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Mathematical Modelling Question

  1. May 28, 2006 #1
    HELP: Mathematical Modelling Question

    Hi

    Given [tex]X_1 \ldots X_n[/tex] be stochastic independent variables with the distribution functions [tex]F_X_{1}, \ldots ,F_X_{n} [/tex]. [tex]U = min(X_1 \ldots X_n)[/tex] and [tex]V = min(X_1 \ldots X_n)[/tex].

    [tex]F_{U}[/tex] and [tex]F_{V}[/tex] for U and V, and let [tex]F_{U,V}[/tex] be simultaneously distribution functions for the stochastic vectors (U,V).

    Then show that

    [tex]F_{V} (s) = \Pi \limit_{i=1} ^{n} F_{X_i} (s)[/tex] where [tex]\forall s \in \mathbb{R}[/tex]

    I can see that if I expand the sum I get

    [tex]F_X_{1}(s) + F_X_{2}(s) + F_X_{3}(s) + \ldots + F_X_{i}(s)[/tex] where [tex]1 \leq i \leq n [/tex]

    Doesn't that mean that

    [tex]F_X_{1}(s) + F_X_{2}(s) + F_X_{3}(s) + \ldots + F_X_{i}(s) = (F_X_{1}(s) \ \mathrm{U} \ F_X_{2}(s) \ \mathrm{U} F_X_{3}(s) \ \mathrm{U} \ \ldots \ \mathrm{U} \ F_X_{n}(s))[/tex] ??

    Since [tex]\sum_{i=1} ^{n} P(A_i) = P(A_1) + P(A_2) + P(A_3) + \ldots + P(A_n) [/tex]

    Sincerely
    Hummingbird
     
    Last edited: May 28, 2006
  2. jcsd
  3. May 28, 2006 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    May we assume that
    [tex]V = min(X_1 \ldots X_n)[/tex]
    was actually supposed to be
    [tex]V = max(X_1 \ldots X_n)[/tex]
     
  4. May 28, 2006 #3
    My assignment uses U and V to distingues between min and max, but I guess it doesn't make that a bit a difference in the final calculation.

    Sincerely Humingbird

     
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