Inner product space
Random variables

For real random variables X {\displaystyle X} and Y, {\displaystyle Y,} the expected value of their product ⟨ X, Y ⟩ = E {\displaystyle \langle X,Y\rangle =\mathbb {E} } is an inner product. In this case, ⟨ X, X ⟩ = 0 {\displaystyle \langle X,X\rangle =0} if and only if P = 1 {\displaystyle \mathbb {P} =1} (that is, X = 0 {\displaystyle X=0} almost surely), where P {\displaystyle \mathbb {P} } denotes the probability of the event. This definition of expectation as inner product can be extended to random vectors as well.
language:response_language[Latin], length_constraints:number_paragraphs[4]