ordered uniform rand var

**Anonymous1** · November 28th 2009, 11:33 AM

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

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

Where $X_{(0)} \equiv 0, X_{(n+1)} \equiv t.$

**mr fantastic** · November 28th 2009, 06:08 PM

Originally Posted by Anonymous1

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

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

Where $X_{(0)} \equiv 0, X_{(n+1)} \equiv t.$

You might start by calculating the joint pdf of $X_{(k)}$ and $X_{(k-1)}$ .

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