How to find the Conditional Expectation of a Discrete Random Variable or Vector or Matrix

I want to find the conditonal expctation of a discete random vriable (or vector) of the following form:

Where <a,b> = a'b, means dot product of two arguments given in the angular brackets.

For evaluation the conidtional expectation, we need to have vlues of conidtion PMF.

But in case of such random variables/vectors/matrices, when we do not have values of conditional PMF for dsicrete signals, how can we find out conditional expecation through numerical solution