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  • Min/Max problem

    I have a question on Actex Prob Set 4, #12. X1 and X2 are 2 indep. random variables, but they have the same density function
    f(x) = 2x for 0<x<1, 0 elsewhere. Find the probability that the max of X1 and X2 is at least .5.


    The solution says P[Max{X1,X2}>=.5]=1-P[(X1<.5) and (X2<.5)]

    Why can't you just evaluate P[(X1>.5) and (X2>.5)]? I don't have a complete understanding of Min/Max problems and Order Statatisics, so if you have any kind of explanation, I'd appreciate it!

  • #2
    Originally posted by ewr
    I have a question on Actex Prob Set 4, #12. X1 and X2 are 2 indep. random variables, but they have the same density function
    f(x) = 2x for 0<x<1, 0 elsewhere. Find the probability that the max of X1 and X2 is at least .5.


    The solution says P[Max{X1,X2}>=.5]=1-P[(X1<.5) and (X2<.5)]

    Why can't you just evaluate P[(X1>.5) and (X2>.5)]? I don't have a complete understanding of Min/Max problems and Order Statatisics, so if you have any kind of explanation, I'd appreciate it!
    Your equation doesn't answer the question. Suppose for a minute that X1 = A and X2 = B (that way, we can use separate letters).

    Your expression answers A and B both being greater than 0.5. The question is asking for the probability of at least one being greater than 0.5. If A was 0.7, but B was 0.3, is this something that you should count as a "success"? In other words, is Max{0.3, 0.7} >= 0.5? Yes. But for your expression, it wouldn't count as a success, since 0.7>=0.5, but not the case for 0.3.

    Is that okay?

    Comment


    • #3
      In order to have Max{X1,X2} > .5 , what we need is
      either X1 > .5 or X2 > .5 , but we do not need both
      to be true. The complement of the event Max{X1,X2} > .5
      is the event that the max is <= .5 , so
      P[Max{X1,X2}>.5] = 1 - P[Max{X1,X2}<=.5 ] .
      In order for Max{X1,X2} <= .5 to be true, we must
      have both X1 <= .5 and X2 <= .5 . This is an intersection
      of the events (X1 <= .5) intersect (X2 <= .5) .
      Because X1 and X2 are independent random variables,
      the probability
      P[(X1 <= .5) intersect (X2 <= .5)] can be written as
      P(X1<=.5) x P(X2<=.5) .
      Sam Broverman

      [email protected]
      www.sambroverman.com

      Comment


      • #4
        Thank you for your responses, I appreciate your help!! The solution makes complete sense now!

        Comment

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