Linked Questions

0
votes
0answers
239 views

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 ...
0
votes
0answers
113 views

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 ...
0
votes
1answer
53 views

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, $c =\sqrt{X_1^2+X_2^2+...X_k^2}$?
31
votes
4answers
19k views

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 ...
36
votes
3answers
14k views

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) $\...
20
votes
3answers
11k views

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 ...
16
votes
3answers
2k views

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 $r$...
11
votes
3answers
11k views

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 ...
13
votes
3answers
3k views

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.
13
votes
5answers
510 views

$(2Y-1)\sqrt X\sim\mathcal N(0,1)$ 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 $Z=(2Y-1)\sqrt X$ ? ...
8
votes
3answers
659 views

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 ...
11
votes
1answer
10k views

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 ...
9
votes
1answer
7k views

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 $\...
9
votes
4answers
507 views

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 ...
7
votes
1answer
7k views

How to use Box-Muller transform to generate n-dimensional normal random variables

I'm trying to generate random variables. I read about Box-Muller transform which is a way to generate a pair of normal variables, 2-d normal distrubution. But how do I expand that transform to ...

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