- #1

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z = x+y

w = x-y

are z and w also independent?

Do I have to actually find the joint pdf of z and w, then find the marginals and then see if they multiply to equal the joint pdf?

Or is there a way to just look at z and w and say whether they are independent or not?

I'm thinking like this: say z = 10... then w could be 5-5=0, but it could also be 10-0=10, or 3-7=-4. So w can be different things when z equals a certain number, but nonetheless it is still constrained by the value of z, so therefore they are not independent. (for example, if z = 10, w could never be 1000)

Is this reasoning correct? I know the definition of independence, but I believe that I have a very poor intuition of it. It's also pretty tedious to do the joint pdf to marginal pdfs comparison, if I could instead figure some of this stuff out by simple argument.