Distribution of distances from the origin of randomly generated points

[Robin04]

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?

 

[marcusl]

How do you write the radial distance from the origin in terms of the 4 coordinates?

 

[Robin04]

How do you write the radial distance from the origin in terms of the 4 coordinates?

$d=\sqrt{x_1^2+x_2^2+x_3^2+x_4^2}$

 

[marcusl]

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.)

 

[Robin04]

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.

 

[marcusl]

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?

 

[Robin04]

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?

 

[marcusl]

Well, this is a starting point, it's the distribution for d^2. You want a related distribution that has the name "chi" in it.

 

[Robin04]

Well, this is a starting point, it's the distribution for d^2. You want a related distribution that has the name "chi" in it.

So, I guess this is what I'm looking for: https://en.wikipedia.org/wiki/Chi_distribution#Probability_density_function
We only learned $\chi^2$ in class

 

[marcusl]

That's it--a chi distribution.

 

[Robin04]

That's it--a chi distribution.

Thank you very much! Sorry for my clumsiness, I need to clean my head about this topic :D

 

[WWGD]

Just to repeat, the sum of squares of normals is Chi-squared -distributed.

 

[Robin04]

Oh, but where do $\sigma$ and $\mu$ come into the game?

 

[marcusl]

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.

 

[WWGD]

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.

 

[WWGD]

Oh, but where do $\sigma$ and $\mu$ come into the game?

Look at the parameters in the right side of the page: https://en.wikipedia.org/wiki/Chi_distribution#Probability_density_function

 

[marcusl]

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.