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A Probability Problem Involving 6 Random Variables

  1. May 11, 2008 #1
    The problem statement, all variables and given/known data
    Let [itex]X_1, \ldots, X_6[/itex] be a sequence of independent and identically distributed continuous random variables. Find

    (a) [itex]P\{X_6 > X_1 \, | \, X_1= \max(X_1, \ldots, X_5)\}[/itex]
    (b) [itex]P\{X_6 > X_2 \, | \, X_1 = \max(X_1, \ldots, X_5)\}[/itex]

    The attempt at a solution
    (a) is the probability that [itex]X_6 = \max(X_1, \ldots, X_6)[/itex] right? How would I determine this probability? In (b), the event [itex]X_6 > X_2[/itex] is independent of [itex]X_1 = \max(X_1, \ldots, X_5)[/itex] right? If it is, the probability is:

    [tex]\int_{-\infty}^{\infty} \int_{x_1}^{\infty} f(x_6, x_1), \, dx_6 \, dx_1[/itex]

    where [itex]f(x_6, x_1) = f(x_6)f(x_1)[/itex] since the random variables are independent. Right?
  2. jcsd
  3. May 12, 2008 #2
    I screwed up. [itex]x_1[/itex] should be [itex]x_2[/itex] in the above.
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