First I would like to clear up some misunderstanding I have about independence. In my text I have:

If , then X and Y are indep.

However, in my notes from class I have:

If , then X and Y are indep.

Which is correct?

Is this the only way to show independence? Does it have anything to do with expectation values? I had a question in which I had:

I was first asked to find the followings: and here is what I did:

, for this I took the 3rd derivative of the moment generating function and got this expression:

I was then asked about whether X and Y are independent. Naturally I attempted to use what I just did: therefore dependent. Is this correct?