Question

From my understanding, the expectation of vector (let’s say nx1) is equivalent to finding the mean. However if we have two vectors x and y, both of which are (nx1), what does it mean to try to find the expectation of the product of these vectors?

e.g:

E[x * y] = ?

Here are we taking the inner product or the outer product? If I was using Matlab, would I be doing:

E[x' * y]

or

E[x * y']

or

E[x .* y]

I’m not really understanding the intuition behind expectation as applied to the product of vectors (my background is not in mathematics), so if someone could shed light on this for me I would really appreciate it. Thanks!

== EDIT ==

You’re right, I wasn’t clear. I came across the definition of the covariance where the formula given was:

Cov[X; Y] = E[X * Y] - E[X] * E[Y]

And the part where E[X * Y] came up is what confused me. I should have put this up on a math site, and will next time. Thanks for the help.

Answer 1

Following on from @woodchips ‘s answer – when it does make sense to multiply two random variables and find the expectation of the product, in the discrete case it depends on whether you have the values for X and Y that correspond with each other i.e. if for each event you have an x and a y. In that case to find the expectation of the product, you simply multiply each pair of x and y you have and find the mean. If they’re independent and you just have two vectors of samples and there is no co-occurrence, the expectation of the product is simply the product of their individual expectations.