There are many question about Euler angles on our sister site, math.stackexchange.com, see e.g. math.stackexchange.com/search?q=Roll-Pitch-Yaw, but I couldn't quickly find one that answers your question. So I do hope you get a good answer here, but it might be worthwhile to take a look over there also.

@RhastaShaman please note that Iosif and I are not the same person; do the edits Iosif made to the original answer answer your questions?

If you read the answer carefully you see that there is only one place where the ball itself (or its center) is used and that is in the definition of $p$. The number $p$ is the probability of any single point drawn from distribution $D$ to land in this ball. For balls at different location you might get a different actual value of $p$, but if you still call this the abstract letter $p$ (and write $q = 1 - p$) all of the answer stays the same.

Welcome to MO. In order to get better responses it would help to add some context: where did you encounter the problem, what have you tried already yourself, and where did you get stuck?

@Adrien you should post this as an answer. Now it is hidden behind the 'show 11 more comments' button

I just wanted to say I already really like the beginners guide you wrote! Please let us know if there will be an improved version

But not all strange normed algebras we can write down do appear as subalgebras of all bigger ones

Yes with that definition these are indeed normed algebras of stange dimension.

Over $\mathbb{R}$ the 'ordinary' octonions are waaaaaaay more famous than their split counterpart, due to being a division algebra. However when people started constructing these algebras over general fields the split ones more or less got their revenge as they exist over every field while division algebras do not.

The split quaternion algebra is isomorphic to the algebra of two by two matrices where the analogue of the square of the norm is the determinant. Inside the two-by-two matrices we have the three dimensional (non-commutative) subalgebra of upper triangular matrices. If we realize the split octonions as the Dickson double of the split quaternion algebra then the Dickson double of the 3D subalg of upper triangular matrices sits inside it as a (nonassociative) 6-dimensional subalgebra

Only in situations where the 'norm' is zero for certain non-zero elements, so i guess in these situation you won't call it a norm. We still have a quadratic form though, that in the division algebra case over $\mathbb{R}$ is the square of the norm.

What is $H$? Any function of the data or something more specific? Also what do we know about the distribution of $(x, y)$?