Question

The expected value of two random variables, E[XY] can be defined as an inner space defined by those variables as . The lecture notes I am looking at say that the norm associated with this inner space is ||Y||=^1/2. Shouldn't it be that the associated norm is ||XY||=^1/2? I thought the norm of an inner space was defined as the square root of that inner space (as in the pythagorean theorem gives the "norm" of the triangle.) Thanks.

Answer 1

XY is not an element of your inner product space, but Y is

Answer 2

Let take an easy inner product space, for example the plane
X=(a,b), Y=(c,d), then
<X,Y>= a c + bd
||Y||^2 =c^2 + d^2 =<Y,Y>

Answer 3

So that means that there is another norm associated with the same space? Specifically ||X||=<X,X?>1/2 right?

Answer 4

I guess that must be true since X and Y are symmetric. I guess I was thrown off partially by the notes saying it was "THE associated norm" in stead of "An associated norm.

Answer 5

For every scalar product space there is a norm associated with it