

A box has 4000 red, 5000 blue and 1000 orange balls. A selection of 70 balls is made, with 25 reds, 35 blues, and 10 oranges being observed. Can one essentially prove that the selection was NOT a ...
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48 views

What's the difference between MCMC and particle MCMC?

Markov chain Monte Carlo (MCMC) methods are a class of algorithms for sampling from a probability distribution based on constructing a Markov chain that has the desired distribution as its equilibrium ...
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39 views

Generating Random Variates from CDF

Suppose I am given a CDF of a distribution, given by $F(x) \propto x + x^2 + x^4 + x^7$. How do I generate a random variable from this distribution?
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27 views

Poisson sampling

Suppose I have a pdf $f(S)$. $f(S)$ describes the size of firms in the economy. Also define the Bernoulli variable $X_{f} \in \{0,1\}$ where $P(X_{f}=1)=g(S_{f})$ and $P(X_{f}=0)=1-g(S_{f})$. $S_{f}$ ...
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95 views

Selecting k distinct numbers from an array with increasing probability distribution

I have to select k distinct numbers from an array such that probability of a number getting selected is more if it is at the end of the array (probability increases linearly). I'm thinking of ...
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112 views

Finding Probability of picking one ball out of N balls.

presented with n identical balls, one with a prize in it. Picks each ball out idependently one at a time till gets prize. I need to find the mean and variance of the number of balls needed to pick ...
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26 views

Biased sample from biased sample

A webpage has users, where each user has a number of projects uniquely assigned to him or her. I want a random sample of users by randomly sampling projects and then taking the users connected to this ...
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53 views

95% Confidence Interval Problem for a random sample

The sample mean of a random sample of $25$ observations is $9.6$ and the sample variance is $22.4$. Derive a $95$ confidence interval for the population mean. I calculated the following: Confidence ...
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2answers
46 views

Resampling operation

I am reading from an arXiv.org paper the following math text: "Let $x\in \{−1, 1\}^I$ be random and uniform, and let $y$ be obtained from $x$ by resampling each coordinate with probability ...
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30 views

Can we use a single sample from a population to create confidence intervals, do hypothesis testing, etc.?

When conducting confidence intervals, hypothesis testing, and ANOVAs are we using the sampling distribution with multiple samples as opposed to a single sample? Are there cases where we use just a ...
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35 views

Distribution or samples of a function of a random variable

OK I edited the question: I have the following setup: Stereo camera setup with two images I, I'. 4 1-dimensional random variables (each corresponding to the inverse depth value of a pixel on an ...
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Intuition of the Hessian of the Log Barrier Function

I have a convex polytope defined by $\mathbf{Ax \leq b}$ (row-wise) The log-barrier function is defined as: $$\phi(x) =-\sum{\log(b_i - a_ix)}$$ The Hessian of the log-barrier is : ...
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31 views

Sampling with independent probabilities

I'm looking for one specific sampling method that decides about inclusion probability of each item regardless of existence of other elements. As an example given 0.5 as the inclusion probability, it ...
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38 views

How does sampling affect the distribution of frequencies of individual types?

Consider a population of size N in which individuals can be of x different types. Take a sample (with replacement) of size ...
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52 views

random sample after adding extra elements

I have N sensor measurements (N=5000000) and a random sample of size s (s=20) from this set of data. For each measurement is computed a rank as being the minimum distance to the sample values. So ...
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29 views

Sampling without replacement from unknown sample size

Five mice are chosen (without replacement) from a litter, three of which are tagged $A$, $B$ and $C$. The probability that all three tagged mice are chosen is twice the probability that $A$ is the ...
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25 views

sampling distributions and test of hypothesis

A manufacturer of a certain type of breakfast cereal claims to produce packets which contain on average 500 grams of cereals. Ten packets were selected at random and the cereals content of each ...
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16 views

Statistics Sampling Type

My question is on Q7. I can't seem to figure this one out. I thought it was a random statified cluster sample because it is breaking down the schools into subsections and then pulling 3 homerooms ...
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33 views

What is the importance of the inclusive range in Reservoir with Random Sort?

I am reading the Reservoir with Random Sort page on Wikipedia, and the algorithm says (copied): ...
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145 views

Analytic Center of Convex Polytope

I have a convex polytope defined by $Ax \leq b$. I want to know how to find the "analytic center" of my convex polytope, because my goal is to sample from the polytope using Monte-Carlo Markov ...
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69 views

Is the Dikin Ellipsoid actually a ball?

I have the inequality (row wise): $Ax \leq b$ The Dikin ellipsoid centered at $x_0$ with radius $r$ is: $$\{z \quad | \quad (z-x_0)^T(z-x_0) \leq \frac{r^2}{H(x_0)}\}$$ where, $$H(x_0) = \sum ...
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13 views

Random sampling-level of significance

Random samples of house selling prices are obtained from the north and south regions of a country. The results are summarized below: ...
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41 views

Sampling distribution of $Y = \frac{\ln U_1}{\ln U_1 + \ln (1 - U_2)}$, where $U_i \sim U(0,1), \forall i$

For this problem I have used the fact, $-2 \ln U \sim \chi^2_{(2)}$. But I have doubt on the independence of numerator and the denominator which are $\ln U_1$ and $\ln U_1 + \ln (1 - U_2)$. If they ...
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37 views

How to sample from a convex hull?

Let us consider a couple of points $x^{(i)}\in \mathbb{R}^m$ where $i=1,\dots,n$. Convex hull is defined as $$ C = \left\{\sum_{i=1}^{m} \alpha_i x^{(i)} \mathrel{\Bigg|} (\forall i: \alpha_i\ge ...
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35 views

Error propagation with dependent errors

I have a function $f(x_1,\ldots,x_n)$ where the variables $x_k$ have errors $\delta_k$. If these errors are independent I can add them root mean square: $\delta ...
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1answer
32 views

Comparing percentages of a sample to that of the population.

This might be stupid question, but I'm in this sort of situation: 60% of people in a city have a pet cat, but the national rate is 50%. So, assuming we have the required bits of information about ...
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33 views

How to describe a frequency spectrum with its samples?

I'm not sure if the following would be more physics-related, but since statistics are involved, I thought I'd post this here... To me the question is pretty straightforward, but nevertheless I have ...