Lucky Hans wrote:
well, I am sure that is changes a lot. In fact, I know that the expectation is k/d, where d is the dimension of the ambient space and k the rank of the orthogonal projection. I also have some deviation estimates that hold with high probability. Nevertheless, I am interested in the actual distribution and was wondering if it is possible to determine it analytically.
Without loss of generality, we can assume P is projection ontothe first k components of
. So changing to spherical polars by
Since r=1, the area element is
, now integrate the
parametrises almost every point on the unit sphere. Finally, change coordinates back.