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difference between P[min(X1, X2, X3) < 3] and P[max(X1, X2, X3) < 3]

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  • difference between P[min(X1, X2, X3) < 3] and P[max(X1, X2, X3) < 3]

    So what exactly is the difference between P[min(X1, X2, X3) < 3] and P[max(X1, X2, X3) < 3]. I am trying to solve problems 4 and 5 from the ACTEX 2010 Edition page 211. The way i see it is

    P[min(X1, X2, X3) < 3] = P(X1 < 3)*P(X2 < 3)*P(X3 < 3)

    which is the same as

    P[max(X1, X2, X3) < 3] = P(X1 < 3)*P(X2 < 3)*P(X3 < 3)

    The solution guide used the survival function for the min, which i dont understand why it couldnt be done my approach. the solution for the max was the way i have it written. it is a uniform distribtion by the way.

  • #2
    Originally posted by rae1989 View Post
    So what exactly is the difference between P[min(X1, X2, X3) < 3] and P[max(X1, X2, X3) < 3]. I am trying to solve problems 4 and 5 from the ACTEX 2010 Edition page 211. The way i see it is

    P[min(X1, X2, X3) < 3] = P(X1 < 3)*P(X2 < 3)*P(X3 < 3)

    which is the same as

    P[max(X1, X2, X3) < 3] = P(X1 < 3)*P(X2 < 3)*P(X3 < 3)

    The solution guide used the survival function for the min, which i dont understand why it couldnt be done my approach. the solution for the max was the way i have it written. it is a uniform distribtion by the way.
    I would advise to pick a forum you want and post in there. There is absolutely no need to post the same question in more than one place.

    With regards to your actual question, the reason why the probability that the min of a set of numbers is less than some other number is NOT the product of those probabilities is because you only need one value in the set to be smaller.

    Think of an example:

    Let X1 = 2, X2 = 99999999999, X3 = 135187518965. The min of that set is (obviously) 2 so your condition is met. But that's not the same as the probability that each of X_i's is less than 3.

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