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11
votes
2answers
436 views

What is the distribution of gaps?

Randomly select $n$ numbers from the universe $\{1,2\dots,m\}$ with or without replacement, and sort the numbers in ascending order. We can get a list of number $\{a_1,a_2,\dots,a_n\}$, and then we ...
0
votes
1answer
154 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} = ...
4
votes
1answer
1k 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
3answers
481 views

estimate population percentage within an interval, given a small sample

Given a small sample from a normally-distributed population, how do I calculate the confidence that a specified percentage of the population is within some bounds [A,B]? To make it concrete, if I ...
1
vote
1answer
155 views

Sampling $Q$ uniformly where $Q^TQ=I$

(This is related to this question) $Q \in \mathbb{R}^{n\times k}$ is a random matrix where $k<n$ and the columns of $Q$ are orthogonal (i.e. $Q^T Q = I$). To examine $E(QQ^T)$, I conducted monte ...
4
votes
4answers
161 views

Sample: don't confuse measurements with actual values?

In Wikipedia's article on Sample there is the following remark: ''Note that a sample of random variables (i.e. a set of measurable functions) must not be confused with the realizations of these ...
1
vote
1answer
50 views

Expected number of distinct nodes visited in a directed bipartite graph

Let $G = (V,E)$ be a directed bipartite graph with $V = \{I \cup O\}$ where $\left\vert{I}\right\vert = n$ and $\left\vert{O}\right\vert = m$. All the edges start from a vertex in $I$ and end on a ...
1
vote
1answer
25 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 ...
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 ...