# Thread: Random vectors & random matrices (1)

1. ## Random vectors & random matrices (1)

1) Theorem: E(X+Y)=E(X)+E(Y)
where X and Y are random (matrices? vectors?).

The source didn't specify the nature of X and Y. Can X and Y be random matrices of any dimension (provided X+Y is defined, of course), or must X and Y be random vectors?

2) Let A be a constant matrix (i.e. all elements are fixed (nonrandom)). Can we now use the E(X+Y)=E(X)+E(Y) with Y=A to show that
E(X+A) = E(X)+E(A) = E(X)+A ?
My main question here is: can we treat A as a special case of a RANDOM matrix even though A is not random, and use the above theorem which requires both X and Y be random?
A similar question: if X constantly takes on the value 5, a lot of times we will say that X is nonrandom, but can X here be treated as a random variable?

Thanks!

note: also under discussion in Talk Stats forum

2. Yes, a constant can be regarded as a random variable having a trivial probability distribution (why not?). Don't confuse "random" with its usual meaning!

It doesn't really matter whether your $X,Y$ are matrices or vectors. In any case matrices can be regarded as special vectors. Start from the definition of the expected value. Now how do you define the expected value of a vector? How are your vector random variables constructed? Do the coordinates of your random vector all have the same probability distribution, or do they have separate probability distributions?