Linked Questions
36 questions linked to/from How to generate uniformly distributed points on the surface of the 3-d unit sphere?
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How does one choose a random isotropic direction and then have the vector have norm 1? [duplicate]
I want to choose a random vector in high dimensions such that it all directions have the same uniform chance (i.e. isotropic in all directions). My current idea is the following algorithm:
sample v ...
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Generate Uniform Random Variates with Constant Norm [duplicate]
How can one generate $k$ uniform random variates centered at zero, $X_1, X_2, ..., X_k$, given a constant Euclidean norm, $c =\sqrt{X_1^2+X_2^2+...X_k^2}$?
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Why are IID normal random variables spherically symmetrical? [duplicate]
Given a finite sequence of $s+1$ IID normal random variables $X_1, \ldots, X_{s+1}$ They are spherically symmetrical.
This means that the radial projection of the point $(X_1, \ldots, X_{s+1}) $ onto ...
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How to create an arbitrary covariance matrix
For example, in R, the MASS::mvrnorm() function is useful for generating data to demonstrate various things in statistics. It ...
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Distribution of scalar products of two random unit vectors in $D$ dimensions
If $\mathbf{x}$ and $\mathbf{y}$ are two independent random unit vectors in $\mathbb{R}^D$ (uniformly distributed on a unit sphere), what is the distribution of their scalar product (dot product) $\...
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Simulating draws from a Uniform Distribution using draws from a Normal Distribution
I recently purchased a data science interview resource in which one of the probability questions was as follows:
Given draws from a normal distribution with known parameters, how can you simulate ...
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Generating random points uniformly on a disk [duplicate]
I have to randomly generate 1000 points over a unit disk such that are uniformly distributed on this disk. Now, for that, I select a radius $r$ and angular orientation $\alpha$ such that the radius $r$...
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How to generate uniformly distributed points in the 3-d unit ball?
I have posted a previous question, this is related but I think it is better to start another thread. This time, I am wondering how to generate uniformly distributed points inside the 3-d unit sphere ...
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How to test uniformity in several dimensions?
Testing for uniformity is something common, however I wonder what are the methods to do it for a multidimensional cloud of points.
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$(2Y-1)\sqrt X\sim\mathcal N(0,1)$ when $X\sim\chi^2_{n-1}$ and $Y\sim\text{Beta}\left(\frac{n}{2}-1,\frac{n}{2}-1\right)$ independently
$X$ and $Y$ are independently distributed random variables where $X\sim\chi^2_{(n-1)}$ and $Y\sim\text{Beta}\left(\frac{n}{2}-1,\frac{n}{2}-1\right)$. What is the distribution of $Z=(2Y-1)\sqrt X$ ?
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Covariance matrix of uniform spherical distribution
I need to figure out the covariance matrix of a uniform spherical distribution. But there I can't even find a closed form of the distribution. This link says it is $\frac{1}{n}\mathbf{I}$, where $\...
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Creating random points in the surface of a n-dimensional sphere
I have a point X in the surface of an n-dimensional sphere with center 0.
I want to create random points following a distribution with center X, the points must be in the surface of the n-dimensional ...
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Distribution of $XY$ if $X \sim$ Beta$(1,K-1)$ and $Y \sim$ chi-squared with $2K$ degrees
Suppose that $X$ has the beta distribution Beta$(1,K-1)$ and $Y$ follows a chi-squared with $2K$ degrees. In addition, we assume that $X$ and $Y$ are independent.
What is the distribution of the ...
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How can I generate uniformly distributed points on a circle?
I am looking to generate 450 data points in R. There are three distinct sets 150 of each distributed in a circular band with different radii (at 1, 2.8 and 5).
In particular, I'm looking to reproduce ...
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Marginal distribution of uniform distribution over sphere
Let $(x_1,…,x_n)$ be a random vector uniformly distributed on the $n$-dimensional unit sphere.
Is there a closed form solution for the joint distribution of $P(x_1, x_2)$?
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