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 ?
