440 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 ...
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1 vote
178 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, $c =\sqrt{X_1^2+X_2^2+...X_k^2}$?
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125 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 ...
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32k 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 ...
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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 ...
986 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 ...
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12k 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 ...
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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 $P(x_1, x_2)$?