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Thread: ordered uniform rand var

  1. #1
    Super Member Anonymous1's Avatar
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    ordered uniform rand var

    Let $\displaystyle X_{(1)} \leq X_{(2)} \leq X_{(3)} \leq...\leq X_{(n)}$ be the ordered values of $\displaystyle n$ independent uniform $\displaystyle (0,1)$ random variables. Prove that for $\displaystyle 1 \leq k \leq n+1,$

    $\displaystyle P(X_{(k)}-X_{(k-1)} > t) = (1 - t)^n$

    Where $\displaystyle X_{(0)} \equiv 0, X_{(n+1)} \equiv t.$
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    Quote Originally Posted by Anonymous1 View Post
    Let $\displaystyle X_{(1)} \leq X_{(2)} \leq X_{(3)} \leq...\leq X_{(n)}$ be the ordered values of $\displaystyle n$ independent uniform $\displaystyle (0,1)$ random variables. Prove that for $\displaystyle 1 \leq k \leq n+1,$

    $\displaystyle P(X_{(k)}-X_{(k-1)} > t) = (1 - t)^n$

    Where $\displaystyle X_{(0)} \equiv 0, X_{(n+1)} \equiv t.$
    You might start by calculating the joint pdf of $\displaystyle X_{(k)}$ and $\displaystyle X_{(k-1)}$.
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