The Independence of z and w: A Scientist's Perspective

[WolfOfTheSteps]

If x and y are independent and identically distributed exponential random variables, and

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.

 

[lanedance]

could you have a look at the covariance? - i think if events are independent the covariance should be zero