# 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?
Homework 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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#### marcusl

Science Advisor
Gold Member
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

Science Advisor
Gold Member
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

Science Advisor
Gold Member
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 learnt 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

Science Advisor
Gold Member
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.

#### marcusl

Science Advisor
Gold Member
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

Science Advisor
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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

Science Advisor
Gold Member
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

Science Advisor
Gold Member
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.

#### marcusl

Science Advisor
Gold Member
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.

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