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my question is related on this one that I have posed on math.stackexchange but is not exactly the same (even I would appreciate receive an answer for the other as well). Since I haven't received answers, I've been told that here could be a better place to ask about statistics.

Let imagine we have a vector $\vec{a} = (1,0,0)$ and that we have a vector $\vec{\lambda}$ uniformly distributed in the unit ball (sphere of radius $1$).

We can then write $$\vec{\lambda} = (r_1.\cos \phi.\sin \theta,r_1.\sin \phi.\sin \theta,r_1.\cos \theta)$$ and $$\vec{a}.\vec{\lambda} = r_1.\cos \phi.\sin \theta$$

But I'm not comfortable with something. If we say that $\vec{\lambda}$ has a uniform distribution in the unit ball, can we say that $$r_1 ~\sqcup_{[0,1]} \\ \phi ~\sqcup_{[0,2\pi]} \\ \theta ~\sqcup_{[0,\pi]}$$

with $\sqcup_{S}$ meaning that the distribution is uniform on the set $S$ ?

If it's the case, I could test the uniformity of $|\vec{a}.\vec{\lambda}|$ by testing this

$$\int_{r_1}^{r_2} \int_{\phi_1}^{\phi_2} \int_{\theta_1}^{\theta_2} |r.\cos \phi.\sin \theta| \,r^2\sin \theta\;dr\,d\phi\,d\theta = \frac{\int_{r_1}^{r_2} \int_{\phi_1}^{\phi_2} \int_{\theta_1}^{\theta_2} \,r^2\sin \theta\;dr\,d\phi\,d\theta}{\frac{4}{3}\pi}$$

right ?

Is it a good way to do it ? How could I do more easily this test ?

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    $\begingroup$ "Uniform on the unit ball" is not equivalent to uniformity of $\phi, \theta,$ and $r$. $\endgroup$
    – Glen_b
    Commented Aug 28, 2013 at 10:37
  • $\begingroup$ And is "Uniform on the unit sphere" equivalent to uniform on $\phi$ and $\theta$ ? $\endgroup$
    – mwoua
    Commented Aug 28, 2013 at 11:44
  • $\begingroup$ No. In both cases you can have your $\phi$ (the one on $[0,2\pi)$) to be uniform, but the other must not be or the points will not be equally dense at the poles as the equator. With $r$ it's more obvious - there's much more volume at large $r$ than small $r$, so the density over $r$ must increase to make the points equally dense at the center and near the surface. $\endgroup$
    – Glen_b
    Commented Aug 28, 2013 at 13:14
  • $\begingroup$ See here, here, here, and here $\endgroup$
    – Glen_b
    Commented Aug 28, 2013 at 13:16
  • $\begingroup$ With the appropriate density for $r$ and your $\theta$, you could do perhaps do three goodness of fit tests, one for each variable. $\endgroup$
    – Glen_b
    Commented Aug 28, 2013 at 13:29

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