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A point is picked randomly in space. Its three coordinates $X$, $Y$, and $Z$ are independent standard normal variables. Let $R = \sqrt{X^2+Y^2+Z^2}$ be the distance from the point from the origin. Find:

a) The density of $R^2$ (don't get how to set up the integral for this)
b) The density of $R$ (don't get part a)
c) $E(R)$
d) $\textrm{Var}(R)$

I don't get how to use the change of variables since we are dealing with $X$, $Y$ and a $Z$. Can you please explain how I can do this? Also, can it be done using spherical coordinates? I am lost on the coordinates available for us on this problem.

J. M. ain't a mathematician
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mary
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2 Answers2

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Hint: Chi-square distribution.

mpiktas
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  • That is probably why such problem was given. So when introducing chi-square distribution you would already be familiar with it. – mpiktas Apr 19 '11 at 09:20
  • can you please solve the problem? – mary Apr 19 '11 at 09:21
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    @user8917, did you bother to read the wikipedia page? Some of the proofs are given there. Do not expect people do your homework for you. – mpiktas Apr 19 '11 at 09:22
  • This happens to involve the chi-square distribution, but the chi-square distribution is just a gamma distribution. – Michael Lugo Apr 19 '11 at 14:13
  • @Lugo, not very specific in terms of what equations to use. – mary Apr 27 '11 at 04:07
  • @mpiktas Regarding the admonestation in you last comment: I agree with its principle, of course. Nevertheless I would advise to wait and see what happens on the page, don't you think... :-) – Did Aug 28 '11 at 17:27
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You can determine the density of the constituent variables simply by using a change of variables. Taking $X^{2}$ as the example:

$$ P(X^{2} \leq u) = P(-\sqrt{u} < X < \sqrt{u}) = \Phi(\sqrt{u}) - \Phi(-\sqrt{u}) = 2\Phi(\sqrt{u}) - 1 $$

You can obtain the density function of $X^{2}$ by differentiation, which will be the same as that of $Y^{2}$ and $Z^{2}$. Finally the density of the sum of two independent random variables is given by the convolution of the two density functions. Apply that formula twice to derive the density of $R^2$.

Macro
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