4
votes
1answer
49 views

Uniform sampling with replacement item frequency

Suppose we are sampling from $N$ distinct items uniformly with replacement $M$ times. What can be said about the distribution of frequencies of items drawn? For example, if I sort all the frequencies ...
1
vote
2answers
14 views

How to pick a random sample of given size from a set of unknown size

I have the following problem: I'm reading huge amounts of data records; when done, I would like to display one or more randomly selected records from the data set. It's easy to do if I can cache the ...
1
vote
0answers
29 views

How many data points from a sample are in the region $(X>F^{-1}_X (p),Y>F^{-1}_Y (q))$? [closed]

Assume that, there are two variables $X$ and $Y$, and there is is a sample consisting of 1000 observations $(x_i,y_i)$ , $i=1,2,\ldots,1000$, $p$ and $q$ are the quantiles of $X$ and $Y$ respectively. ...
1
vote
1answer
30 views

Computing P-value

In a book, from a sample they derived Mantel-Haenszel chi-square statistic $$\chi_1^2=1.41$$ And it is written that : this $\chi_1^2=1.41$ is associated with a one-sided P-value between $0.10$ and ...
2
votes
4answers
129 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 $$\mathbb E\left[\frac{X}{Y}\right]={\frac{\mathbb E[X]}{\mathbb E[Y]}}$$ ??
0
votes
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
vote
1answer
45 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 ...
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 ...
4
votes
2answers
95 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 ...
0
votes
1answer
68 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
39 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
votes
0answers
18 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 ...
0
votes
1answer
23 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 ...
0
votes
1answer
67 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 ...
2
votes
1answer
93 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) ...
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 ...
2
votes
0answers
25 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 ...
0
votes
1answer
40 views

the random heights of north american women

The heights of North American women are normally distributed with a mean of 64 inches and a standard deviation of 2 inches. A random sample of four women is selected. What is the probability that the ...
0
votes
0answers
29 views

Derivation of F distribution

Prove that the PDF of Snecdor's F distribution, given by: $$F=\frac{U/n_1}{V/n_2}$$ Where $U=\chi^2(n_1)$ and $V=\chi^2(n_2)$, is given by: ...
4
votes
2answers
206 views

Monte Carlo estimator of the number of 1's in a very long binary sequence

Preface The question below is related to a problem I am working on, which requires counting the number of times a logic-valued function evaluates "TRUE" given an input value. The size of my input set ...
0
votes
0answers
13 views

Indepedence of sample means of two orthogonal Gaussian vectors?

Suppose $\boldsymbol{x}_{1}$ and $\boldsymbol{x}_{2}$ are Gaussian vectors with each distinct but arbitrary means and covariances, i.e., the elements of each vector are generally intra-correlated. ...
1
vote
1answer
68 views

The mean and variance of the sample median

The population and the median of a sample sized $2k+1$ should have the same mean and variance. Why is that? Will the result still be so tidy for a sample sized $2k$?
0
votes
1answer
49 views

Sampling without replacement

I've recently started studying statistics again, and I've just come across sampling without replacement. My book states that if we have a elements of type I and b elements of type II, then the ...
1
vote
1answer
73 views

Uniformly distributed points over the surface of the standard simplex

I would like to generate points that are uniformly distributed over the SURFACE of a standard $k$-simplex ($k$ dimensions, $k+1$ vertices). One way to efficiently generate points that are uniformly ...
0
votes
0answers
47 views

Minimum sample size to obtain population mean?

Knowing the average of a population, how could I evaluate an expected sample average based on k_sample? Data Info: Variance and standard deviation: $\sigma^2= 0.18500975256337462$ ...
0
votes
0answers
38 views

Bootstrap sampling (i.e. sample N with replacement) - distribution of histogram

In bootstrap sampling, we have $N$ items and we perform random sampling with replacement $N$ times. The resulting sample could be summarised by a histogram illustrating the number of items which were ...
2
votes
0answers
107 views

Central Limit Theorem Clarification [duplicate]

The Central Limit Theorem states that the sampling distribution of the sample mean: Converges in distribution to a normal distribution. Has an expected value (mean of the sampling distribution of ...
0
votes
0answers
23 views

Taylor Series Expansion for Sampling Theory

There is a question which I am not particularly comfortable with doing. It is related to the Taylor Expansion for Random Variables: Let $\overline{y}_S$ and $\overline{x}_S$ be sample means, $m = ...
1
vote
0answers
30 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
vote
1answer
58 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 $p_1$, $p_2$, ...
1
vote
1answer
229 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
vote
0answers
21 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 ...
1
vote
1answer
71 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 ...
2
votes
1answer
222 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
62 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 ...
0
votes
2answers
80 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
vote
0answers
150 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 ...
1
vote
1answer
580 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
votes
0answers
67 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 ...
3
votes
0answers
111 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 ...
0
votes
0answers
229 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 ...
1
vote
1answer
43 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 ...
1
vote
2answers
909 views

Sampling problem with and without replacement

Find the probability of the event that number $1$ and number $7$ were chosen first and third, respectively at the experiment of choosing five numbers from this set of numbers $\{1,2,3,4,5,6,7,8,9\}$ ...
4
votes
1answer
946 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 ...
2
votes
1answer
478 views

unbiased estimator of sample variance using two samples

I have a couple questions, I'm hoping someone can help! Let $X_1...X_n$ is a random i.i.d. sample from a $N(\mu,\sigma^2)$ distribution, and $Y_1...Y_m$ is a random i.i.d. sample from a ...
1
vote
0answers
92 views

Is it possible to calculate/impute the sample size “N” from a given mean and standard deviation?

Can anyone provide with some advice? - thank you I am required to calculate the sample size for two groups, given the following data: Total sample N (group1+group2)=583 group 1: mean=8.35, SD1.07, ...
1
vote
0answers
69 views

Estimating the number of observations from a set of samples

I repeatedly measure a value $S_n$ which is the sum of a set of $n$ hidden inputs. The goal is to identify the number of hidden inputs. All of the hidden inputs are driven by an experimenter ...
1
vote
0answers
40 views

Uniformly sample points over a circular patch of a sphere without rejection [duplicate]

Possible Duplicate: Generate a random direction within a cone A point on a unit sphere $(x,y,z)$ and an maximal angular separation $\theta$ defines a patch with an area of $\Omega = 2 \pi ...
2
votes
0answers
100 views

How to sample from a product-of-sums distribution?

$A$ is a $M$x$N$ matrix whose entries are positive. $x$ is a $N$ dimensional binary (i.e. consisting of $0$s and $1$s) vector and the number of $1$s in $x$ is constant. Let $y = Ax$. The distribution ...