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Homework Help: Independent Random Variables

  1. Oct 25, 2009 #1
    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.
  2. jcsd
  3. Oct 25, 2009 #2


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    Homework Helper

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