How can I calculate the Expectation for a particular known vector?
First off, I would read Examples of Independent and uncorrelated data in real-life, and ways to measure/detect them.
It will likely prove useful to you.
If you are looking for something quick, easy, and useful, I would simply cross-correlate the two vectors and normalize the result.  A small result (less that 0.05?) indicates, but does not prove, independence.  A large result (more than 0.2?) would indicate some amount of dependence.  If the vectors have a non-zero mean, you may need to subtract out the mean before doing the correlation, depending on why they are non-zero mean.
You could also go a more academic route and use the method pichenettes outlines (estimate the probability distribution using a kernel density estimator) in the thread that I linked to at the beginning of the post.