Random vectors & random matrices

kingwinner

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!

statdad

1) Yes, the quantities can be random vectors or random vectors.

2) Yes, one of the two vectors (matrices) can be a constant vector (matrix). The notation seems different, but this is the same basic principle used when working with random variables. If you want to think of a scalar, vector, or matrix, that is a constant in terms of probability, think of it as one that has all probability concentrated on that constant.

kingwinner

E(X+Y)=E(X)+E(Y)
If X and Y are random matrices, does the above property still hold?

Thank you!

EnumaElish

Yes, it does.