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Probability proof question

  1. Feb 2, 2010 #1
    Good evening:

    I have a probability proof question that is driving me crazy. I feel
    like I must have forgot an easy trick. Any help is GREATLY
    appreciated. Here's the setup:

    Let's assume a,b are indepedent random variables from cummulative
    distribution F.


    I think it's safe to say:


    P( a > b) = .5


    Now, let's assume x,y are independent random variables from CDF G.
    Again:


    P(x > y) = 0.5


    Assume CDFs G and F are indepedent. Now it seems straightforward that:


    P(x + a > y + b) = 0.5


    but I don't know how to show it without assuming a distribution type.


    Again, any help is appreciated.


    Thank you,
    Chris
     
  2. jcsd
  3. Feb 3, 2010 #2

    mathman

    User Avatar
    Science Advisor

    If you are familiar with characteristic functions it is simple.
    Let f and g be the characteristic functions of F and G.
    Then S= x - y + a - b will have a ch. fcn. f(t)f(-t)g(t)g(-t).
    This means that S has a symmetric distribution.
     
    Last edited: Feb 3, 2010
  4. Feb 6, 2010 #3
    Thank you for the tip and the response - I think I have it solved. On an aside, that trick wouldn't work for P(ax < by) right? It seems obvious that the P(ax - by < 0 ) = P (by - ax < 0)= .5, but of course characteristic functions are most useful for sums.

    Thanks again for your assistance.
    Chris
     
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