Angular probability distribution between 3D vectors? The angle $\theta$ between two 3D-vectors with a uniform random orientation in space is distributed according to $sin(\theta)$. In Fig. 1 I have simulated random points on a sphere (like in How to find a random axis or unit vector in 3D?) and calculated the angle between the vectors to the points and the z-axis:
Fig 1: http://imgur.com/a/HDi13
NOW MY QUESTION:
I am looking for a general expression if the vectors are not uniformly oriented in space anymore but rather look all in one direction with just small, gaussian distributed deviations in the angles (e.g. like molecules in a crystal lattice). I have the feeling that it should look something like $$ p_n(\theta)\cdot p_s(\theta), \text{ where } p_n \sim \mathcal{N}(\mu, \sigma), p_s \sim sin(\cdot)$$ (see simulation in Fig. 2):
Fig 2: http://imgur.com/a/r79oe
Can that be true and how can I argue that from a probability theory (and also intuitive) point of view?
The expression $ p(\theta) = p_s(\theta)\cdot p_n(\theta) $ looks somewhat strange to me as I am multiplying two distributions of the same variable.
 A: In case anyone is once interested in this topic, I think I was now able to answer my question. I had especially problems with finding the right description for a "random orientation" when asking this question. The clue idea was that we can normalize, transform and turn two vectors in space, such that both vectors start from the origin and one of the two vectors is aligned with the z-axis, without affecting the probability distribution of the angle between the two vectors. The "random-orientation" of the second vector is then a point density on the unit sphere and the angle between the two vectors coincides with the polar angle of the  spherical coordinate system. 
If we now assume the two independent random angles $\Theta'$ and $\Phi'$, where $\Theta'$ is distributed in the yz-plane according to $f_{\Theta'}(\theta')$ and $\Phi'$ is distributed in the xy-plane according to $f_{\Phi'}(\phi')$, we find the probability of a point on the unit sphere contained in a differential area according to
$$f(\theta', \phi') sin(\theta') d\theta' d\phi' = f_{\Theta'}(\theta') f_{\Phi'}(\phi') sin(\theta') d\theta' d\phi',$$
where $sin(\theta') d\theta' d\phi'$ is the surface area element of the spherical coordinates. Therefore we can get the probability distribution of $\theta$ from
$$f_{\Theta}(\theta) d\theta = \int_{\phi} f_{\Theta'}(\theta) f_{\Phi'}(\phi') sin(\theta) d\theta' d\phi' = f_{\Theta'}(\theta) sin(\theta) d\theta$$ as $$f_{\Theta}(\theta) = f_{\Theta'}(\theta) sin(\theta).$$
In particular we can asume $f_{\Theta'}(\theta')$ to approximately be Gaussian in the interval $[0,\pi]$.
Please correct me if this is wrong..
