Random Unit Vector From a uniform Distribution

In summary, the conversation discusses the problem of randomly choosing a unit n-dimensional vector from a uniform distribution. It is mentioned that this can be solved by switching to generalized spherical coordinates or by generating a uniform distribution from a normal distribution of the vector's coordinates and dividing by the norm. The conversation also brings up the question of why the covariance matrix of a multivariate unit normal follows a spherical probability distribution and how it can be used to obtain expectation values.
  • #1
emob2p
56
1
Hi,

I have encountered the following problem in my research. As I do not have a strong background in probability theory, I was wondering if anyone here could help me through the following.

I would like to know how one makes rigorous the problem of randomly choosing a unit n-dimensional vector from a uniform distribution.

This is like choosing an point on the n-sphere in which the problem can be solved by switching to generalized spherical coordinates. However, I have read that one can also generate a uniform distribution from a normal distribution of the vector's coorindates, and then dividing by the norm. It is not clear to me why this method produces a uniform distribution.

Thanks Much,
Eric
 
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  • #2
The covariance matrix of a multivariate unit normal expressed in cartesian coordinates is a constant times the identity matrix. In other words, the multivariate unit normal has a spherical probability distribution. The direction is uniformly distributed over the unit n-sphere.
 
  • #3
D H said:
In other words, the multivariate unit normal has a spherical probability distribution.

Why does this follow? And if so, how does one use the covariance matrix to obtain a probability density that can be integrated to find expectation values?
 

FAQ: Random Unit Vector From a uniform Distribution

1. What is a random unit vector from a uniform distribution?

A random unit vector from a uniform distribution is a vector that has a length of 1 and is selected randomly from a set of possible unit vectors. The selection is done using a uniform distribution, meaning that each unit vector has an equal chance of being selected.

2. Why is a random unit vector from a uniform distribution important in scientific research?

Random unit vectors from a uniform distribution are important in scientific research because they can be used to simulate random directionality in various experiments and simulations. This can help researchers to better understand and model natural phenomena that exhibit random directionality, such as the movement of particles in a fluid or the orientation of magnetic fields.

3. How is a random unit vector from a uniform distribution generated?

A random unit vector from a uniform distribution is generated by first selecting a random point on the surface of a unit sphere using a uniform distribution. This point is then used as the direction for the unit vector, which is created by scaling the point to have a length of 1.

4. Can a random unit vector from a uniform distribution be used in any dimension?

Yes, a random unit vector from a uniform distribution can be used in any dimension. The process of generating a random unit vector from a uniform distribution is the same regardless of the dimension, as long as the vector has a length of 1.

5. How is a random unit vector from a uniform distribution different from a random vector?

A random unit vector from a uniform distribution is different from a random vector in that it has a fixed length of 1, while a random vector can have any length. Additionally, a random unit vector is selected from a set of possible unit vectors using a uniform distribution, while a random vector can be selected from a larger set of possible vectors using a different distribution.

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