Distribution of distances from the origin of randomly generated points

In summary: Start with indepent ID Normals. The sum of their squares is Chi-square-distributed. The square root in previous step is Chi-distributed.In summary, the radial distance from the origin in terms of the 4 coordinates is d = \sqrt{x_1^2+x_2^2+x_3^2+x_4^2}.
  • #1
Robin04
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Homework Statement
We randomly generate points in 4 dimensional Euclidean space. The expecte value ##\mu## of the coordinates is 0 and the standard deviation is ##\sigma = 2.5##. Their distribution is normal.
What's the distribution of the distance of these points from the origin?
Relevant Equations
Density of the normal distribution: ##\rho (x)=\frac{1}{\sqrt{2 \pi \sigma}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}##
I'm not really sure how to do this. Maybe somehow I should transform the density function. Can you give me a hint?
 
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  • #2
How do you write the radial distance from the origin in terms of the 4 coordinates?
 
  • #3
marcusl said:
How do you write the radial distance from the origin in terms of the 4 coordinates?
##d=\sqrt{x_1^2+x_2^2+x_3^2+x_4^2}##
 
  • #4
Right. You know that each variable is IID random with the same normal distribution. What distribution applies? (You may have seen the more usual case for 2 dimensions.)
 
  • #5
marcusl said:
Right. You know that each variable is IID random with the same normal distribution. What distribution applies? (You may have seen the more usual case for 2 dimensions.)
I'm not sure what you mean by that. I think I haven't seen the problem for 2 dims.
 
  • #6
Hmm, we at PF are supposed to guide you to the answer without giving it outright, but I'm not sure what to do here...
The distribution you're looking for is related to chi-squared. Have you seen that?
 
  • #7
marcusl said:
Hmm, we at PF are supposed to guide you to the answer without giving it outright, but I'm not sure what to do here...
The distribution you're looking for is related to chi-squared. Have you seen that?
Oh yes, we learned an equation for that: ##\rho_n (x) = \frac{1}{\Gamma (n/2) 2^{n/2}}x^{\frac{n}{2}-1}e^{-\frac{x}{2}}##
In my case n would be 4. And I also have to transform this as I need the square root, right?
 
  • #8
Well, this is a starting point, it's the distribution for [itex]d^2[/itex]. You want a related distribution that has the name "chi" in it.
 
  • #10
That's it--a chi distribution.
 
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  • #11
marcusl said:
That's it--a chi distribution.
Thank you very much! Sorry for my clumsiness, I need to clean my head about this topic :D
 
  • #12
Just to repeat, the sum of squares of normals is Chi-squared -distributed.
 
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  • #13
Oh, but where do ##\sigma## and ##\mu## come into the game?
 
  • #14
WWGD said:
Just to repeat, the sum of squares of normals is Chi-squared -distributed.
But the problem is looking for the square-root of the sum of squares, which has a Chi distribution.
 
  • #15
marcusl said:
But the problem is looking for the square-root of the sum of squares, which has a Chi distribution.
You're right, I should have completed it. I was trying to do it step-by-step, but I did not finish--my bad:
Start with indepent ID Normals. The sum of their squares is Chi-square-distributed. The square root in previous step is Chi-distributed.
 
  • #17
Robin04 said:
Oh, but where do ##\sigma## and ##\mu## come into the game?
Take a look at the first equation in the Wikipedia article and you'll see that the pdf is for normalized, zero-mean random variables (just as is the pdf you quoted for Chi-squared in post #7). You need to "unnormalize" the variables and you'll find sigma shows up in the pdf expression.
 
Last edited:

1. What is the significance of the distribution of distances from the origin of randomly generated points?

The distribution of distances from the origin of randomly generated points is important in understanding the randomness and uniformity of the points. It can also provide insights into the underlying pattern or structure of the data.

2. How is the distribution of distances from the origin of randomly generated points calculated?

The distribution of distances from the origin of randomly generated points is typically calculated by measuring the distance of each point from the origin and then plotting these distances on a graph. This creates a distribution curve that shows the frequency of points at different distances from the origin.

3. What does a normal distribution of distances from the origin indicate?

A normal distribution of distances from the origin indicates that the randomly generated points are evenly spread out around the origin, with a majority of points falling within a certain range of distances. This suggests a uniform distribution and no clear pattern or structure in the data.

4. How does the distribution of distances from the origin change with different sample sizes?

The distribution of distances from the origin can change with different sample sizes. With a larger sample size, the distribution may become more uniform and evenly spread out, while a smaller sample size may result in a more skewed distribution with fewer points at certain distances from the origin.

5. Can the distribution of distances from the origin be used to make predictions about future data points?

The distribution of distances from the origin can provide insights into the randomness and uniformity of the data, but it cannot be used to make predictions about future data points. The distribution only shows the pattern of the current data and cannot account for any potential changes or patterns in future data points.

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